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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/2024_DOEPy_iminuit/talk.html b/2024_DOEPy_iminuit/talk.html new file mode 100644 index 0000000..7d26d70 --- /dev/null +++ b/2024_DOEPy_iminuit/talk.html @@ -0,0 +1,5512 @@ + + + + + + + + + +Hans Dembinski’s blog - iminuit: A foundational flexible library for fitting + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ +
+
+

iminuit: A foundational flexible library for fitting

+
+ + + +
+ + + + +
+ + +
+ +

Author Hans Dembinski, TU Dortmund, Germany

+

Abstract I will start with a general introduction into the theory of fitting and why you may want to use iminuit. I will show a few difficult nuts to crack with iminuit, and generally, how you get the best performance for a particular fit. Fitting speed in research matters: if an analysis runs quickly, one can do more checks and variations, and it makes computationally-intensive methods like the bootstrap for error estimation feasible.

+
+

About me

+
    +
  • Particle physicist working with the LHCb experiment
  • +
  • Formerly astroparticle physicist working with Pierre Auger Observatory and IceCube
  • +
  • Statistics convener in LHCb (2019-2021, 2024-now)
  • +
  • Interests +
      +
    • Statistics
    • +
    • High-performance computing
    • +
    • API design: “Make Interfaces Easy to Use Correctly and Hard to Use Incorrectly” Scott Meyers
    • +
  • +
  • Memberships +
  • +
  • Author/maintainer +
  • +
  • Contributor to scipy and matplotlib
  • +
+
+
+

Introduction

+
    +
  • Fitting?

    +
      +
    • Adjust a statistical model to data set, find model parameters which match data best
    • +
    • AI community calls this “learning”
    • +
  • +
  • Dataset: samples \(\{ \vec x_i \}\)

    +
      +
    • Sample can be one-dimensional or multidimensional
    • +
    • Values can be continuous (e.g. candidate mass) or discrete (e.g. bin counts)
    • +
  • +
  • Model: Probability density or probability mass function \(f(\vec x; \vec p)\) which depends on unknown parameters \(\vec p\)

  • +
  • Conjecture: maximum-likelihood estimate (MLE) \(\hat{\vec p} = \text{argmax}_{\vec p} L(\vec p)\) is optimal, with \(L(\vec p) = \prod_i f(\vec x_i; \vec p)\)

    +
      +
    • Proven: Asymptotically optimal, unbiased with minimum variance +
        +
      • No proof for finite data sets, but unchallenged
      • +
    • +
    • Universal: MLE can be computed for any problem
    • +
    • Equivalent: minimize negative log-likelihood (NLL), \(\hat{\vec p} = \text{argmin}_{\vec p}(-\ln L(\vec p))\)
    • +
  • +
  • Limiting case of maximum-likelihood fit: least-squares fit aka chi-square fit

  • +
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import norm
+
+dist = norm(0, 1)
+x1 = dist.rvs(size=100, random_state=1)
+plt.plot(x1, np.zeros_like(x1), "|", ms=15, label="data")
+w, xe, *_ = plt.hist(x1, label="histogram", facecolor="0.8")
+xm = np.linspace(-3, 3, 1000)
+plt.plot(xm, dist.pdf(xm) * len(x1) * (xe[1]-xe[0]), label="model (scaled)")
+plt.xlabel("x")
+plt.ylabel("density")
+plt.legend();
+
+

+
+
+
+
# model pdf
+def model1(x, mu, sigma):
+    return norm.pdf(x, mu, sigma)
+
+# negative log-likelihood
+def nll1(mu, sigma):
+    logL = np.log(model1(x1, mu, sigma))
+    return -np.sum(logL)
+
+
+
mu = np.linspace(-1, 1, 10)
+sigma = np.linspace(0.5, 1.5, 10)
+mu2, sigma2 = np.meshgrid(mu, sigma)
+z = np.vectorize(nll1)(mu2, sigma2)
+
+plt.figure(figsize=(10, 4))
+ax = plt.subplot(121, projection="3d")
+ax.view_init(elev=20, azim=40, roll=0)
+ax.plot_surface(mu2, sigma2, z)
+ax.set(xlabel="mu", ylabel="sigma")
+plt.sca(ax)
+plt.subplots_adjust(left=0.1, right=1);
+
+ax = plt.subplot(122)
+plt.pcolormesh(mu2, sigma2, z)
+plt.colorbar()
+ax.set(xlabel="mu", ylabel="sigma");
+
+

+
+
+
+
+

iminuit

+
    +
  • Python frontend for C++ Minuit2 library maintained by CERN ROOT team
  • +
  • Minuit2 is a minimizer developed for negative likelihoods and chi-square functions
  • +
  • Computes parameter errors via HESSE or MINOS methods
  • +
  • You can write your own cost function in plain Python
  • +
  • Jupyter notebook friendly object displays
  • +
+
+
from iminuit import Minuit
+
+m = Minuit(nll1, mu=0, sigma=1)
+m.errordef = Minuit.LIKELIHOOD
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 129.7Nfcn = 32
EDM = 3.42e-06 (Goal: 0.0001)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0mu0.060.09
1sigma0.890.06
+ + + + + + + + + + + + + + + + + + +
musigma
mu0.007840.000
sigma0.0000.00392
+
+
+
    +
  • Bundled extensions +
      +
    • Library of common cost functions for various fitting problems +
        +
      • UnbinnedNLL: maximum-likelihood estimation on unbinned data
      • +
      • BinnedNLL: maximum-likelihood estimation on binned data
      • +
      • ExtendedUnbinnedNLL and ExtendedBinnedNLL: fit shape and amplitude
      • +
      • Template: fit non-parametric models
      • +
      • NormalConstraint: penalty terms aka soft constraints
      • +
      • LeastSquares
      • +
    • +
    • Wrapper for scipy minimizers that handle e.g. constrained problems
    • +
    • Interactive fitting widget for Jupyter notebooks
    • +
  • +
+
+
from iminuit import Minuit
+from iminuit.cost import UnbinnedNLL
+
+c = UnbinnedNLL(x1, model1)
+
+m = Minuit(c, mu=0, sigma=1)
+m.limits["mu"] = (-2, 2)
+m.limits["sigma"] = (0.01, 1.5)
+m.interactive()
+
+ +
+
+
+
m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 259.4Nfcn = 32
EDM = 2.83e-06 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0mu0.060.09-22
1sigma0.890.060.011.5
+ + + + + + + + + + + + + + + + + + +
musigma
mu0.007840.000
sigma0.0000.00392
+ + + + + + + + 2024-01-31T20:41:27.598494 + image/svg+xml + + + Matplotlib v3.7.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
    +
  • iminuit computes parameter errors with the Hesse method and the Minos method
  • +
  • Hesse method produces symmetric error intervals and covariance matrix
  • +
  • Minos method produces asymmetric error intervals and 2D contours
  • +
  • Neither method is superior: use Hesse unless you have strongly asymmetric errors
  • +
+
+
# 1, 2, 3 sigma contours
+m.draw_mncontour("mu", "sigma", cl=(1, 2, 3));
+
+

+
+
+
+
+

iminuit vs. other packages

+
    +
  • pyhf +
      +
    • Better choice when you need to compute limits and preserve the likelihood
    • +
  • +
  • zfit and RooFit +
      +
    • Help you build statistical models: automatic normalization & convolutions +
        +
      • This has to be done “by hand” in iminuit
      • +
    • +
    • Good for standard tasks, but restrict your freedom
    • +
  • +
  • iminuit +
      +
    • Has very good documentation
    • +
    • Well-designed API
    • +
    • Is very flexible and open
    • +
    • Is very fast with numba
    • +
  • +
+
+
+

Fast fits with iminuit + numba

+
    +
  • Fast fits are important when… +
      +
    • You need an unbinned fit on large sample
    • +
    • You have to repeat the same fit many times on different data +
        +
      • E.g. fitting many kinematic bins, bootstrapping uncertainties, studying systematic variations
      • +
    • +
  • +
  • Performance insights +
      +
    • Time spend in minimizer is negligible
    • +
    • Bottleneck is evaluating cost function
    • +
    • Bottleneck inside cost function is usually evaluation the pdf
    • +
    • Cost function is trivially parallelizable: log(pdf) computed independently for each data point +
        +
      • Exploit SIMD instructions and parallel computing on multiple cores
      • +
    • +
  • +
+
+
def make_data(size, seed=1):
+    rng = np.random.default_rng(seed)
+    s = rng.normal(0.5, 0.1, size=size // 2)
+    b = rng.exponential(1, size=2 * size)
+    b = b[b < 1]
+    b = b[: size // 2]
+    x = np.append(s, b)
+    return x
+
+x2 = make_data(1_000_000)
+plt.hist(x2, bins=100);
+
+

+
+
+
+
from numba_stats import norm, truncexpon
+
+
+def model2(x, z, mu, sigma, slope):
+    b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)
+    s = norm.pdf(x, mu, sigma)
+    return (1 - z) * b + z * s
+
+
+def nll2(z, mu, sigma, slope):
+    p = model2(x2, z, mu, sigma, slope)
+    return -np.sum(np.log(p))
+
+
+
m = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -2.108e+05Nfcn = 163
EDM = 2.8e-07 (Goal: 0.0002)time = 6.4 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.49970.0013
1mu499.98e-30.29e-3
2sigma99.59e-30.29e-3
3slope0.9970.008
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z1.77e-06-0.02e-6 (-0.049)0.21e-6 (0.548)-2.4e-6 (-0.229)
mu-0.02e-6 (-0.049)8.36e-08-0 (-0.060)-0.51e-6 (-0.226)
sigma0.21e-6 (0.548)-0 (-0.060)8.18e-08-0.37e-6 (-0.164)
slope-2.4e-6 (-0.229)-0.51e-6 (-0.226)-0.37e-6 (-0.164)6.12e-05
+
+
+
+
import numba as nb
+
+@nb.njit(inline="always")
+def model2(x, z, mu, sigma, slope):
+    b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)
+    s = norm.pdf(x, mu, sigma)
+    return (1 - z) * b + z * s
+
+
+@nb.njit(parallel=True, fastmath=True)
+def nll2(z, mu, sigma, slope):
+    p = model2(x2, z, mu, sigma, slope)
+    return -np.sum(np.log(p))
+
+
+m = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)
+m.migrad()
+
+
OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -2.108e+05Nfcn = 163
EDM = 2.8e-07 (Goal: 0.0002)time = 9.0 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.49970.0013
1mu499.98e-30.29e-3
2sigma99.59e-30.29e-3
3slope0.9970.008
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z1.77e-06-0.02e-6 (-0.049)0.21e-6 (0.548)-2.4e-6 (-0.229)
mu-0.02e-6 (-0.049)8.36e-08-0 (-0.060)-0.51e-6 (-0.226)
sigma0.21e-6 (0.548)-0 (-0.060)8.18e-08-0.37e-6 (-0.164)
slope-2.4e-6 (-0.229)-0.51e-6 (-0.226)-0.37e-6 (-0.164)6.12e-05
+
+
+
+
m = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -2.108e+05Nfcn = 163
EDM = 2.8e-07 (Goal: 0.0002)time = 1.1 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.49970.0013
1mu499.98e-30.29e-3
2sigma99.59e-30.29e-3
3slope0.9970.008
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z1.77e-06-0.02e-6 (-0.049)0.21e-6 (0.548)-2.4e-6 (-0.229)
mu-0.02e-6 (-0.049)8.36e-08-0 (-0.060)-0.51e-6 (-0.226)
sigma0.21e-6 (0.548)-0 (-0.060)8.18e-08-0.37e-6 (-0.164)
slope-2.4e-6 (-0.229)-0.51e-6 (-0.226)-0.37e-6 (-0.164)6.12e-05
+
+
+

Plots from the iminuit benchmarks

+

+

+
+
+

Binned vs. unbinned fits

+
    +
  • Idea: don’t fit original data sample, fit histogram of data sample +
      +
    • Histogram: lossy compression of original data
    • +
    • Another way to speed up fitting
    • +
  • +
  • Fit results based on histograms not more biased, but may have larger variance
  • +
  • Loss can be made negligible with sufficiently fine binning
  • +
  • Binned fits require the modeling cdf instead of pdf
  • +
  • Further benefit: iminuit cost functions compute chi-square goodness-of-fit test statistic for free, based on S. Baker and RD. Cousins, Nucl.Instrum.Meth. 221 (1984) 437-442
  • +
+
+
n = 100_000
+truth = np.array((1.0, 1.0, 1.0, 0.1, 1.0))
+rng = np.random.default_rng(1)
+s = rng.normal(truth[2], truth[3], size=int(n * truth[0]))
+b = rng.exponential(truth[4], size=int(n * truth[1]))
+x3 = np.append(s, b)
+x3 = x3[(x3 > 0) & (x3 < 2)]
+
+w, xe = np.histogram(x3, bins=200)  # for more complex histograms use boost-histogram
+plt.stairs(w, xe);
+
+

+
+
+
+
from numba_stats import norm, expon
+from iminuit.cost import ExtendedBinnedNLL
+
+def model3(xe, s, b, mu, sigma, tau):
+    return s * n * norm.cdf(xe, mu, sigma) + b * n * expon.cdf(xe, 0, tau)
+
+
+
c = ExtendedBinnedNLL(w, xe, model3)
+m = Minuit(c, *truth)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 190.9 (χ²/ndof = 1.0)Nfcn = 110
EDM = 2.26e-06 (Goal: 0.0002)time = 0.5 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0s0.9950.004
1b1.0080.005
2mu999.3e-30.4e-3
3sigma99.22e-30.34e-3
4tau1.0020.007
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
sbmusigmatau
s1.32e-05-0.005e-3 (-0.284)-0.04e-6 (-0.029)0.30e-6 (0.248)-0.004e-3 (-0.161)
b-0.005e-3 (-0.284)2.37e-050-0.47e-6 (-0.286)0.020e-3 (0.587)
mu-0.04e-6 (-0.029)01.46e-07-0.01e-6 (-0.046)-0.14e-6 (-0.054)
sigma0.30e-6 (0.248)-0.47e-6 (-0.286)-0.01e-6 (-0.046)1.12e-07-0.37e-6 (-0.159)
tau-0.004e-3 (-0.161)0.020e-3 (0.587)-0.14e-6 (-0.054)-0.37e-6 (-0.159)4.77e-05
+ + + + + + + + 2024-01-31T20:41:46.336920 + image/svg+xml + + + Matplotlib v3.7.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
    +
  • Full study can be reviewed in iminuit docs
  • +
  • 100k data points
  • +
  • Normal signal + truncated exponential background
  • +
  • Bins: infinite (unbinned likelihood), 200, 100, 50, 20, 10, 5
  • +
  • Study bias and variance of fit results +
      +
    • Repeatedly fit random data sets of equal size
    • +
  • +
+

+

+

+
+
+

Template fits

+
    +
  • Sometimes PDF/CDF not known in parametric form, but can be Monte-Carlo simulated
  • +
  • Idea: fit a histogram of simulation output (template) to data histogram
  • +
  • Additional source of uncertainty from finite size of simulation
  • +
  • Correct error propagation with Barlow-Beeston method, Barlow and Beeston, Comput.Phys.Commun. 77 (1993) 219-228
  • +
  • iminuit.cost.Template +
  • +
+
+
def generate(rng, nmc, truth, bins):
+    xe = np.linspace(0, 2, bins + 1)
+    b = np.diff(truncexpon.cdf(xe, 0, 2, 0, 1))
+    s = np.diff(norm.cdf(xe, 1, 0.1))
+    n = rng.poisson(b * truth[0]) + rng.poisson(s * truth[1])
+    t = np.array([rng.poisson(b * nmc), rng.poisson(s * nmc)])
+    return xe, n, t
+
+rng = np.random.default_rng(1)
+truth = 750, 250
+xe, n, t = generate(rng, 200, truth, 15)
+
+_, ax = plt.subplots(1, 2, figsize=(10, 4))
+
+ax[0].stairs(n, xe, color="k", label="data")
+ax[0].legend();
+ax[1].stairs(t[0], xe, label="background")
+ax[1].stairs(t[1], xe, label="signal")
+ax[1].legend(title="templates");
+
+

+
+
+
+
from iminuit.cost import Template
+
+c = Template(n, xe, t)
+m = Minuit(c, 1, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 8.511 (χ²/ndof = 0.7)Nfcn = 124
EDM = 3.22e-05 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0760600
1x1249340
+ + + + + + + + + + + + + + + + + + +
x0x1
x03.98e+03-0.6e3 (-0.274)
x1-0.6e3 (-0.274)1.13e+03
+ + + + + + + + 2024-01-31T20:41:46.828903 + image/svg+xml + + + Matplotlib v3.7.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
+
xe, n, t = generate(rng, 1_000_000, truth, 15)
+
+c = Template(n, xe, t)
+m = Minuit(c, 1, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 12.32 (χ²/ndof = 0.9)Nfcn = 110
EDM = 5.76e-05 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0782310
1x1238200
+ + + + + + + + + + + + + + + + + + +
x0x1
x0960-0.2e3 (-0.281)
x1-0.2e3 (-0.281)416
+ + + + + + + + 2024-01-31T20:41:47.026893 + image/svg+xml + + + Matplotlib v3.7.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
+
+

Summary

+
    +
  • iminuit is a versatile, fast, easy-to-use library for fitting
  • +
  • Builtin cost functions cover most common use-cases
  • +
  • Emphasis is put on having excellent documentation and being bug-free
  • +
  • Due to its focussed open design +
      +
    • Preferred by statistics experts (e.g. Glen Cowan)
    • +
    • Excellent for teaching
    • +
  • +
  • Excellent performance in combination with numba and numba-stats
  • +
  • Easy installation: binary wheels for all major platforms (Python-3.12 will be added in next release)
  • +
+ + +
+ +
+ +
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zt{wUbBm8j^QZux}sx3U|;5c+sFWd!w=6s&q@m0WGy@}a+q89x!Y!Mfmfu|X$G1lTf;6rD zc<0BK{8+-jEkz2cx@s0L6d(MDGeV;INe(!}XPv(fSpMg6XhWf%a*LtHfL7`sKg0M8 z#F+zZqW>Ue`}^kyL2Jd(oIjxO+sb}?>-P+v@QR_)Kgb{d<=IFTJLt}=yXT)j|K#%* z@Pu;KQ~%^7$SysQhb+II3mO`H$;| z0_VRo>p%JDj{^b@?l^4moEUxfFCFHO^}`{2`adK1KO^}6zzF=l?@#1?9IpFTJ?7 literal 0 HcmV?d00001 diff --git a/2024_LHCb_Starterkit/lecture_1.html b/2024_LHCb_Starterkit/lecture_1.html new file mode 100644 index 0000000..7f35e56 --- /dev/null +++ b/2024_LHCb_Starterkit/lecture_1.html @@ -0,0 +1,3861 @@ + + + + + + + + + +Hans Dembinski’s blog - Statistics I + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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Statistics I

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Hans Dembinski, TU Dortmund

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  • Scikit-HEP project
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  • Reading simple ROOT files
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  • Histograms
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  • Data visualization
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  • Fitting
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About me

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  • Working with the LHCb experiment since 2016
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  • Before astroparticle physicist working with Pierre Auger Observatory and IceCube
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  • Statistics convener in LHCb (2019-2021, 2024-now)
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  • Interests +
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    • Statistics
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    • High-performance computing
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    • API design: “Make Interfaces Easy to Use Correctly and Hard to Use Incorrectly” Scott Meyers
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Scikit-HEP project

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  • Provides high-performance data analysis tools in Python
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  • Partially funded by NSF grants
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  • Several LHCb developers: Eduardo Rodrigues, Chris Burr, myself, …
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  • Core values +
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    • Specialized tools that integrate well with existing Python ecosystem
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    • Easy to use for beginners
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    • Flexible for power users
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    • Good documentation
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    • Easy installation
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The power of combinatorics

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Build everything from basic components that fit together

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Image credit: Toni Zaat on Unsplash

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Important packages

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  • Python scientific stack

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    • Numpy: fast computation with arrays
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    • Scipy: integration, statistical distributions, special functions, …
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    • Matplotlib: scientific plotting
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    • Numba: JIT Compiler for Python (very fast)
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    • Pandas: Convenient processing of tabular data
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  • Scikit-HEP Project

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    • uproot: fast reading and writing of ROOT Trees
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    • particle: get particle properties from PDG IDs
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    • boost-histogram: multi-dimensional generalised histograms
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    • iminuit: fitting and error computation package
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    • resample: Easy bootstrapping to compute uncertainty and bias
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    • pyhf: for histogram fitting/limit setting and preserving likelihoods
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  • Other packages

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    • jacobi: Easy numerical error propagation
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    • numba-stats: Fast statistical distributions for fitting
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    • tabulate: Convert data tables to various formats (LaTeX, HTML, Markdown, …)
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    • scikit-learn: Basic machine learning tools
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    • xgboost: Gradient-boosting machine library that won many Kaggle competitions
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Install everything with pip install LIBRARY

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Working with Jupyter notebooks

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  • I recommend to use vscode +
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    • Very good Jupyter notebook support
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    • Remote explorer: edit and run code on remote machines
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Prepare computing environment

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  • In terminal, create and enter virtual environment (you need Python-3.8 or later)

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    python3 -m venv .venv
    +source .venv/bin/activate
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  • Install packages

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    python -m pip install --upgrade numba matplotlib uproot boost-histogram iminuit scipy particle numba_stats ipywidgets
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  • Download this notebook from indico and example.root, put latter into same folder

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  • Run vscode on the current folder, then open the notebook inside the editor OR

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  • Run

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    python -m pip install ipykernel
    +python -m ipykernel install --name "starterkit"
    +jupyter notebook lecture_1.ipynb
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    Then select starterkit as kernel

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  • uproot provides fast and convenient access to ROOT trees
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import uproot
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+print(f"uproot {uproot.__version__}")
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uproot 5.2.2
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f = uproot.open("example.root")
+event = f["event"]
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+event.show()
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name                 | typename                 | interpretation                
+---------------------+--------------------------+-------------------------------
+trk_len              | int32_t                  | AsDtype('>i4')
+mc_trk_len           | int32_t                  | AsDtype('>i4')
+trk_imc              | int32_t[]                | AsJagged(AsDtype('>i4'))
+trk_px               | float[]                  | AsJagged(AsDtype('>f4'))
+trk_py               | float[]                  | AsJagged(AsDtype('>f4'))
+trk_pz               | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_px            | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_py            | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_pz            | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_pid           | int32_t[]                | AsJagged(AsDtype('>i4'))
+
+
+
    +
  • Tree contains fake simulated LHCb events

  • +
  • Reconstructed tracks in simulation, branches with trk_*

  • +
  • Truth in simulation, branches with mc_trk_*

  • +
  • Some branches are 2D (e.g. trk_px)

    +
      +
    • First index iterates over events
    • +
    • Second index iterates over tracks per event
    • +
  • +
  • Mathematical operators and slicing works on these arrays similar to Numpy

  • +
  • Many numpy functions work on awkward arrays

  • +
+
+
import numpy as np
+np.__version__
+
+
'1.24.4'
+
+
+
+
data = event.arrays()
+
+np.sum(data["trk_len"])
+
+
5032
+
+
+

Exercise

+
    +
  • Use np.sum and an array mask to count how many events have zero tracks
  • +
+
+
# do exercise here
+
+
    +
  • For some operations and awkward arrays, we need the awkward library
  • +
+
+
import awkward as ak
+ak.__version__
+
+
'2.6.1'
+
+
+
    +
  • For plotting, we use matplotlib
  • +
+
+
import matplotlib
+import matplotlib.pyplot as plt
+matplotlib.__version__
+
+
'3.8.2'
+
+
+

Exercise * Plot the branch trk_px with matplotlib.pyplot.hist * Can you figure out from the error message why it does not work? * Fix the issue with the function ak.flatten

+
+
# do exercise here
+
+
+
+

Reading data from a large tree

+
    +
  • Tree often do not fit into memory (tree size > 100 GB and more possible)
  • +
  • Iterate over chunks
  • +
  • Chunk-wise processing works very well with histograms
  • +
+

+
    +
  • Python frontend to Boost.Histogram library in C++ from the Boost project
  • +
  • Very fast and flexible
  • +
  • Multi-dimensional histograms and other binned statistics (profiles!)
  • +
  • Supports weighted data
  • +
  • And much much more, see docs
  • +
+
+
import boost_histogram as bh
+bh.__version__
+
+
'1.4.0'
+
+
+
+
# make an axis
+xaxis = bh.axis.Regular(20, -2, 2)
+
+# easy to make several histograms with same binning by reusing axis
+h_px = bh.Histogram(xaxis)
+h_mc_px = bh.Histogram(xaxis)
+
+
+
# incremental filling, only read branch "trk_px" from tree
+for data in event.iterate(["trk_px"]):
+    h_px.fill(ak.flatten(data["trk_px"]))
+
+
+
h_px
+
+
Histogram(Regular(20, -2, 2), storage=Double()) # Sum: 5032.0
+
+
+
+
print(h_px)
+
+
                 ┌───────────────────────────────────────────────────────────┐
+[-inf,   -2) 0   │                                                           │
+[  -2, -1.8) 1   │▏                                                          │
+[-1.8, -1.6) 3   │▎                                                          │
+[-1.6, -1.4) 17  │█▎                                                         │
+[-1.4, -1.2) 25  │█▊                                                         │
+[-1.2,   -1) 78  │█████▌                                                     │
+[  -1, -0.8) 181 │████████████▉                                              │
+[-0.8, -0.6) 308 │█████████████████████▊                                     │
+[-0.6, -0.4) 453 │████████████████████████████████▏                          │
+[-0.4, -0.2) 647 │█████████████████████████████████████████████▉             │
+[-0.2,    0) 819 │██████████████████████████████████████████████████████████ │
+[   0,  0.2) 782 │███████████████████████████████████████████████████████▍   │
+[ 0.2,  0.4) 629 │████████████████████████████████████████████▌              │
+[ 0.4,  0.6) 480 │██████████████████████████████████                         │
+[ 0.6,  0.8) 323 │██████████████████████▉                                    │
+[ 0.8,    1) 159 │███████████▎                                               │
+[   1,  1.2) 77  │█████▌                                                     │
+[ 1.2,  1.4) 37  │██▋                                                        │
+[ 1.4,  1.6) 7   │▌                                                          │
+[ 1.6,  1.8) 3   │▎                                                          │
+[ 1.8,    2) 3   │▎                                                          │
+[   2,  inf) 0   │                                                           │
+                 └───────────────────────────────────────────────────────────┘
+
+
+
+
# plot histogram with plt.stairs
+plt.stairs(h_px.values(), h_px.axes[0].edges);
+
+

+
+
+
    +
  • Histograms can be cut down to size after filling +
      +
    • Shrink axis range with slices and using loc
    • +
    • Make binning coarser with rebin
    • +
    • See docs of boost-histogram or extended lecture
    • +
  • +
  • Beyond 1D histograms +
      +
    • Supports advanced axis types (e.g. category axis)
    • +
    • Supports higher dimensional histograms
    • +
    • Supports generalized histograms with binned statistics (aka “profile” in ROOT)
    • +
    • See docs of boost-histogram or extended lecture
    • +
  • +
+
+
+

Fits

+
    +
  • Typical analysis work flow (often automated with Snakemake)

    +
      +
    1. Pre-select data
    2. +
    3. Make histograms or profiles from pre-selected data
    4. +
    5. Fit histograms or profiles to extract physical parameters
    6. +
  • +
  • Many specialized fitting tools for individual purposes, e.g.: scipy.optimize.curve_fit

  • +
  • Generic libraries

    +
  • +
  • Fitting (“estimation” in statistics)

    +
      +
    • Adjust parametric statistical model to dataset and find model parameters which match dataset best
    • +
    • AI community calls this “learning”
    • +
    • Estimate: result of fit
    • +
  • +
  • Dataset: samples \(\{ \vec x_i \}\)

  • +
  • Model: Probability density or probability mass function \(f(\vec x; \vec p)\) which depends on unknown parameters \(\vec p\)

  • +
  • Conjecture: maximum-likelihood estimate (MLE) \(\hat{\vec p} = \text{argmax}_{\vec p} L(\vec p)\) is optimal, with \(L(\vec p) = \prod_i f(\vec x_i; \vec p)\)

    +
      +
    • Equivalent: minimize negative log-likelihood (NLL), \(\hat{\vec p} = \text{argmin}_{\vec p}(-\ln L(\vec p))\)
    • +
  • +
  • Limiting case of maximum-likelihood fit: least-squares fit aka chi-square fit

  • +
+

+
    +
  • Python frontend to Minuit2 C++ library maintained by ROOT team at CERN
  • +
  • Maximum-likelihood and least-squares fits with error estimation
  • +
+
+
import iminuit
+from iminuit import Minuit
+iminuit.__version__
+
+
'2.25.2'
+
+
+
    +
  • iminuit vs. other packages +
      +
    • pyhf +
        +
      • Better choice when you need to compute limits and preserve the likelihood
      • +
    • +
    • zfit and RooFit +
        +
      • Help you build statistical models: automatic normalization & convolutions +
          +
        • This has to be done “by hand” in iminuit
        • +
      • +
      • Good for standard tasks, but restrict your freedom
      • +
    • +
    • iminuit +
        +
      • Has good documentation
      • +
      • Well-designed API
      • +
      • Is very flexible and open
      • +
      • Is very fast with numba
      • +
    • +
  • +
  • Typical fit in particle physics: mass distribution of decay candidates with signal peak over smooth background
  • +
  • Task: fit signal fraction to calculate signal yield (fraction * number of candidates)
  • +
+
+
# fast implementations of statistical distributions, API similar to scipy.stats
+import numba_stats
+from numba_stats import norm, truncexpon
+numba_stats.__version__
+
+
'1.7.0'
+
+
+
+
def make_data(size, seed=0):
+    z=0.5
+    mu=0.5
+    sigma=0.1
+    slope=0.5
+    s = norm.rvs(mu, sigma, size=int(z * size), random_state=seed)
+    b = truncexpon.rvs(0, 1, 0, slope, size=int((1-z) * size), random_state=seed)
+    return np.append(s, b)
+
+x1 = make_data(10_000)
+plt.hist(x1, bins=100);
+
+

+
+
+
    +
  • Statistical model assumptions +
      +
    • Full sample is additive mixture of signal sample and background sample
    • +
    • Signal is normal-distributed; pdf is \(\mathcal{N}(\mu, \sigma)\) with parameters \(\mu\) and \(\sigma\)
    • +
    • Background is truncated exponential distribution
    • +
  • +
+
+
# model pdf
+def model1(x, z, mu, sigma, slope):
+    s = norm.pdf(x, mu, sigma)
+    b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)
+    return (1 - z) * b + z * s
+
+# we use NLL implementation provided by iminuit for convenience
+from iminuit.cost import UnbinnedNLL
+
+nll1 = UnbinnedNLL(x1, model1)
+m = Minuit(nll1, z=0.5, mu=0.5, sigma=0.1, slope=1)
+
+
+
# need to set parameter limits or fit might fail
+m.limits["z", "mu"] = (0, 1)
+m.limits["sigma", "slope"] = (0, None)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -4741Nfcn = 99
EDM = 2.03e-05 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.4890.00901
1mu0.49830.002001
2sigma0.09770.00190
3slope0.5050.0160
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.48e-05-2e-6 (-0.102)9e-6 (0.535)-0.06e-3 (-0.395)
mu-2e-6 (-0.102)3.94e-06-0e-6 (-0.124)-5e-6 (-0.153)
sigma9e-6 (0.535)-0e-6 (-0.124)3.7e-06-9e-6 (-0.280)
slope-0.06e-3 (-0.395)-5e-6 (-0.153)-9e-6 (-0.280)0.000268
+ + + + + + + + 2024-02-13T09:27:46.977103 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

Exercise

+
    +
  • Use interactive mode via m.interactive() to check how fit reacts to parameter changes
  • +
  • Useful for debugging and to find starting values by hand
  • +
+
+
# do exercise here
+
+
    +
  • you can compute 2D confidence regions with Minuit.mncontour
  • +
+

Exercise * Draw the likelihood profile contour of parameters z and sigma with Minuit.draw_mncontour * Compute matrix of likelihood profile contours with Minuit.draw_mnmatrix

+
+
# do exercise here
+
+
+
+

Faster fits?

+
    +
  • Option 1: accelerate NLL computation with numba
  • +
  • Option 2: use binned data
  • +
+
+
x2 = make_data(1_000_000)
+
+
    +
  • Option 1 +
      +
    • NLL computation is trivially parallelizable: log(pdf) computed independently for each data point
    • +
    • Numba allows us to exploit auto-vectorization (SIMD instructions) and parallel computing on multiple cores
    • +
    • See full lecture for an example to numba-JIT compile the NLL function
    • +
    • See benchmark of iminuit + numba vs. RooFit
    • +
  • +
  • Option 2 +
      +
    • Often simpler: fit histograms
    • +
    • No bias from fitting histograms if done correctly, only loss in precision
    • +
    • Need model cdf instead of model pdf: \(F(x; \vec p) = \int_{-\infty}^x f(x'; \vec p) \text{d}x'\)
    • +
  • +
+
+
from iminuit.cost import BinnedNLL
+
+# model is now a cdf!
+def model3(x, z, mu, sigma, slope):
+    s = norm.cdf(x, mu, sigma)
+    b = truncexpon.cdf(x, 0.0, 1.0, 0.0, slope)
+    return z * s + (1 - z) * b
+
+w, xe = np.histogram(x2, bins=100, range=(0, 1))
+nll3 = BinnedNLL(w, xe, model3)
+
+
+
m = Minuit(nll3, z=0.5, mu=0.5, sigma=1, slope=1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 94.55 (χ²/ndof = 1.0)Nfcn = 192
EDM = 1.76e-07 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z501.0e-30.9e-3
1mu500.3e-30.2e-3
2sigma100.06e-30.20e-3
3slope0.49820.0017
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.91e-07-0.02e-6 (-0.103)0.10e-6 (0.557)-0.7e-6 (-0.422)
mu-0.02e-6 (-0.103)4.03e-08-0 (-0.125)-0.05e-6 (-0.155)
sigma0.10e-6 (0.557)-0 (-0.125)3.98e-08-0.10e-6 (-0.307)
slope-0.7e-6 (-0.422)-0.05e-6 (-0.155)-0.10e-6 (-0.307)2.72e-06
+ + + + + + + + 2024-02-13T09:27:47.235298 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+
+

Other fits

+
    +
  • To fit yields (counts) instead of fractions, use ExtendedUnbinnedNLL or ExtendedBinnedNLL
  • +
  • To fit (x, y) pairs, use LeastSquares
  • +
  • To fit simulated templates, use Template
  • +
  • See documentation how models should be constructed then
  • +
+
+
+

Template fits

+ +
+
def make_data2(rng, nmc, truth, bins):
+    xe = np.linspace(0, 1, bins + 1)
+    b = np.diff(truncexpon.cdf(xe, 0, 1, 0, 1))
+    s = np.diff(norm.cdf(xe, 0.5, 0.05))
+    n = rng.poisson(b * truth[0]) + rng.poisson(s * truth[1])
+    t = np.array([rng.poisson(b * nmc), rng.poisson(s * nmc)])
+    return xe, n, t
+
+rng = np.random.default_rng(1)
+truth = 750, 250
+xe, n, t = make_data2(rng, 200, truth, 15)
+
+_, ax = plt.subplots(1, 2, figsize=(10, 4))
+ax[0].stairs(n, xe, color="k", label="data")
+ax[0].legend();
+ax[1].stairs(t[0], xe, label="background")
+ax[1].stairs(t[1], xe, label="signal")
+ax[1].legend(title="templates");
+
+

+
+
+
+
from iminuit.cost import Template
+
+c = Template(n, xe, t)
+m = Minuit(c, 1, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 8.352 (χ²/ndof = 0.6)Nfcn = 132
EDM = 2.75e-07 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0770700
1x1220400
+ + + + + + + + + + + + + + + + + + +
x0x1
x04.3e+03-0.8e3 (-0.354)
x1-0.8e3 (-0.354)1.32e+03
+ + + + + + + + 2024-02-13T09:27:47.726719 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
    +
  • With Template you can also fit a mix of template and parametric model
  • +
  • Often template only needed for background(s)
  • +
+

Exercise

+
    +
  • With make_data2 generate templates with 1 000 000 simulated points
  • +
  • Fit this data with the Template cost function
  • +
  • Compare parameter uncertainties
  • +
+
+
# do exercise here
+
+ + +
+ +
+ +
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+Hans Dembinski’s blog - Statistics I + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ +
+
+

Statistics I

+
+ + + +
+ + + + +
+ + +
+ +

Hans Dembinski, TU Dortmund

+
    +
  • Scikit-HEP project
  • +
  • Reading simple ROOT files
  • +
  • Histograms
  • +
  • Data visualization
  • +
  • Fitting
  • +
+
+

About me

+
    +
  • Working with the LHCb experiment since 2016
  • +
  • Before astroparticle physicist working with Pierre Auger Observatory and IceCube
  • +
  • Statistics convener in LHCb (2019-2021, 2024-now)
  • +
  • Interests +
      +
    • Statistics
    • +
    • High-performance computing
    • +
    • API design: “Make Interfaces Easy to Use Correctly and Hard to Use Incorrectly” Scott Meyers
    • +
  • +
+
+
+

Scikit-HEP project

+
    +
  • Provides high-performance data analysis tools in Python
  • +
  • Partially funded by NSF grants
  • +
  • Many LHCb developers: Eduardo Rodrigues, Chris Burr, myself, …
  • +
  • Core values +
      +
    • Specialized tools that integrate well with existing Python ecosystem
    • +
    • Easy to use for beginners
    • +
    • Flexible for power users
    • +
    • Good documentation
    • +
    • Easy installation
    • +
  • +
+
+
+

The power of combinatorics

+

Build everything from basic components that fit together

+

+

Image credit: Toni Zaat on Unsplash

+
+
+

Important packages

+
    +
  • Python scientific stack

    +
      +
    • Numpy: fast computation with arrays
    • +
    • Scipy: integration, statistical distributions, special functions, …
    • +
    • Matplotlib: scientific plotting
    • +
    • Numba: JIT Compiler for Python (very fast)
    • +
    • Pandas: Convenient processing of tabular data
    • +
  • +
  • Scikit-HEP Project

    +
      +
    • uproot: fast reading and writing of ROOT Trees
    • +
    • particle: get particle properties from PDG IDs
    • +
    • boost-histogram: multi-dimensional generalised histograms
    • +
    • iminuit: fitting and error computation package
    • +
    • resample: Easy bootstrapping to compute uncertainty and bias
    • +
    • pyhf: for histogram fitting/limit setting and preserving likelihoods
    • +
  • +
  • Other packages

    +
      +
    • jacobi: Easy numerical error propagation
    • +
    • numba-stats: Fast statistical distributions for fitting
    • +
    • tabulate: Convert data tables to various formats (LaTeX, HTML, Markdown, …)
    • +
    • scikit-learn: Basic machine learning tools
    • +
    • xgboost: Gradient-boosting machine library that won many Kaggle competitions
    • +
  • +
+

Install everything with pip install LIBRARY

+
+
+

Working with Jupyter notebooks

+
    +
  • I recommend to use vscode +
      +
    • Very good Jupyter notebook support
    • +
    • Remote explorer: edit and run code on remote machines
    • +
  • +
+
+
+

Prepare computing environment

+
    +
  • In terminal, create and enter virtual environment (you need Python-3.8 or later)

    +
    python3 -m venv .venv
    +source .venv/bin/activate
  • +
  • Install packages

    +
    python -m pip install --upgrade numba matplotlib uproot boost-histogram iminuit scipy particle numba_stats ipywidgets
  • +
+

+
    +
  • uproot provides fast and convenient access to ROOT trees
  • +
+

+
+
import uproot
+
+print(f"uproot {uproot.__version__}")
+
+
uproot 5.2.2
+
+
+
+
f = uproot.open("example.root")
+event = f["event"]
+ 
+event.show()
+
+
name                 | typename                 | interpretation                
+---------------------+--------------------------+-------------------------------
+trk_len              | int32_t                  | AsDtype('>i4')
+mc_trk_len           | int32_t                  | AsDtype('>i4')
+trk_imc              | int32_t[]                | AsJagged(AsDtype('>i4'))
+trk_px               | float[]                  | AsJagged(AsDtype('>f4'))
+trk_py               | float[]                  | AsJagged(AsDtype('>f4'))
+trk_pz               | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_px            | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_py            | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_pz            | float[]                  | AsJagged(AsDtype('>f4'))
+mc_trk_pid           | int32_t[]                | AsJagged(AsDtype('>i4'))
+
+
+
    +
  • Tree contains fake simulated LHCb events

  • +
  • Truth in simulation

    +
      +
    • mc_trk_len: number of true particles in event
    • +
    • mc_trk_px: x-component of true momentum of particle
    • +
    • mc_trk_py: y-component of true momentum of particle
    • +
    • mc_trk_pz: z-component of true momentum of particle
    • +
    • mc_trk_pid: particle ID (PDG ID)
    • +
  • +
  • Reconstructed simulation

    +
      +
    • trk_len: number reconstructed tracks in event
    • +
    • trk_px: x-component of momentum of reconstructed track
    • +
    • trk_py: y-component of momentum of reconstructed track
    • +
    • trk_pz: z-component of momentum of reconstructed track
    • +
  • +
  • Special

    +
      +
    • trk_imc: index of matched true particle or -1
    • +
  • +
+
+
# get branches as arrays
+arr = event.arrays(["trk_len", "trk_px", "mc_trk_len", "mc_trk_px"])
+
+trk_len = arr["trk_len"]
+trk_px = arr["trk_px"]
+mc_trk_len = arr["mc_trk_len"]
+mc_trk_px = arr["mc_trk_px"]
+
+
+
trk_len
+
+
[6,
+ 7,
+ 2,
+ 6,
+ 7,
+ 7,
+ 6,
+ 7,
+ 4,
+ 6,
+ ...,
+ 2,
+ 4,
+ 13,
+ 3,
+ 6,
+ 3,
+ 4,
+ 6,
+ 3]
+------------------
+type: 1000 * int32
+
+
+

Exercise: Explore some other arrays from event

+
+
# do exercise here
+
+
    +
  • Some branches are 2D (e.g. trk_px)
  • +
  • First index iterates over events
  • +
  • Second index iterates over tracks per event
  • +
+
+
for ievent in range(5):
+    print(ievent, trk_len[ievent], trk_px[ievent])
+
+
0 6 [-0.979, 0.232, -0.464, 0.629, 0.0287, 0.156]
+1 7 [-0.59, 0.102, -0.282, -0.585, 0.0525, -0.249, -0.0836]
+2 2 [0.124, 0.309]
+3 6 [-0.287, 0.0082, -0.671, -0.933, -0.011, 0.0323]
+4 7 [-0.086, -0.101, 0.0523, 0.204, -0.0287, 0.115, 0.0954]
+
+
+
    +
  • Mathematical operators and slicing works on these arrays similar to Numpy
  • +
+
+
trk_len[:10]
+
+
[6,
+ 7,
+ 2,
+ 6,
+ 7,
+ 7,
+ 6,
+ 7,
+ 4,
+ 6]
+----------------
+type: 10 * int32
+
+
+
+
trk_len > 10
+
+
[False,
+ False,
+ False,
+ False,
+ False,
+ False,
+ False,
+ False,
+ False,
+ False,
+ ...,
+ False,
+ False,
+ True,
+ False,
+ False,
+ False,
+ False,
+ False,
+ False]
+-----------------
+type: 1000 * bool
+
+
+
    +
  • Many numpy functions work on awkward arrays
  • +
+
+
import numpy as np
+print(np.__version__)
+
+
1.24.4
+
+
+
+
np.sum(trk_len)
+
+
5032
+
+
+

Exercise

+
    +
  • Use numpy functions to count how many events have zero tracks
  • +
  • Hint: use np.sum and array mask
  • +
+
+
# do exercise here
+
+
+
# solution
+
+f"{np.sum(trk_len == 0)} events with zero tracks"
+
+
'5 events with zero tracks'
+
+
+
+
# let's plot some things
+import matplotlib.pyplot as plt
+
+plt.hist(trk_len);
+
+

+
+
+
    +
  • For some operations and awkward arrays, we need the awkward library
  • +
+
+
import awkward as ak
+print(ak.__version__)
+
+
2.6.1
+
+
+

Exercise * Try to plot trk_px * Can you figure out from the error message why it does not work? * Fix the issue with the function ak.flatten

+
+
# do exercise here
+
+
+
# solution
+
+# plt.hist(trk_px) fails with helpful error message
+plt.hist(ak.flatten(trk_px));
+
+

+
+
+
+
+

Reading data from a large tree

+
    +
  • Tree often do not fit into memory (tree size > 100 GB and more possible)
  • +
  • Iterate over chunks
  • +
+
+
for chunk in event.iterate(["trk_px", "mc_trk_px"]):
+
+    plt.hist(ak.flatten(chunk["trk_px"]), label="rec")
+    plt.hist(ak.flatten(chunk["mc_trk_px"]), label="gen")
+    plt.xlabel("px / MeVc-1")
+    plt.legend()
+
+    # stop after reading one chunk
+    break
+
+

+
+
+
+
+

Accelerate Python code with Numba

+
    +
  • Just-In-Time compiler for scientific Python code
  • +
  • Very powerful and very fast
  • +
  • Fast for loops
  • +
  • Not all Python code supported, works best with numerical code
  • +
  • Only use when needed
  • +
+
+
import numba as nb
+print(f"numba {nb.__version__}")
+
+
numba 0.57.1
+
+
+
    +
  • let’s subtract reconstructed from true momentum to check momentum resolution
  • +
  • need to select tracks matched to MC particles
  • +
  • this is relatively slow in Python, even if we use array computations
  • +
+
+
%%timeit -n 1 -r 3
+
+delta_px = []
+for chunk in event.iterate(["trk_px", "trk_imc", "mc_trk_px"]):
+    for px, imc, mc_px in zip(chunk["trk_px"], chunk["trk_imc"], chunk["mc_trk_px"]):
+        # select only tracks with associated true particle
+        mask = imc >= 0
+        subset_px = px[mask]
+        subset_mc_px = mc_px[imc[mask]]
+        # conversion to numpy needed
+        delta_px = np.append(delta_px, subset_px - subset_mc_px) 
+
+
963 ms ± 35.4 ms per loop (mean ± std. dev. of 3 runs, 1 loop each)
+
+
+
+
# awkward arrays can be used in Numba
+
+@nb.njit
+def calc(chunk):
+    r = []
+    # loop over events
+    for px, imc, mc_px in zip(chunk["trk_px"], chunk["trk_imc"], chunk["mc_trk_px"]):
+        # loop over reconstructed tracks per event
+        for px_i, imc_i in zip(px, imc):
+            if imc_i < 0:
+                continue
+            mc_px_i = mc_px[imc_i]
+            r.append(px_i - mc_px_i)
+    return r
+
+
+
# function is JIT-compiled on first call with new data types
+for chunk in event.iterate(["trk_px", "trk_imc", "mc_trk_px"]):
+    calc(chunk)
+    break
+
+
+
%%timeit -n 1 -r 3
+delta_px = []
+for data in event.iterate(["trk_px", "trk_imc", "mc_trk_px"]):
+    d = calc(data)
+    delta_px = np.append(delta_px, d)
+
+
4.17 ms ± 1.17 ms per loop (mean ± std. dev. of 3 runs, 1 loop each)
+
+
+
    +
  • Created array delta_px could still become very large in memory
  • +
  • Better: Create histogram object first and then fill iteratively with chunks
  • +
+

+
    +
  • Python frontend to Boost.Histogram library in C++ from the Boost project
  • +
  • Very fast and flexible
  • +
  • Multi-dimensional histograms and other binned statistics (profiles!)
  • +
  • Supports weighted data
  • +
  • And much much more, see docs
  • +
+
+
import boost_histogram as bh
+print(f"boost-histogram {bh.__version__}")
+
+
boost-histogram 1.4.0
+
+
+
+
# make an axis
+xaxis = bh.axis.Regular(20, -2, 2)
+
+# easy to make several histograms with same binning by reusing axis
+h_px = bh.Histogram(xaxis)
+h_mc_px = bh.Histogram(xaxis)
+
+
+
# let's use a finer binning for this one
+h_delta_px = bh.Histogram(bh.axis.Regular(100, -2, 2))
+
+
+
# incremental filling
+for data in event.iterate(["trk_px", "trk_imc", "mc_trk_px"]):
+    h_px.fill(ak.flatten(trk_px))
+
+
+
h_px
+
+
Histogram(Regular(20, -2, 2), storage=Double()) # Sum: 5032.0
+
+
+
+
print(h_px)
+
+
                 ┌───────────────────────────────────────────────────────────┐
+[-inf,   -2) 0   │                                                           │
+[  -2, -1.8) 1   │▏                                                          │
+[-1.8, -1.6) 3   │▎                                                          │
+[-1.6, -1.4) 17  │█▎                                                         │
+[-1.4, -1.2) 25  │█▊                                                         │
+[-1.2,   -1) 78  │█████▌                                                     │
+[  -1, -0.8) 181 │████████████▉                                              │
+[-0.8, -0.6) 308 │█████████████████████▊                                     │
+[-0.6, -0.4) 453 │████████████████████████████████▏                          │
+[-0.4, -0.2) 647 │█████████████████████████████████████████████▉             │
+[-0.2,    0) 819 │██████████████████████████████████████████████████████████ │
+[   0,  0.2) 782 │███████████████████████████████████████████████████████▍   │
+[ 0.2,  0.4) 629 │████████████████████████████████████████████▌              │
+[ 0.4,  0.6) 480 │██████████████████████████████████                         │
+[ 0.6,  0.8) 323 │██████████████████████▉                                    │
+[ 0.8,    1) 159 │███████████▎                                               │
+[   1,  1.2) 77  │█████▌                                                     │
+[ 1.2,  1.4) 37  │██▋                                                        │
+[ 1.4,  1.6) 7   │▌                                                          │
+[ 1.6,  1.8) 3   │▎                                                          │
+[ 1.8,    2) 3   │▎                                                          │
+[   2,  inf) 0   │                                                           │
+                 └───────────────────────────────────────────────────────────┘
+
+
+
+
# plot histogram with plt.stairs
+plt.stairs(h_px.values(), h_px.axes[0].edges);
+
+

+
+
+

Exercise * Use iterative loop to fill h_mc_px and h_delta_px, use calc from before * Plot all three histograms overlaid into one figure with plt.stairs

+
+
# do exercise here
+
+
+
# solution
+
+for chunk in event.iterate(["trk_px", "trk_imc", "mc_trk_px"]):
+    h_px.fill(ak.flatten(chunk["trk_px"]))
+    h_mc_px.fill(ak.flatten(chunk["trk_px"]))
+    h_delta_px.fill(calc(chunk))
+    
+for h, label in ((h_px, "px"), (h_mc_px, "mc_px"), (h_delta_px, "delta_px")):
+    plt.stairs(h.values(), h.axes[0].edges, label=label)
+    
+plt.legend();
+
+

+
+
+
    +
  • Histograms can be reduced
  • +
  • Shrink axis range with slices and using loc
  • +
  • Make binning coarser with rebin
  • +
+
+
from boost_histogram.tag import loc, rebin
+
+h_delta_px_2 = h_delta_px[loc(-0.5):loc(0.5):rebin(2)]
+
+for h in (h_delta_px_2, h_delta_px):
+    plt.stairs(h.values(), h.axes[0].edges,
+               label=f"{len(h.values())} bins")
+plt.legend();
+
+

+
+
+
+
+

Beyond 1D histograms

+
    +
  • Supports advanced axis types (e.g. category axis)
  • +
  • Supports higher dimensional histograms
  • +
  • Supports generalized histograms with binned statistics (aka “profile” in ROOT)
  • +
+
+
# check detector momentum resolution in x for different particle species
+
+p_axis = bh.axis.Regular(50, -2, 2)
+pid_axis = bh.axis.IntCategory([], growth=True)
+
+
+
# profile: compute mean of reconstructed px in bins of mc_px for different particle species
+
+p_2d = bh.Histogram(p_axis, pid_axis, storage=bh.storage.Mean())
+
+
+
branches = ["trk_px", "trk_imc", "mc_trk_px", "mc_trk_pid"]
+for data in event.iterate(branches):
+    for px, imc, mc_px, mc_pid in zip(*(data[b] for b in branches)):
+        # select only tracks with associated true particle
+        mask = imc >= 0
+        px = px[mask]
+        associated = imc[mask]
+        mc_px = mc_px[associated]
+        mc_pid = mc_pid[associated]
+
+        p_2d.fill(mc_px, mc_pid, sample=px)
+
+p_2d
+
+
Histogram(
+  Regular(50, -2, 2),
+  IntCategory([-321, 211, 2212, -2212, -211, 321], growth=True),
+  storage=Mean()) # Sum: Mean(count=3240, value=0.000433322, variance=0.263444)
+
+
+
    +
  • We use particle library to look-up properties of particles based on their particle ID
  • +
  • Get name, mass, life-time, quark content, …
  • +
+
+
import particle
+print(f"particle {particle.__version__}")
+
+
particle 0.23.1
+
+
+
+
pids = p_2d.axes[1]
+for i, pid in enumerate(pids):
+    part = particle.Particle.from_pdgid(pid)
+
+    x = p_2d.axes[0].centers
+    y = p_2d.values()[:, i]
+    ye = p_2d.variances()[:, i] ** 0.5
+    plt.errorbar(x, y, ye, fmt="o", label=f"${part.latex_name}$ pid={int(part.pdgid)}")
+
+plt.xlabel("mc_px")
+plt.ylabel("px")
+plt.legend();
+
+

+
+
+

Exercise * Only show bins with sufficient entries, use p_2d.counts()

+
+
# do exercise here
+
+
+
# solution
+
+pids = p_2d.axes[1]
+for i, pid in enumerate(pids):
+    part = particle.Particle.from_pdgid(pid)
+
+    x = p_2d.axes[0].centers
+    y = p_2d.values()[:, i]
+    ye = p_2d.variances()[:, i] ** 0.5
+
+    ma = (p_2d.counts() > 5)[:, i]
+    
+    plt.errorbar(x[ma], y[ma], ye[ma], fmt="o",
+                 label=f"${part.latex_name}$ pid={int(part.pdgid)}")
+
+plt.xlabel("mc_px")
+plt.ylabel("px")
+plt.legend();
+
+

+
+
+
    +
  • We find that the reconstruction resolution is the same for all particle species
  • +
  • Which means we can remove this axis by summing over its entries (called projection in ROOT)
  • +
  • boost-histogram keeps track of events in overflow and underflow bins by default to make summing exact
  • +
+
+
p_1d = p_2d[:, sum] # remove PID axis by summing
+
+
+
x = p_1d.axes[0].centers
+y = p_1d.values()
+ye = p_1d.variances() ** 0.5
+m = p_1d.counts() >= 5
+
+plt.errorbar(x[m], y[m], ye[m], fmt="o")
+
+plt.xlabel("mc_px")
+plt.ylabel("px");
+
+

+
+
+
+
+

Fits

+
    +
  • Typical analysis work flow (often automated with Snakemake)

    +
      +
    1. Pre-select data
    2. +
    3. Make histograms or profiles from pre-selected data
    4. +
    5. Fit histograms or profiles to extract physical parameters
    6. +
  • +
  • Many specialized fitting tools for individual purposes, e.g.: scipy.optimize.curve_fit

  • +
  • Generic libraries

    +
  • +
  • Fitting (“estimation” in statistics)

    +
      +
    • Adjust parametric statistical model to dataset and find model parameters which match dataset best
    • +
    • AI community calls this “learning”
    • +
    • Estimate: result of fit
    • +
  • +
  • Dataset: samples \(\{ \vec x_i \}\)

    +
      +
    • Sample can be one-dimensional or multidimensional
    • +
    • Values can be continuous (e.g. candidate mass) or discrete (e.g. bin counts)
    • +
  • +
  • Model: Probability density or probability mass function \(f(\vec x; \vec p)\) which depends on unknown parameters \(\vec p\)

  • +
  • Conjecture: maximum-likelihood estimate (MLE) \(\hat{\vec p} = \text{argmax}_{\vec p} L(\vec p)\) is optimal, with \(L(\vec p) = \prod_i f(\vec x_i; \vec p)\)

    +
      +
    • Proven: Asymptotically optimal, unbiased with minimum variance +
        +
      • No proof for finite data sets, but unchallenged
      • +
    • +
    • Universal: MLE can be computed for any problem
    • +
    • Equivalent: minimize negative log-likelihood (NLL), \(\hat{\vec p} = \text{argmin}_{\vec p}(-\ln L(\vec p))\)
    • +
  • +
  • Limiting case of maximum-likelihood fit: least-squares fit aka chi-square fit

  • +
+

+
    +
  • Python frontend to Minuit2 C++ library maintained by ROOT team at CERN
  • +
  • Maximum-likelihood and least-squares fits with error estimation
  • +
  • Easy to use for beginners, flexible for power users
  • +
  • Comes with builtin cost functions for common fits
  • +
  • Large number of tutorials
  • +
+
+
import iminuit
+from iminuit import Minuit
+
+print(f"iminuit {iminuit.__version__}")
+
+
iminuit 2.25.2
+
+
+
+
+

iminuit vs. other packages

+
    +
  • pyhf +
      +
    • Better choice when you need to compute limits and preserve the likelihood
    • +
  • +
  • zfit and RooFit +
      +
    • Help you build statistical models: automatic normalization & convolutions +
        +
      • This has to be done “by hand” in iminuit
      • +
    • +
    • Good for standard tasks, but restrict your freedom
    • +
  • +
  • iminuit +
      +
    • Has good documentation
    • +
    • Well-designed API
    • +
    • Is very flexible and open
    • +
    • Is very fast with numba
    • +
  • +
  • Typical fit in particle physics: mass distribution of decay candidates with signal peak over smooth background
  • +
  • Task: get the number of signal candidates = signal yield
  • +
  • Can be solved with a fit
  • +
+
+
def make_data(size, seed=1):
+    rng = np.random.default_rng(seed)
+    s = rng.normal(0.5, 0.1, size=size // 2)
+    b = rng.exponential(1, size=2 * size)
+    b = b[b < 1]
+    b = b[: size // 2]
+    x = np.append(s, b)
+    return x
+
+x1 = make_data(10_000)
+plt.hist(x1, bins=100);
+
+

+
+
+
    +
  • To apply maximum-likelihood method, we need statistical model
  • +
  • Full sample is additive mixture of signal sample and background sample
  • +
  • Assumption 1: signal is normal-distributed; pdf is \(\mathcal{N}(\mu, \sigma)\) with parameters \(\mu\) and \(\sigma\)
  • +
  • Assumption 2: background is truncated exponential distribution
  • +
+
+
# fast implementations of statistical distributions, API similar to scipy.stats
+from numba_stats import norm, truncexpon
+
+# model pdf
+def model1(x, z, mu, sigma, slope):
+    s = norm.pdf(x, mu, sigma)
+    b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)
+    return (1 - z) * b + z * s
+
+# negative log-likelihood
+def nll1(z, mu, sigma, slope):
+    logL = np.log(model1(x1, z, mu, sigma, slope))
+    return -np.sum(logL)
+
+
+
m = Minuit(nll1, z=0.5, mu=0.5, sigma=1, slope=1)
+m.errordef = Minuit.LIKELIHOOD
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -1790Nfcn = 209
EDM = 762 (Goal: 0.0001)
INVALID MinimumABOVE EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.5000.012
1mu0.5000.003
2sigma0.1550.005
3slope1.000.07
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z0.000149-1e-6 (-0.037)0.033e-3 (0.508)-0.25e-3 (-0.304)
mu-1e-6 (-0.037)9.19e-06-0e-6 (-0.025)-68e-6 (-0.335)
sigma0.033e-3 (0.508)-0e-6 (-0.025)2.91e-05-0.053e-3 (-0.147)
slope-0.25e-3 (-0.304)-68e-6 (-0.335)-0.053e-3 (-0.147)0.00442
+
+
+
    +
  • Fit did not work
  • +
  • Plotting 1D or 2D slices of likelihood may help
  • +
+
+
m.draw_profile("z");
+
+

+
+
+
    +
  • Increase print level
  • +
+
+
m.print_level = 1
+m.migrad()
+m.print_level = 0
+
+
W FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W VariableMetricBuilder No improvement in line search
+W VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001
+W VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001
+W FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W VariableMetricBuilder No improvement in line search
+W VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001
+W VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001
+W FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W VariableMetricBuilder No improvement in line search
+W VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001
+W VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001
+W FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W VariableMetricBuilder No improvement in line search
+W VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001
+W VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001
+W FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W FCN result is NaN for [ nan nan nan nan ]
+W VariableMetricBuilder No improvement in line search
+W VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001
+W VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001
+
+
+
    +
  • fit failed because parameters need limits
  • +
+

Exercise

+
    +
  • Repeat fit, but with parameter limits.
  • +
  • Set parameter limits by calling m.limits["parameter"] = (lower, upper) before m.migrad()
  • +
  • You can set identical limits for multiple parameters, for example, m.limits["a", "b"] = (0, 1)
  • +
+
+
# do exercise here
+
+
+
# solution
+
+m.limits["z", "mu"] = (0, 1)
+m.limits["sigma", "slope"] = (0, None)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -2095Nfcn = 481
EDM = 1.78e-06 (Goal: 0.0001)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.4980.00901
1mu0.49680.002001
2sigma0.09940.00200
3slope1.030.060
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.94e-05-1e-6 (-0.063)10e-6 (0.547)-0.12e-3 (-0.213)
mu-1e-6 (-0.063)4.21e-06-0e-6 (-0.061)-27e-6 (-0.225)
sigma10e-6 (0.547)-0e-6 (-0.061)3.94e-06-18e-6 (-0.156)
slope-0.12e-3 (-0.213)-27e-6 (-0.225)-18e-6 (-0.156)0.00345
+
+
+
    +
  • compute asymmetric errors with Minos algorithm (profile likelihood method)
  • +
+
+
m.minos()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -2095Nfcn = 692
EDM = 1.78e-06 (Goal: 0.0001)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.4980.009-0.0090.00901
1mu0.49680.0021-0.00200.002001
2sigma0.09940.0020-0.00190.00200
3slope1.030.06-0.060.060
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
Error-0.0090.009-0.0020.002-0.00190.0020-0.060.06
ValidTrueTrueTrueTrueTrueTrueTrueTrue
At LimitFalseFalseFalseFalseFalseFalseFalseFalse
Max FCNFalseFalseFalseFalseFalseFalseFalseFalse
New MinFalseFalseFalseFalseFalseFalseFalseFalse
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.94e-05-1e-6 (-0.063)10e-6 (0.547)-0.12e-3 (-0.213)
mu-1e-6 (-0.063)4.21e-06-0e-6 (-0.061)-27e-6 (-0.225)
sigma10e-6 (0.547)-0e-6 (-0.061)3.94e-06-18e-6 (-0.156)
slope-0.12e-3 (-0.213)-27e-6 (-0.225)-18e-6 (-0.156)0.00345
+
+
+
    +
  • you can compute 2D confidence regions with mncontour and draw them with draw_mncontour
  • +
+
+
m.draw_mncontour("z", "sigma");
+
+

+
+
+

Exercise

+
    +
  • Draw 1,2,3 sigma contours, use keyword cl
  • +
+
+
# do exercise here
+
+
+
# solution
+
+m.draw_mncontour("z", "sigma", cl=(1, 2, 3));
+
+

+
+
+
    +
  • for nice overview: compute matrix of likelihood profiles with draw_mnmatrix
  • +
+
+
m.draw_mnmatrix(figsize=(9,7));
+
+ +
+
+

+
+
+
    +
  • Writing unbinned negative-log-likelihood is repetitive, only model ever changes
  • +
  • Use built-in cost function from iminuit to get nice extras
  • +
+
+
from iminuit.cost import UnbinnedNLL
+
+nll1 = UnbinnedNLL(x1, model1)
+m = Minuit(nll1, z=0.5, mu=0.5, sigma=1, slope=1)
+m.limits["z", "mu"] = (0, 1)
+m.limits["sigma", "slope"] = (0, None)
+
+
+
m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -4189Nfcn = 169
EDM = 8.37e-05 (Goal: 0.0002)time = 0.4 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z0.4980.00901
1mu0.49680.002001
2sigma0.09940.00200
3slope1.030.060
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.62e-05-1e-6 (-0.037)10e-6 (0.529)-0.11e-3 (-0.211)
mu-1e-6 (-0.037)4.17e-06-0e-6 (-0.039)-28e-6 (-0.230)
sigma10e-6 (0.529)-0e-6 (-0.039)3.83e-06-17e-6 (-0.152)
slope-0.11e-3 (-0.211)-28e-6 (-0.230)-17e-6 (-0.152)0.00345
+ + + + + + + + 2024-02-09T11:12:58.377610 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

Exercise

+
    +
  • Use interactive mode via m.interactive() to check how fit reacts to parameter changes
  • +
  • Useful for debugging and to find starting values by hand
  • +
+
+
# do exercise here
+
+
+
+

Faster fits?

+
    +
  • Option 1: accelerate NLL computation with Numba
  • +
  • Option 2: use binned data
  • +
+
+
x2 = make_data(1_000_000)
+
+
+
m = Minuit(UnbinnedNLL(x2, model1), z=0.5, mu=0.5, sigma=1, slope=1)
+m.limits["z", "mu"] = (0, 1)
+m.limits["sigma", "slope"] = (0, None)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -4.216e+05Nfcn = 169
EDM = 0.000135 (Goal: 0.0002)time = 5.9 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z499.7e-30.9e-301
1mu499.98e-30.20e-301
2sigma99.59e-30.20e-30
3slope0.9970.0060
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.87e-07-0.01e-6 (-0.049)0.10e-6 (0.548)-1.2e-6 (-0.229)
mu-0.01e-6 (-0.049)4.18e-08-0 (-0.060)-0.26e-6 (-0.226)
sigma0.10e-6 (0.548)-0 (-0.060)4.09e-08-0.18e-6 (-0.164)
slope-1.2e-6 (-0.229)-0.26e-6 (-0.226)-0.18e-6 (-0.164)3.06e-05
+ + + + + + + + 2024-02-09T11:13:04.515566 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

General insights * Time spend in minimizer is negligible * Bottleneck is evaluating NLL function * Bottleneck inside NLL function is usually evaluation of model * NLL computation is trivially parallelizable: log(pdf) computed independently for each data point * Numba allows us to exploit auto-vectorization (SIMD instructions) and parallel computing on multiple cors

+
+
@nb.njit(parallel=True, fastmath=True, error_model="numpy")
+def nll2(z, mu, sigma, slope):
+    def model(x, z, mu, sigma, slope):
+        s = norm.pdf(x, mu, sigma)
+        b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)
+        return (1 - z) * b + z * s
+
+    logL = np.log(model(x2, z, mu, sigma, slope))
+    return -np.sum(logL)
+
+# compile function
+nll2(0.5, 0.5, 0.1, 0.1);
+
+
OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.
+
+
+
+
m = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)
+m.errordef = Minuit.LIKELIHOOD
+m.limits["z", "mu"] = (0, 1)
+m.limits["sigma", "slope"] = (0, None)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = -2.108e+05Nfcn = 242
EDM = 2.48e-06 (Goal: 0.0001)time = 1.4 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z499.7e-30.9e-301
1mu499.98e-30.20e-301
2sigma99.59e-30.20e-30
3slope0.9970.0060
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.87e-07-0.01e-6 (-0.049)0.10e-6 (0.548)-1.2e-6 (-0.229)
mu-0.01e-6 (-0.049)4.18e-08-0 (-0.060)-0.26e-6 (-0.226)
sigma0.10e-6 (0.548)-0 (-0.060)4.09e-08-0.18e-6 (-0.164)
slope-1.2e-6 (-0.229)-0.26e-6 (-0.226)-0.18e-6 (-0.164)3.06e-05
+
+
+

+
    +
  • Often simpler: fit histograms

  • +
  • No bias from fitting histograms if done correctly, only loss in precision

  • +
  • Need model cdf instead of model pdf: \(F(x; \vec p) = \int_{-\infty}^x f(x'; \vec p) \text{d}x'\)

  • +
  • Recommended to use iminuit.cost.BinnedNLL, comes with many useful features

  • +
+
+
from iminuit.cost import BinnedNLL
+
+# model is now a cdf!
+def model3(x, z, mu, sigma, slope):
+    s = norm.cdf(x, mu, sigma)
+    b = truncexpon.cdf(x, 0.0, 1.0, 0.0, slope)
+    return (1 - z) * b + z * s
+
+w, xe = np.histogram(x2, bins=100, range=(0, 1))
+nll3 = BinnedNLL(w, xe, model3)
+
+
+
m = Minuit(nll3, z=0.5, mu=0.5, sigma=1, slope=1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 84.34 (χ²/ndof = 0.9)Nfcn = 162
EDM = 1.36e-07 (Goal: 0.0002)time = 0.4 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0z499.7e-30.9e-3
1mu499.99e-30.20e-3
2sigma99.58e-30.20e-3
3slope0.9970.006
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
zmusigmaslope
z8.98e-07-0.01e-6 (-0.049)0.10e-6 (0.535)-1.2e-6 (-0.236)
mu-0.01e-6 (-0.049)4.22e-08-0 (-0.052)-0.26e-6 (-0.228)
sigma0.10e-6 (0.535)-0 (-0.052)4.15e-08-0.18e-6 (-0.161)
slope-1.2e-6 (-0.236)-0.26e-6 (-0.228)-0.18e-6 (-0.161)3.09e-05
+ + + + + + + + 2024-02-09T11:13:17.369404 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+
+

How well is model matching data?

+
    +
  • Judgement requires binning of data
  • +
  • Look at pull distribution +
      +
    • Compute (data - model) / error for each bin
    • +
  • +
  • Compare \(\chi^2\) value (goodness-of-fit test statistic) against the degrees of freedom +
      +
    • Simple check: reduced chi-square \(\chi^2/n_\text{dof}\) should be about 1
    • +
    • Better check: p-value \(= \int_{\chi^2_\text{observed}}^{\infty} f(\chi^2; n_\text{dof}) \, \text{d}\chi^2\) (survival function)
    • +
    • If p-value very small (< 1%), wrong model or underestimated errors
    • +
  • +
+
+
from scipy.stats import chi2
+
+pulls = nll3.pulls(m.values)
+cx = xe[:-1] + np.diff(xe)
+plt.errorbar(cx, pulls, np.ones_like(pulls), fmt="o");
+
+

+
+
+
+
# m.fval is chi-square distributed for all built-in cost functions
+# except UnbinnedNLL and ExtendedUnbinnedNLL
+pvalue = chi2(m.ndof).sf(m.fval)
+f"chi2/ndof = {m.fmin.reduced_chi2:.2f} p-value = {pvalue:.2f}"
+
+
'chi2/ndof = 0.88 p-value = 0.80'
+
+
+

Exercise

+
    +
  • Plot histogram of pulls, should look like standard normal distribution
  • +
  • Compute mean and standard deviation (use np.mean and np.std)
  • +
+
+
# do exercise here
+
+
+
# solution
+
+xe = plt.hist(pulls, bins=20)[1]
+plt.title(f"mean = {np.mean(pulls):.2f} std = {np.std(pulls):.2f}");
+
+

+
+
+
+
+

Other fits

+
    +
  • To fit yields (counts) instead of fractions, use ExtendedUnbinnedNLL or ExtendedBinnedNLL
  • +
  • To fit (x, y) pairs, use LeastSquares
  • +
  • To fit simulated templates, use Template
  • +
  • See documentation how models should be constructed then
  • +
+
+
+

Template fits

+
    +
  • Sometimes PDF/CDF not known in parametric form, but can be Monte-Carlo simulated
  • +
  • Idea: fit a histogram of simulation output (template) to data histogram
  • +
  • Additional source of uncertainty from finite size of simulation
  • +
  • Correct error propagation with Barlow-Beeston method, Barlow and Beeston, Comput.Phys.Commun. 77 (1993) 219-228
  • +
  • iminuit.cost.Template +
  • +
+
+
def make_data2(rng, nmc, truth, bins):
+    xe = np.linspace(0, 1, bins + 1)
+    b = np.diff(truncexpon.cdf(xe, 0, 1, 0, 1))
+    s = np.diff(norm.cdf(xe, 0.5, 0.05))
+    n = rng.poisson(b * truth[0]) + rng.poisson(s * truth[1])
+    t = np.array([rng.poisson(b * nmc), rng.poisson(s * nmc)])
+    return xe, n, t
+
+rng = np.random.default_rng(1)
+truth = 750, 250
+xe, n, t = make_data2(rng, 200, truth, 15)
+
+_, ax = plt.subplots(1, 2, figsize=(10, 4))
+ax[0].stairs(n, xe, color="k", label="data")
+ax[0].legend();
+ax[1].stairs(t[0], xe, label="background")
+ax[1].stairs(t[1], xe, label="signal")
+ax[1].legend(title="templates");
+
+

+
+
+
+
from iminuit.cost import Template
+
+c = Template(n, xe, t)
+m = Minuit(c, 1, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 8.352 (χ²/ndof = 0.6)Nfcn = 132
EDM = 2.75e-07 (Goal: 0.0002)time = 0.5 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0770700
1x1220400
+ + + + + + + + + + + + + + + + + + +
x0x1
x04.3e+03-0.8e3 (-0.354)
x1-0.8e3 (-0.354)1.32e+03
+ + + + + + + + 2024-02-09T11:13:24.390185 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

Exercise

+
    +
  • With make_data2 generate templates with 1 000 000 simulated points
  • +
  • Fit this data with the Template cost function
  • +
  • Compare parameter uncertainties
  • +
+
+
# do exercise here
+
+
+
# solution
+
+xe, n, t = make_data2(rng, 1_000_000, truth, 15)
+c = Template(n, xe, t)
+m = Minuit(c, 1, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 4.329 (χ²/ndof = 0.3)Nfcn = 110
EDM = 2.66e-06 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0820320
1x1201200
+ + + + + + + + + + + + + + + + + + +
x0x1
x01.03e+03-0.2e3 (-0.326)
x1-0.2e3 (-0.326)415
+ + + + + + + + 2024-02-09T11:13:24.547539 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
    +
  • With Template you can also fit a mix of template and parametric model
  • +
  • Often template only needed for background(s)
  • +
+
+
# density model for signal component, as for ExtendedBinnedNLL
+def model5(x, s, mu, sigma):
+    return s * norm.cdf(x, mu, sigma)
+
+c = Template(n, xe, (t[0], model5))
+m = Minuit(c, x0=700, x1_s=300, x1_mu=0.5, x1_sigma=0.1)
+m.limits["x1_mu"] = (0, 1)
+m.limits["x1_sigma"] = (0, None)
+
+
+
m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 2.526 (χ²/ndof = 0.2)Nfcn = 106
EDM = 1.52e-06 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0819340
1x1_s20223
2x1_mu0.4920.00601
3x1_sigma0.0490.0070
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
x0x1_sx1_mux1_sigma
x01.16e+03-0.3e3 (-0.428)8.13e-3 (0.038)-79.31e-3 (-0.330)
x1_s-0.3e3 (-0.428)541-8.17e-3 (-0.056)79.27e-3 (0.483)
x1_mu8.13e-3 (0.038)-8.17e-3 (-0.056)3.99e-05-0 (-0.069)
x1_sigma-79.31e-3 (-0.330)79.27e-3 (0.483)-0 (-0.069)4.98e-05
+ + + + + + + + 2024-02-09T11:13:24.717132 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
+
+

Overfitting

+
    +
  • Do not use models with degenerate parameters
  • +
  • Avoid overfitting
  • +
+
+
from iminuit.cost import LeastSquares
+from numba_stats import bernstein  # Bernstein polynomials > Chebychev polynomials
+
+x = [-0.37, -0.25, -0.15, -0.05, 0.047, 0.147, 0.243, 0.374]
+y = [0.96, 0.90, 0.85, 0.81, 0.75, 0.66, 0.42, 0.14]
+ey = [0.01, 0.01, 0.01, 0.02, 0.02, 0.03, 0.04, 0.05]
+
+def model(x, p):
+    return bernstein.density(x, p, -0.4, 0.4)
+
+
+
start = np.ones(3)
+m = Minuit(LeastSquares(x, y, ey, model), start)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 23.81 (χ²/ndof = 4.8)Nfcn = 51
EDM = 4.25e-21 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x00.9500.011
1x10.9200.027
2x20.210.04
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
x0x1x2
x00.000127-0.21e-3 (-0.680)0.14e-3 (0.321)
x1-0.21e-3 (-0.680)0.000737-0.7e-3 (-0.666)
x20.14e-3 (0.321)-0.7e-3 (-0.666)0.00147
+ + + + + + + + 2024-02-09T11:19:07.003025 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

Exercise

+
    +
  • Fit with increasing number of parameters (increase the length of the starting parameters)

    +
      +
    • What do you observe? Look out for changes in chi2/ndof and the parameter correlations
    • +
  • +
  • Try fitting this model

    +
    def model(x, p):
    +    return p[0] + bernstein.density(x, p[1:], -0.4, 0.4)
    +
      +
    • What do you observe?
    • +
  • +
+
+
# do exercise here
+
+
+
# solution
+
+# def model(x, p):
+#       return p[0] + bernstein.density(x, p[1:], -0.4, 0.4)
+
+start = np.ones(6)
+m = Minuit(LeastSquares(x, y, ey, model), start)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 1.069 (χ²/ndof = 0.5)Nfcn = 152
EDM = 2.41e-15 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x00.9560.026
1x11.020.13
2x20.470.27
3x31.300.34
4x40.290.25
5x50.100.08
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
x0x1x2x3x4x5
x00.000693-3.0e-3 (-0.885)5.7e-3 (0.800)-6.2e-3 (-0.687)3.6e-3 (0.551)-0.7e-3 (-0.307)
x1-3.0e-3 (-0.885)0.0161-0.033 (-0.961)0.037 (0.852)-0.022 (-0.698)0.004 (0.394)
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z`6W!_8mQvgYW@dJ4u|Pxe9r-T{R_A{jaYxP}8Qhh26M+HHNT5VWi3UJbV|> hokyzwKc&qk?3bz@UE>j-{)&ux>IJd$F=y59{6BbI%b)-N literal 0 HcmV?d00001 diff --git a/2024_LHCb_Starterkit/lecture_2.html b/2024_LHCb_Starterkit/lecture_2.html new file mode 100644 index 0000000..037e53e --- /dev/null +++ b/2024_LHCb_Starterkit/lecture_2.html @@ -0,0 +1,2120 @@ + + + + + + + + + +Hans Dembinski’s blog - Statistics II + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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Statistics II

+
+ + + +
+ + + + +
+ + +
+ +

Hans Dembinski, TU Dortmund

+
    +
  • Uncertainties of estimators
  • +
  • Bootstrapping of uncertainties
  • +
  • Error propagation
  • +
  • Estimation of systematic uncertainties
  • +
  • Resources for further learning
  • +
+
+
+

+
image.png
+
+
+
+

What is an uncertainty, really?

+
    +
  • Example: \(a = 3 \pm 0.5\), where 0.5 is the standard error (jargon: “error” = uncertainty)

  • +
  • What we would like it to mean: “The true value is between 2.5 and 3.5 with a probability of 68 %”

  • +
  • What it technically means (in most cases): “The interval 2.5 to 3.5, if always constructed with the same procedure, contains the true value in 68 % of identically repeated random experiments.”

  • +
  • Both statements coincide only in the asymptotic limit of infinitely large samples

  • +
+
+
+

Statistical and systematic uncertainties

+
+

Statistical uncertainty

+
    +
  • Origin: We use finite sample instead of infinite distribution
  • +
  • Goes down as simple size increases
  • +
  • Example +
      +
    • Arithmetic mean: \(\bar x = \frac 1N\sum_i x_i\)
    • +
    • Variance of \(\bar x\): \(\hat V_{\bar x} = \frac{1}{N - 1} \frac 1 N\sum_i (x_i - \bar x)^2\)
    • +
  • +
  • We have reliable standard recipes to calculate these
  • +
+
+

Variance or standard deviation?

+
    +
  • Standard deviation \(\sigma = \sqrt{V}\)
  • +
  • Variance is more fundamental +
      +
    • Second central moment of distribution
    • +
    • \(V = V_1 + V_2\), but \(\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}\)
    • +
  • +
  • Communicate standard deviations, compute with variances
  • +
+
+
+
+

Systematic uncertainty

+
    +
  • Origin: Imperfect appartus or technique, deviations of reality from simplified models
  • +
  • Quantifies potential mistakes in analysis
  • +
  • Does not go down as sample size increases
  • +
  • Often correlated +
      +
    • Real-life: Measured the length of ten shoes with ruler than has factory tolerance of 0.1 mm
    • +
    • LHCb: Measured cross-sections with value of luminosity that has uncertainty of 1-2%
    • +
  • +
  • Not well covered in statistics books, but some theoretical results available
  • +
+
+
+
+

Estimators

+
    +
  • Estimator maps dataset to value \(\{x_i \} \to \hat p\)

    +
      +
    • Notation: \(\hat p\) is estimate of true value \(p\)
    • +
    • Estimator can have functional form or be an algorithm
    • +
    • Arithmetic mean \(\hat \mu = \frac 1N \sum_i x_i\)
    • +
    • Sample median: Sort \(\{ x_i \}\), pick value in the middle
    • +
  • +
  • Many kinds of estimators, some better than others

  • +
  • Optimal estimators

    +
      +
    • As close to true value as possible
    • +
    • Easy to compute and/or easy to apply to any problem
    • +
    • Robustness against outliers
    • +
  • +
  • In HEP, we mainly use

    +
      +
    • Plug-in estimator
    • +
    • Maximum-likelihood estimator (MLE)
    • +
  • +
+
+
+

Bias and variance

+
    +
  • In one experiment, we obtain sample \(\{ x_i \}\) with size \(N\) and compute estimate \(\hat a\)
  • +
  • Sample \(\{ x_i \}\) is random, estimate \(\hat a\) also random
  • +
  • If experiment is identically repeated \(K\) times, we get sample \(\{ x_i \}_k\) and estimate \(\hat a_k\) each time
  • +
  • Derive accuracy of estimator from estimates \(\{ \hat a_1, \hat a_2, \dots, \hat a_K \}\) +
      +
    • Bias of \(\hat a\) = \(\frac1K \sum_i (\hat a_i - a)\)
    • +
    • Variance of \(\hat a\) = \(\frac 1 {K -1}\sum_i (\hat a_i - \frac 1K \sum_k \hat a_k)^2\)
    • +
  • +
  • Example for biased estimator: \(\hat V = \frac1N\sum_i (x_i - \hat\mu)^2\) is biased if \(\hat \mu\) is estimated from same sample
  • +
+
+
from matplotlib import pyplot as plt
+import numpy as np
+
+mu = 1
+sigma = 1
+
+N = 10
+K = 10000
+var_1, var_2 = [], []
+for k in range(K):
+    rng = np.random.default_rng(seed=k)
+    x_k = rng.normal(mu, sigma, size=N)
+    v1 = np.var(x_k)
+    v2 = np.mean((x_k - mu) ** 2)
+    var_1.append(v1)
+    var_2.append(v2)
+
+
+
fig, ax = plt.subplots(1, 2, sharex=True, sharey=True, figsize=(10, 4))
+ax[0].hist(var_1, bins=50, label="$\hat V_1$");
+ax[1].hist(var_2, bins=50, label="$\hat V_2$");
+for axi in ax:
+    axi.axvline(sigma ** 2, color="C3", label="true V")
+ax[0].axvline(np.mean(var_1), color="C1", ls="--", label="mean of $\hat V_1$")
+ax[1].axvline(np.mean(var_2), color="C1", ls="--", label="mean of $\hat V_2$")
+for axi in ax:
+    axi.legend()
+
+

+
+
+
    +
  • Well-known bias correction for this case: \[ +\hat V_{1,\text{corr}} = \frac N{N-1}\hat V_1 +\] For derivation see e.g. F. James book (references at end)
  • +
+

Exercise * Compute \(\hat V_{1, \text{corr}}\) and compare its mean with true \(V\)

+
+
# do exercise here
+
+
+

Plug-in estimator

+
    +
  • Plug-in principle: replace true values in some formula with estimates, e.g.
  • +
+

\[ +c = g(a, b, \dots) \to \hat c = g(\hat a, \hat b, \dots) +\]

+
    +
  • Justification +
      +
    • For \(N\to \infty\), \(\Delta a = \hat a - a \to 0\) for any reasonable estimator
    • +
    • \(g(\hat a) = g(a + \Delta a) = g(a) + O(\Delta a) \to g(a)\) for smooth \(g\)
    • +
  • +
  • Example: \(\sqrt{N}\) uncertainty estimator for Poisson-distributed count \(N\) +
      +
    • \(P(N; \lambda) = e^{-\lambda} \lambda^N / N!\)
    • +
    • Variance of counts: \(V_N = \lambda\)
    • +
    • Estimator \(\hat \lambda = \frac 1 K \sum_k N_k\); for \(K = 1\), \(\hat \lambda = N\)
    • +
    • Plug-in principle: \(\hat V_N = \hat \lambda = N\)
    • +
    • Standard deviation: \(\hat \sigma_N = \sqrt{\hat V_N} = \sqrt{N}\)
    • +
  • +
  • Plug-in estimators can behave poorly in small samples
  • +
  • Poisson example: \(\lambda = 0.01\), sample \(N = 0 \to \hat \sigma_N = 0\)
  • +
+
+
+
+

Bootstrapping bias and variance of estimators

+
    +
  • Bootstrap method: generic way to compute bias and variance of any estimator
  • +
+
+

Parametric bootstrap

+
    +
  • Available: +
      +
    • Sample \(\{ x_i \}\) distributed along \(f(x; a)\), \(a\) is unknown
    • +
    • Estimator for \(\hat a\)
    • +
  • +
  • Want to know bias and variance of estimator
  • +
  • Simulate experiment \(K\) times +
      +
    • Draw samples \(\{ x_i \}_k\) of equal size from \(\hat f(x; \hat a)\) (plugin-estimate)
    • +
    • Compute \(\hat a_k\)
    • +
    • Bias \(\frac 1 K \sum_k (\hat a_k - \hat a)\)
    • +
    • Variance \(\frac 1 {K-1} \sum_k (\hat a_k - \frac 1K \sum_k \hat a_k)^2\)
    • +
  • +
+
+
from numba_stats import norm, truncexpon, poisson
+
+
+def make_data(s, mu, sigma, b, slope, seed):
+    sr = poisson.rvs(s, size=1, random_state=seed)[0]
+    br = poisson.rvs(b, size=1, random_state=seed)[0]
+    s = norm.rvs(mu, sigma, size=sr, random_state=seed)
+    b = truncexpon.rvs(0, 1, 0, slope, size=br, random_state=seed)
+    return np.append(s, b)
+
+truth = {
+    "s": 1000,
+    "mu": 0.5,
+    "sigma": 0.05,
+    "b": 500,
+    "slope": 1,
+}
+
+
+
x = make_data(**truth, seed=0)
+
+w, xe = plt.hist(x, bins=100)[:2];
+
+

+
+
+
+
from iminuit import Minuit
+from iminuit.cost import ExtendedBinnedNLL
+from numba_stats import norm, truncexpon
+
+def model(x, s, mu, sigma, b, slope):
+    return s * norm.cdf(x, mu, sigma) + b * truncexpon.cdf(x, 0, 1, 0, slope)
+
+nll = ExtendedBinnedNLL(w, xe, model)
+m = Minuit(nll, **truth)
+m.limits["mu"] = (0, 1)
+m.limits["s", "b", "sigma", "slope"] = (0, None)
+
+
+
m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 106.9 (χ²/ndof = 1.1)Nfcn = 101
EDM = 9.95e-07 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0s1000350
1mu0.49830.001801
2sigma0.04820.00140
3b510270
4slope0.950.150
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
smusigmabslope
s1.2e+03-48.4e-68.9237e-3 (0.177)-0.2e3 (-0.213)-0.368 (-0.073)
mu-48.4e-63.09e-06-0 (-0.005)61.5e-6 (0.001)-17.0e-6 (-0.066)
sigma8.9237e-3 (0.177)-0 (-0.005)2.12e-06-8.9641e-3 (-0.231)-16.5e-6 (-0.078)
b-0.2e3 (-0.213)61.5e-6 (0.001)-8.9641e-3 (-0.231)7110.354 (0.091)
slope-0.368 (-0.073)-17.0e-6 (-0.066)-16.5e-6 (-0.078)0.354 (0.091)0.0213
+ + + + + + + + 2024-02-13T09:47:06.936461 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
    +
  • Fit can be written as estimator
  • +
+
+
def estimator(x):
+    w = np.histogram(x, bins=xe)[0]
+    nll = ExtendedBinnedNLL(w, xe, model)
+    m = Minuit(nll, **truth)
+    m.limits["mu"] = (0, 1)
+    m.limits["s", "b", "sigma", "slope"] = (0, None)
+    m.migrad()
+    assert m.valid
+    return m.values
+
+
+
replicates = []
+for seed in range(100):
+    x_k = make_data(*m.values, seed=seed+1)
+    p_k = estimator(x_k)
+    replicates.append(p_k)
+
+replicates = np.transpose(replicates)
+
+

Exercise

+
    +
  • Compute bias and standard deviation of replicates for each parameter
  • +
  • Compare standard deviation of replicates with Minuit’s uncertainty estimate, accessible via m.errors
  • +
+
+
# do exercise here
+
+
+
+
+

Nonparametric bootstrap

+
    +
  • Can do the same by sampling directly from \(\{ x_i \}\)
  • +
  • Standard method for samples of fixed size \(N\): +
      +
    • Pick \(N\) random samples from \(\{ x_i \}\) with replacement to obtain \(\{ x_i \}_k\)
    • +
    • Compute \(\hat a_k\) from \(\{ x_i \}_k\) and proceed as before
    • +
  • +
  • “Extended method” for samples of variable size (Poisson): +
      +
    • Randomly draw \(M\) from Poisson distribution with \(\lambda = N\)
    • +
    • Pick \(M\) random samples from \(\{ x_i \}\) with replacement to obtain \(\{ x_i \}_k\)
    • +
    • +
  • +
  • Convenient: use same algorithms for any distribution
  • +
  • Efficient implementations for both algorithms in resample
  • +
+

+
+
import resample
+
+resample.__version__
+
+
'1.7.1'
+
+
+
+
from resample.bootstrap import resample
+
+# generate replicated samples of varying size (Poisson distributed)
+for x_k in resample(x, size=100, method="extended", random_state=1):
+    print(x_k)
+    break
+
+
[0.58820262 0.58820262 0.52000786 ... 0.38532216 0.26892968 0.26892968]
+
+
+

Exercise

+
    +
  • Again compute replicated fit results using resampe and store results in a list
  • +
  • Copy previous code where possible
  • +
  • Again compute bias and standard deviation of replicates and compare with Minuit’s uncertainty estimate
  • +
+
+
# do exercise here
+
+
    +
  • resample provides shortcut for computing variance
  • +
+
+
from resample.bootstrap import variance
+
+var = variance(estimator, x, method="extended", size=100, random_state=1)
+
+for i, name in enumerate(m.parameters):
+    std = var[i] ** 0.5
+    print(f"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g}")
+
+
s        𝜎(Minuit) =     35 𝜎(bootstrap) =     33
+mu       𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017
+sigma    𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0014
+b        𝜎(Minuit) =     27 𝜎(bootstrap) =     27
+slope    𝜎(Minuit) =   0.15 𝜎(bootstrap) =   0.17
+
+
+

Exercise

+
    +
  • Repeat the variance calculation without using method="extended"
  • +
  • Compare the uncertainties of \(s\) and \(b\)
  • +
+
+
# do exercise here
+
+
    +
  • Bootstrap can be applied to most statistical problems in HEP
  • +
  • Restriction: samples must be identically and independently distributed
  • +
  • May work well even with small samples (< 10)
  • +
+
+
+

Error propagation

+
    +
  • Full analysis typically combines many intermediate results with uncertainties into final result
  • +
  • Consider \(\vec b = g(\vec a)\) with uncertainties in \(\vec a\)
  • +
  • Two ways of computing uncertainties of \(\vec b\) +
      +
    • Monte-Carlo simulation
    • +
    • Error propagation
    • +
  • +
  • Can be equally used for statistical and systematic uncertainties
  • +
+
+

Monte-Carlo simulation

+
    +
  • Draw variations of fit result from multivariate normal distribution

  • +
  • Compute variable of interest from variations

  • +
  • Compute standard deviation or covariance of

  • +
  • Example: compute uncertainty of \(\hat N = \hat s + \hat b\)

  • +
+
+
rng = np.random.default_rng(1)
+a_k = rng.multivariate_normal(m.values, m.covariance, size=1000)
+
+def g(a):
+    return a[0] + a[3]
+
+b_k = np.apply_along_axis(g, -1, a_k)
+
+print(f"𝜎(N, simulation) = {np.std(b_k):.3g}")
+print(f"𝜎(N, naive)      = {(m.errors[0]**2 + m.errors[3]**2)**0.5:.3g}")
+print(f"𝜎(N, expected)   = {np.sqrt(len(x)):.3g}")
+
+
𝜎(N, simulation) = 37.7
+𝜎(N, naive)      = 43.7
+𝜎(N, expected)   = 38.8
+
+
+
    +
  • \(\sigma_{N,\text{simulation}}^2 < \sigma_s^2 + \sigma_b^2\), because \(s_k\) and \(b_k\) are anti-correlated
  • +
+
+
+

Error propagation

+
    +
  • Given: \(\vec a\) with covariance matrix \(C_a\) and differentiable function \(\vec b = g(\vec a)\)

  • +
  • \(\vec b\) can have different dimensionality from \(\vec a\)

  • +
  • One can derive via first-order Taylor expansion: \[ +C_b = J \, C_a \, J^T \text{ with } J_{ik} = \frac{\partial g_i}{\partial a_k} +\]

  • +
  • Useful special cases

    +
      +
    • Sum of scaled independent random values \(c = \alpha a + \beta b\) (\(\alpha, \beta\) are constant) \[ + \sigma^2_c = \alpha^2 \sigma^2_a + \beta^2 \sigma^2_b + \]
    • +
    • Product of powers of independent random values \(c = a^\alpha \, b^\beta\) \[ + \left(\frac{\sigma_c}{c}\right)^2 = \alpha^2 \left(\frac{\sigma_a}{a}\right)^2 + \beta^2 \left(\frac{\sigma_b}{b}\right)^2 + \]
    • +
  • +
  • Final uncertainty often dominated by one term

  • +
  • When estimating systematic uncertainties: put your effort in estimating largest term and be lax on others

  • +
  • Computation of \(J\) by hand error-prone and tedious

  • +
  • jacobi library offers automatic error propagation based on a robust numerical derivative

  • +
+

+
+
import jacobi
+
+jacobi.__version__
+
+
'0.9.2'
+
+
+
+
b, cov_b = jacobi.propagate(g, m.values, m.covariance)
+
+f"𝜎(N, error propagation) = {cov_b ** 0.5:.3g} 𝜎(N, expected) = {np.sqrt(len(x)):.3g}"
+
+
'𝜎(N, error propagation) = 38.9 𝜎(N, expected) = 38.8'
+
+
+
    +
  • jacobi.propagate works for any differentiable mapping, which even includes most fits!
  • +
+
+
from iminuit.cost import LeastSquares
+
+def h(pvec):
+    a, b = pvec
+    x = np.array([1, 2, 3])
+    y = a + b * x
+    m = Minuit(LeastSquares(x, y, 1, lambda x, a, b: a + b * x), a=0, b=0)
+    m.strategy = 0
+    m.migrad()
+    assert m.valid
+    return m.values
+
+jacobi.propagate(h, (1, 2), (1, 1))
+
+
(array([1., 2.]),
+ array([[ 1.00000000e+00, -1.84267416e-12],
+        [-1.84267416e-12,  1.00000000e+00]]))
+
+
+ +
+
+
+

Systematic uncertainties: Known unknowns

+

“There are known knowns, things we know that we know;
and there are known unknowns, things that we know we don’t know.
But there are also unknown unknowns, things we do not know we don’t know.”
Donald Rumsfeld

+
    +
  • Why data analyses take so long: study data, detector, and methods to turn unknown unknowns into known knowns or known unknowns
  • +
  • known unknowns = systematic uncertainties
  • +
  • No formal way, but generally…
  • +
+

Worry * What could go wrong? * What are my assumptions?

+

Example * You take an efficiency or correction factor from simulation, but simulation differs from real experiment. * Solution: Perform control measurement to estimate how much simulation differs from real experiment.

+

Methods * Use methods that are known to be optimal for your problem or derive them from first principles * General purpose methods often safer to use than special-case methods * Maximum-likelihood fit > least-squares fit * Bootstrap estimate of uncertainty > other uncertainty estimates

+

Apply checks * Validate whole analysis on simulated data (sometimes you can use toy simulation) * Split data by magnet polarity, fill, etc., analyze splits separately, check for agreement

+

Guidelines for handling systematic errors * Roger Barlow: Systematic Errors: Facts and Fictions (2002) * Roger Barlow, “Practical Statistics for Particle Physics” (2019), Section 6.5

+
+
+

Further resources for learning

+
+

LHCb Statistics and ML WG

+ +
+
+

Introductory statistics books

+
    +
  • List of books on TWiki pages of Statistis and Machine Learning WG

  • +
  • Books I personally like a lot

    +
      +
    • Glen Cowan, “Statistical Data Analysis”, Oxford University Press, 1998
    • +
    • F. James, “Statistical Methods in Experimental Physics”, 2nd edition, World Scientific, 2006
    • +
    • R. Barlow, “Statistics: a Guide to the Use of Statistical Methods in the Physical Sciences”, Wiley, 1989
    • +
    • B. Efron, R.J. Tibshirani, “An Introduction to the Bootstrap”, CRC Press, 1994
    • +
  • +
+
+
+

Lectures, papers, notebooks

+ + + +
+
+ +
+ +
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b/2024_LHCb_Starterkit/lecture_2_full.html @@ -0,0 +1,2188 @@ + + + + + + + + + +Hans Dembinski’s blog - Statistics II + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Statistics II

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Hans Dembinski, TU Dortmund

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  • Uncertainties of estimators
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  • Bootstrapping of uncertainties
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  • Error propagation
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  • Estimation of systematic uncertainties
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  • Resources for further learning
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What is an uncertainty, really?

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  • Example: \(a = 3 \pm 0.5\), where 0.5 is the standard error (jargon: “error” = uncertainty).

  • +
  • What we would like it to mean: “The true value is between 2.5 and 3.5 with a probability of 68 %”.

  • +
  • What it technically means (in most cases): “The interval 2.5 to 3.5, if always constructed with the same procedure, contains the true value in 68 % of identically repeated random experiments.”

  • +
  • Both statements coincide only in the asymptotic limit of infinitely large samples.

  • +
+
+
+

Statistical and systematic uncertainties

+
+

Statistical uncertainty

+
    +
  • Origin: We use finite sample instead of infinite distribution
  • +
  • Goes down as simple size increases
  • +
  • Example +
      +
    • Arithmetic mean: \(\bar x = \frac 1N\sum_i x_i\)
    • +
    • Variance of \(\bar x\): \(\hat V_{\bar x} = \frac{1}{N - 1} \frac 1 N\sum_i (x_i - \bar x)^2\)
    • +
  • +
  • We have reliable standard recipes to calculate these
  • +
+
+

Variance or standard deviation?

+
    +
  • Standard deviation \(\sigma = \sqrt{V}\)
  • +
  • Variance is more fundamental +
      +
    • Second central moment of distribution
    • +
    • \(V = V_1 + V_2\), but \(\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}\)
    • +
  • +
  • Communicate standard deviations, compute with variances
  • +
+
+
+
+

Systematic uncertainty

+
    +
  • Origin: Imperfect appartus or technique, deviations of reality from simplified models
  • +
  • Quantifies potential mistakes in analysis
  • +
  • Does not go down as sample size increases
  • +
  • Often correlated +
      +
    • Real-life: Measured the length of ten shoes with ruler than has factory tolerance of 0.1 mm
    • +
    • LHCb: Measured cross-sections with value of luminosity that has uncertainty of 1-2%
    • +
  • +
  • Not well covered in statistics books, but some theoretical results available
  • +
+
+
+
+

Estimators

+
    +
  • Estimator maps dataset to value \(\{x_i \} \to \hat p\)

    +
      +
    • Notation: \(\hat p\) is estimate of true value \(p\)
    • +
    • Estimator can have functional form or be an algorithm
    • +
    • Arithmetic mean \(\hat \mu = \frac 1N \sum_i x_i\)
    • +
    • Sample median: Sort \(x_i\), pick value in the middle
    • +
  • +
  • Many kinds of estimators, some better than others

  • +
  • Optimal estimators

    +
      +
    • As close to true value as possible
    • +
    • Easy to compute and/or easy to apply to any problem
    • +
    • Robustness against outliers
    • +
  • +
  • In HEP, we mainly use

    +
      +
    • Plug-in estimator
    • +
    • Maximum-likelihood estimator (MLE)
    • +
  • +
+
+
+

Bias and variance

+
    +
  • In one experiment, we obtain sample \(\{ x_i \}\) with size \(N\) and compute estimate \(\hat a\)
  • +
  • Sample \(\{ x_i \}\) is random, estimate \(\hat a\) also random
  • +
  • If experiment is identically repeated \(K\) times, we get sample \(\{ x_i \}_k\) and estimate \(\hat a_k\) each time
  • +
  • Derive accuracy of estimator from estimates \(\{ \hat a_1, \hat a_2, \dots, \hat a_K \}\) +
      +
    • Bias of \(\hat a\) = \(\frac1K \sum_i (\hat a_i - a)\)
    • +
    • Variance of \(\hat a\) = \(\frac1{K-1}\sum_i (\hat a_i - \frac 1 K \sum_k \hat a_k)^2\)
    • +
  • +
  • Example for a biased estimator: \(\hat V = \frac1N\sum_i (x_i - \hat\mu)^2\) is biased, if \(\hat \mu\) is estimated from same sample
  • +
+
+
from matplotlib import pyplot as plt
+import numpy as np
+
+mu = 1
+sigma = 1
+
+N = 10
+K = 10000
+var_1, var_2 = [], []
+for k in range(K):
+    rng = np.random.default_rng(seed=k)
+    x_k = rng.normal(mu, sigma, size=N)
+    v1 = np.var(x_k)
+    v2 = np.mean((x_k - mu) ** 2)
+    var_1.append(v1)
+    var_2.append(v2)
+
+
+
fig, ax = plt.subplots(1, 2, sharex=True, sharey=True, figsize=(10, 4))
+ax[0].hist(var_1, bins=50, label="$\hat V_1$");
+ax[1].hist(var_2, bins=50, label="$\hat V_2$");
+for axi in ax:
+    axi.axvline(sigma ** 2, color="C3", label="true V")
+ax[0].axvline(np.mean(var_1), color="C1", ls="--", label="mean of $\hat V_1$")
+ax[1].axvline(np.mean(var_2), color="C1", ls="--", label="mean of $\hat V_2$")
+for axi in ax:
+    axi.legend()
+
+

+
+
+
    +
  • Well-known bias correction for this case: \[ +\hat V_{1,\text{corr}} = \frac N{N-1}\hat V_1 +\] For derivation see e.g. F. James book (references at end)
  • +
+

Exercise * Compute \(\hat V_{1, \text{corr}}\) and compare its mean with true \(V\)

+
+
# do exercise here
+
+
+
# solution
+
+var_1_corr = np.multiply(var_1, N / (N - 1))
+
+print(f"<hat V_1>            = {np.mean(var_1):.2f}")
+print(f"<hat V_1(corrected)> = {np.mean(var_1_corr):.2f}")
+print(f"truth                = {sigma**2:.2f}")
+
+
<hat V_1>            = 0.91
+<hat V_1(corrected)> = 1.01
+truth                = 1.00
+
+
+
+

Plug-in estimator

+
    +
  • Plug-in principle: replace true values in some formula with estimates, e.g.
  • +
+

\[ +c = g(a, b, \dots) \to \hat c = g(\hat a, \hat b, \dots) +\]

+
    +
  • Justification +
      +
    • For \(N\to \infty\), \(\Delta a = \hat a - a \to 0\) for any reasonable estimator
    • +
    • \(g(\hat a) = g(a + \Delta a) = g(a) + O(\Delta a) \to g(a)\) for smooth \(g\)
    • +
  • +
  • Example: \(\sqrt{N}\) uncertainty estimator for Poisson-distributed count \(N\) +
      +
    • \(P(N; \lambda) = e^{-\lambda} \lambda^N / N!\)
    • +
    • Variance of counts: \(V_N = \lambda\)
    • +
    • Estimator \(\hat \lambda = \frac 1 K \sum_k N_k\); for \(K = 1\), \(\hat \lambda = N\)
    • +
    • Plug-in principle: \(\hat V_N = \hat \lambda = N\)
    • +
    • Standard deviation: \(\hat \sigma_N = \sqrt{\hat V_N} = \sqrt{N}\)
    • +
  • +
  • Plug-in estimators can behave poorly in small samples
  • +
  • Poisson example: \(\lambda = 0.01\), sample \(N = 0 \to \hat \sigma_N = 0\)
  • +
+
+
+
+

Bootstrapping bias and variance of estimators

+
    +
  • Bootstrap method: generic way to compute bias and variance of any estimator
  • +
+
+

Parametric bootstrap

+
    +
  • Available: +
      +
    • Sample \(\{ x_i \}\) distributed along \(f(x; a)\), \(a\) is unknown
    • +
    • Estimator for \(\hat a\)
    • +
  • +
  • Want to know bias and variance of estimator
  • +
  • Simulate experiment \(K\) times +
      +
    • Draw samples \(\{ x_i \}_k\) of equal size from \(\hat f(x; \hat a)\) (plugin-estimate)
    • +
    • Compute \(\hat a_k\)
    • +
    • Bias \(\frac 1 K \sum_k (\hat a_k - \hat a)\)
    • +
    • Variance \(\frac 1 {K-1} \sum_k (\hat a_k - \frac 1K \sum_k \hat a_k)^2\)
    • +
  • +
+
+
from numba_stats import norm, truncexpon, poisson
+
+
+def make_data(s, mu, sigma, b, slope, seed):
+    sr = poisson.rvs(s, size=1, random_state=seed)[0]
+    br = poisson.rvs(b, size=1, random_state=seed)[0]
+    s = norm.rvs(mu, sigma, size=sr, random_state=seed)
+    b = truncexpon.rvs(0, 1, 0, slope, size=br, random_state=seed)
+    return np.append(s, b)
+
+truth = {
+    "s": 1000,
+    "mu": 0.5,
+    "sigma": 0.05,
+    "b": 500,
+    "slope": 1,
+}
+
+
+
x = make_data(**truth, seed=0)
+
+w, xe = plt.hist(x, bins=100)[:2];
+
+

+
+
+
+
from iminuit import Minuit
+from iminuit.cost import ExtendedBinnedNLL
+from numba_stats import norm, truncexpon
+
+def model(x, s, mu, sigma, b, slope):
+    return s * norm.cdf(x, mu, sigma) + b * truncexpon.cdf(x, 0, 1, 0, slope)
+
+nll = ExtendedBinnedNLL(w, xe, model)
+m = Minuit(nll, **truth)
+m.limits["mu"] = (0, 1)
+m.limits["s", "b", "sigma", "slope"] = (0, None)
+
+
+
m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 106.9 (χ²/ndof = 1.1)Nfcn = 101
EDM = 9.95e-07 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0s1000350
1mu0.49830.001801
2sigma0.04820.00140
3b510270
4slope0.950.150
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
smusigmabslope
s1.2e+03-48.4e-68.9237e-3 (0.177)-0.2e3 (-0.213)-0.368 (-0.073)
mu-48.4e-63.09e-06-0 (-0.005)61.5e-6 (0.001)-17.0e-6 (-0.066)
sigma8.9237e-3 (0.177)-0 (-0.005)2.12e-06-8.9641e-3 (-0.231)-16.5e-6 (-0.078)
b-0.2e3 (-0.213)61.5e-6 (0.001)-8.9641e-3 (-0.231)7110.354 (0.091)
slope-0.368 (-0.073)-17.0e-6 (-0.066)-16.5e-6 (-0.078)0.354 (0.091)0.0213
+ + + + + + + + 2024-02-12T18:09:38.188990 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
    +
  • Fit can be written as estimator
  • +
+
+
def estimator(x):
+    w = np.histogram(x, bins=xe)[0]
+    nll = ExtendedBinnedNLL(w, xe, model)
+    m = Minuit(nll, **truth)
+    m.limits["mu"] = (0, 1)
+    m.limits["s", "b", "sigma", "slope"] = (0, None)
+    m.migrad()
+    assert m.valid
+    return m.values
+
+
+
replicates = []
+for seed in range(100):
+    x_k = make_data(*m.values, seed=seed+1)
+    p_k = estimator(x_k)
+    replicates.append(p_k)
+
+replicates = np.transpose(replicates)
+
+

Exercise

+
    +
  • Compute bias and standard deviation of replicates for each parameter
  • +
  • Compare standard deviation of replicates with Minuit’s uncertainty estimate, accessible via m.errors
  • +
+
+
# do exercise here
+
+
+
# solution
+
+for i, name in enumerate(m.parameters):
+    p = m.values[i]
+    r = replicates[i]
+    bias = np.mean(r - p)
+    std = np.std(r, ddof=1)
+    print(f"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g} "
+          f"bias/𝜎 = {bias / std:5.2f}")
+
+
s        𝜎(Minuit) =     35 𝜎(bootstrap) =     34 bias/𝜎 = -0.06
+mu       𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017 bias/𝜎 = -0.07
+sigma    𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0016 bias/𝜎 = -0.03
+b        𝜎(Minuit) =     27 𝜎(bootstrap) =     24 bias/𝜎 = -0.12
+slope    𝜎(Minuit) =   0.15 𝜎(bootstrap) =   0.16 bias/𝜎 = -0.09
+
+
+
+
+
+

Nonparametric bootstrap

+
    +
  • Can do the same by sampling directly from \(\{ x_i \}\)
  • +
  • Standard method for samples of fixed size \(N\): +
      +
    • Pick \(N\) random samples from \(\{ x_i \}\) with replacement to obtain \(\{ x_i \}_k\)
    • +
    • Compute \(\hat a_k\) from \(\{ x_i \}_k\) and proceed as before
    • +
  • +
  • “Extended method” for samples of variable size (Poisson): +
      +
    • Randomly draw \(M\) from Poisson distribution with \(\lambda = N\)
    • +
    • Pick \(M\) random samples from \(\{ x_i \}\) with replacement to obtain \(\{ x_i \}_k\)
    • +
    • +
  • +
  • Convenient: use same algorithms for any distribution
  • +
  • Efficient implementations for both algorithms in resample
  • +
+

+
+
import resample
+
+resample.__version__
+
+
'1.7.1'
+
+
+
+
from resample.bootstrap import resample
+
+# generate replicated samples of varying size (Poisson distributed)
+for x_k in resample(x, size=100, method="extended", random_state=1):
+    print(x_k)
+    break
+
+
[0.58820262 0.58820262 0.52000786 ... 0.38532216 0.26892968 0.26892968]
+
+
+

Exercise

+
    +
  • Again compute replicated fit results using resampe and store results in a list
  • +
  • Copy previous code where possible
  • +
  • Again compute bias and standard deviation of replicates and compare with Minuit’s uncertainty estimate
  • +
+
+
# do exercise here
+
+
+
# solution
+
+replicates = []
+for x_k in resample(x, size=100, method="extended", random_state=1):
+    replicates.append(estimator(x_k))
+
+replicates = np.transpose(replicates)
+
+for i, name in enumerate(m.parameters):
+    p = m.values[i]
+    r = replicates[i]
+    bias = np.mean(r - p)
+    std = np.std(r, ddof=1)
+    print(f"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g}"
+          f" bias/𝜎 = {bias / std:5.2f}")
+
+
s        𝜎(Minuit) =     35 𝜎(bootstrap) =     33 bias/𝜎 = -0.18
+mu       𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017 bias/𝜎 =  0.10
+sigma    𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0014 bias/𝜎 = -0.05
+b        𝜎(Minuit) =     27 𝜎(bootstrap) =     27 bias/𝜎 = -0.04
+slope    𝜎(Minuit) =   0.15 𝜎(bootstrap) =   0.17 bias/𝜎 =  0.18
+
+
+
    +
  • resample provides shortcut for computing variance
  • +
+
+
from resample.bootstrap import variance
+
+var = variance(estimator, x, method="extended", size=100, random_state=1)
+
+for i, name in enumerate(m.parameters):
+    std = var[i] ** 0.5
+    print(f"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g}")
+
+
s        𝜎(Minuit) =     35 𝜎(bootstrap) =     33
+mu       𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017
+sigma    𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0014
+b        𝜎(Minuit) =     27 𝜎(bootstrap) =     27
+slope    𝜎(Minuit) =   0.15 𝜎(bootstrap) =   0.17
+
+
+

Exercise

+
    +
  • Repeat the variance calculation without using method="extended"
  • +
  • Compare the uncertainties of \(s\) and \(b\)
  • +
+
+
# do exercise here
+
+
+
# solution
+
+var2 = variance(estimator, x, size=100, random_state=1)
+
+# uncertainties of s and b are too small, if normal bootstrap is used,
+# since size variation is not simulated
+for i, name in enumerate(m.parameters):
+    std = var[i] ** 0.5
+    std2 = var2[i] ** 0.5
+    print(f"{name:8} 𝜎(extended bootstrap) = {std:6.2g} "
+          f"𝜎(normal bootstrap) = {std2:6.2g}")
+
+
s        𝜎(extended bootstrap) =     33 𝜎(normal bootstrap) =     24
+mu       𝜎(extended bootstrap) = 0.0017 𝜎(normal bootstrap) = 0.0018
+sigma    𝜎(extended bootstrap) = 0.0014 𝜎(normal bootstrap) = 0.0016
+b        𝜎(extended bootstrap) =     27 𝜎(normal bootstrap) =     24
+slope    𝜎(extended bootstrap) =   0.17 𝜎(normal bootstrap) =   0.17
+
+
+
    +
  • Bootstrap can be applied to most statistical problems in HEP
  • +
  • Restriction: samples must be identically and independently distributed
  • +
  • May work well even with small samples (< 10)
  • +
+
+
+

Error propagation

+
    +
  • Full analysis typically combines many intermediate results with uncertainties into final result
  • +
  • Consider \(\vec b = g(\vec a)\) with uncertainties in \(\vec a\)
  • +
  • Two ways of computing uncertainties of \(\vec b\) +
      +
    • Monte-Carlo simulation
    • +
    • Error propagation
    • +
  • +
  • Can be equally used for statistical and systematic uncertainties
  • +
+
+

Monte-Carlo simulation

+
    +
  • Draw variations of fit result from multivariate normal distribution

  • +
  • Compute variable of interest from variations

  • +
  • Compute standard deviation or covariance of

  • +
  • Example: compute uncertainty of \(\hat N = \hat s + \hat b\)

  • +
+
+
rng = np.random.default_rng(1)
+a_k = rng.multivariate_normal(m.values, m.covariance, size=1000)
+
+def g(a):
+    return a[0] + a[3]
+
+b_k = np.apply_along_axis(g, -1, a_k)
+
+print(f"𝜎(N, simulation) = {np.std(b_k):.3g}")
+print(f"𝜎(N, naive)      = {(m.errors[0]**2 + m.errors[3]**2)**0.5:.3g}")
+print(f"𝜎(N, expected)   = {np.sqrt(len(x)):.3g}")
+
+
𝜎(N, simulation) = 37.7
+𝜎(N, naive)      = 43.7
+𝜎(N, expected)   = 38.8
+
+
+
    +
  • \(\sigma_{N,\text{simulation}}^2 < \sigma_s^2 + \sigma_b^2\), because \(s_k\) and \(b_k\) are anti-correlated
  • +
+
+
+

Error propagation

+
    +
  • Given: \(\vec a\) with covariance matrix \(C_a\) and differentiable function \(\vec b = g(\vec a)\)

  • +
  • \(\vec b\) can have different dimensionality from \(\vec a\)

  • +
  • One can derive via first-order Taylor expansion: \[ +C_b = J \, C_a \, J^T \text{ with } J_{ik} = \frac{\partial g_i}{\partial a_k} +\]

  • +
  • Useful special cases

    +
      +
    • Sum of scaled independent random values \(c = \alpha a + \beta b\) (\(\alpha, \beta\) are constant) \[ + \sigma^2_c = \alpha^2 \sigma^2_a + \beta^2 \sigma^2_b + \]
    • +
    • Product of powers of independent random values \(c = a^\alpha \, b^\beta\) \[ + \left(\frac{\sigma_c}{c}\right)^2 = \alpha^2 \left(\frac{\sigma_a}{a}\right)^2 + \beta^2 \left(\frac{\sigma_b}{b}\right)^2 + \]
    • +
  • +
  • Final uncertainty often dominated by one term

  • +
  • When estimating systematic uncertainties: put your effort in estimating largest term and be lax on others

  • +
  • Computation of \(J\) by hand error-prone and tedious

  • +
  • jacobi library offers automatic error propagation based on a robust numerical derivative

  • +
+

+
+
import jacobi
+
+jacobi.__version__
+
+
'0.9.2'
+
+
+
+
b, cov_b = jacobi.propagate(g, m.values, m.covariance)
+
+f"𝜎(N, error propagation) = {cov_b ** 0.5:.3g} 𝜎(N, expected) = {np.sqrt(len(x)):.3g}"
+
+
'𝜎(N, error propagation) = 38.9 𝜎(N, expected) = 38.8'
+
+
+
    +
  • jacobi.propagate works for any differentiable mapping, which even includes most fits!
  • +
+
+
from iminuit.cost import LeastSquares
+
+def h(pvec):
+    rng = np.random.default_rng(1)
+    a, b = pvec
+    x = np.array([1, 2, 3])
+    y = rng.normal(a + b * x, 0.1)
+    m = Minuit(LeastSquares(x, y, 0.1, lambda x, a, b: a + b * x), a=0, b=0)
+    m.strategy = 0
+    m.migrad()
+    assert m.valid
+    return m.values
+
+jacobi.propagate(h, (1, 2), (1, 1))
+
+
(array([1.05143603, 1.99924264]),
+ array([[1.00000000e+00, 1.44351198e-12],
+        [1.44351198e-12, 1.00000000e+00]]))
+
+
+
+
+
+

Systematic uncertainties: Known unknowns

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“There are known knowns, things we know that we know;
and there are known unknowns, things that we know we don’t know.
But there are also unknown unknowns, things we do not know we don’t know.”
Donald Rumsfeld

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  • Why data analyses take so long: study data, detector, and methods to turn unknown unknowns into known knowns or known unknowns
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  • known unknowns = systematic uncertainties
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  • No formal way, but generally…
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Worry * What could go wrong? * What are my assumptions?

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Example * Efficiency or correction factor taken from simulation, but simulation differs from real experiment * Solution: perform control measurement to estimate how much simulation differs from real experiment

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Methods * Use methods that are known to be optimal for your problem or derive them from first principles * General purpose methods often safer to use than special-case methods * Maximum-likelihood fit > least-squares fit * Bootstrap estimate of uncertainty > other uncertainty estimates

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Apply checks * Validate whole analysis on simulated data (sometimes you can use toy simulation) * Split data by magnet polarity, fill, etc., analyze splits separately, check for agreement

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Guidelines for handling systematic errors * Roger Barlow: Systematic Errors: Facts and Fictions (2002) * Roger Barlow, “Practical Statistics for Particle Physics” (2019), Section 6.5

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Further resources for learning

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LHCb Statistics and ML WG

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Introductory statistics books

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  • List of books on TWiki pages of Statistis and Machine Learning WG

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  • Books I personally like a lot

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    • Glen Cowan, “Statistical Data Analysis”, Oxford University Press, 1998
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    • F. James, “Statistical Methods in Experimental Physics”, 2nd edition, World Scientific, 2006
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    • R. Barlow, “Statistics: a Guide to the Use of Statistical Methods in the Physical Sciences”, Wiley, 1989
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    • B. Efron, R.J. Tibshirani, “An Introduction to the Bootstrap”, CRC Press, 1994
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Lectures, papers, notebooks

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b/2024_LHCb_Starterkit/roofit_vs_iminuit.svg @@ -0,0 +1,2586 @@ + + + + + + + + 2023-12-21T11:57:41.899742 + image/svg+xml + + + Matplotlib v3.7.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/about.html b/about.html new file mode 100644 index 0000000..6e993fa --- /dev/null +++ b/about.html @@ -0,0 +1,357 @@ + + + + + + + + + +Hans Dembinski’s blog - About + + + + + + + + + + + + + + + + + + + + + + + + + + +
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About

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I have curious mind and a broad range of interests, professionally they circle around data analysis, statistics, software development, machine learning, and artificial intelligence. In this blog, I write about these topics.

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I have a PhD in astroparticle physics. If you want to know more about my professional career, read more about me on LinkedIn. I wrote and maintain a couple of data science libraries. Check out my Github page if you want to know more. I am a member of the Boost C++ Project, to which I contributed the Histogram library. My C++ skill is in the grandmaster area, but recently I only program in Python, since it is so much more enjoyable and with Numba one can write numerical code in Python with performance on par with C++ in a fraction of the time.

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+ + + + \ No newline at end of file diff --git a/index.html b/index.html new file mode 100644 index 0000000..d9c6cca --- /dev/null +++ b/index.html @@ -0,0 +1,1354 @@ + + + + + + + + + +Hans Dembinski’s blog + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + \ No newline at end of file diff --git a/listings.json b/listings.json new file mode 100644 index 0000000..3dd0ae8 --- /dev/null +++ b/listings.json @@ -0,0 +1,52 @@ +[ + { + "listing": "/index.html", + "items": [ + "/posts/Wilson Score Interval with Weighted Histograms.html", + "/posts/exceptions.html", + "/posts/Sleep mode power consumption.html", + "/posts/Simple parallelization in Jupyter Notebooks.html", + "/posts/parse_latex.html", + "/posts/look_elsewhere_effect.html", + "/posts/cows.html", + "/posts/RooFit.html", + "/posts/Interactive plotting in Jupyter with matplotlib.html", + "/posts/hyperon_feed_down.html", + "/posts/Cross-section.html", + "/posts/benchmark of building an array with numba.html", + "/posts/comparison_chisquare_test_statistics.html", + "/posts/regression.html", + "/posts/template_with_distortion.html", + "/posts/From fixed target to sqrt_s_nn and back.html", + "/posts/invariant mass combinatorial background.html", + "/posts/error_propagation_with_sympy.html", + "/posts/MachineLearning.html", + "/posts/render_latex_to_svg.html", + "/posts/tracking_efficiency.html", + "/posts/naive_calibration_bias.html", + "/posts/duffing.html", + "/posts/visual_cross_section.html", + "/posts/Leave-one-out cross-validation.html", + "/posts/Fit weighted histograms with SPD method.html", + "/posts/plots_with_hyperlinks.html", + "/posts/ratio bias.html", + "/posts/MCMC.html", + "/posts/New iminuit displays.html", + "/posts/struct_of_arrays_vs_arrays_of_structs.html", + "/posts/UncertaintiesLecture.html", + "/posts/p-value conversion.html", + "/posts/Numerically stable calculation of invariant mass.html", + "/posts/all_KK_decays.html", + "/posts/SPD with negative weights.html", + "/posts/HESSE_vs_MINOS_TwoGauss.html", + "/posts/deep_sets/fast_deep_set.html", + "/posts/UnbiasedEML.html", + "/posts/Correlation.html", + "/posts/resample and numba.html", + "/posts/logaddexp.html", + "/posts/chance_of_deviations_in_random_splits.html", + "/posts/Uncertainty of efficiency computed from fitted yields.html", + "/posts/error_propagation_ratio.html" + ] + } +] \ No newline at end of file diff --git a/posts/Correlation.html b/posts/Correlation.html new file mode 100644 index 0000000..8f7cc29 --- /dev/null +++ b/posts/Correlation.html @@ -0,0 +1,840 @@ + + + + + + + + + +Hans Dembinski’s blog - Particle correlations in proton-proton collisions + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+

Particle correlations in proton-proton collisions

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Here, I run simulations of proton-proton collisions with Chromo to investigate two-particle correlations in pseudorapidity space. With Chromo, it is very easy to study the output of different event generators.

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I generate collisions at 13 TeV center-of-mass energy. For each event, I form all unique pairs of final-state charged particles, these are effectively long-lived prompt charged particles. Prompt here means that they either have no ancestors or that their ancestors are short-lived. I then fill the pseudorapidity pairs into a two-dimensional histogram.

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We can see interesting patterns in these plots when different generators are compared. SIBYLL-2.3d shows a stronger concentration around mid-rapidity and has more pairs which are close in rapidity than the other generators.

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+
import impy.models as im
+from impy.kinematics import CenterOfMass
+from impy.constants import TeV
+import boost_histogram as bh
+from matplotlib import pyplot as plt
+import numba as nb
+import numpy as np
+
+
+
ekin = CenterOfMass(13 * TeV, "proton", "proton")
+
+@nb.njit
+def pairs(eta, seed):
+    r1 = []
+    r2 = []
+    np.random.seed(seed)
+    for i, etai in enumerate(eta):
+        for j in range(i):
+            etaj = eta[j]
+            # avoid structures that arise from implementation-specific
+            # ordering of particles in HEPEVT record
+            if np.random.uniform(0, 1) < 0.5:
+                r1.append(etai)
+                r2.append(etaj)
+            else:
+                r1.append(etaj)
+                r2.append(etai)
+    return np.array(r1), np.array(r2)
+
+nevent = 1000
+
+histograms = {}
+for Model in (im.Sibyll21, im.Sibyll23d, im.QGSJetII04, im.EposLHC, im.Pythia6):
+    h = bh.Histogram(bh.axis.Regular(300, -15, 15), bh.axis.Regular(300, -15, 15))
+    for i, ev in enumerate(Model(ekin)(nevent)):
+        fsc = ev.final_state_charged()
+        with np.errstate(invalid="ignore", divide="ignore"):
+            eta1, eta2 = pairs(fsc.eta, i)
+        h.fill(eta1, eta2)
+    histograms[Model] = h
+
+
 ====================================================
+ |                                                  |
+ |                 S I B Y L L  2.1                 |
+ |                                                  |
+ |         HADRONIC INTERACTION MONTE CARLO         |
+ |                        BY                        |
+ |                   Ralph ENGEL                    |
+ |           R.S. FLETCHER, T.K. GAISSER            |
+ |               P. LIPARI, T. STANEV               |
+ |                                                  |
+ | Publication to be cited when using this program: |
+ | R. Engel et al., Proc. 26th ICRC, 1 (1999) 415   |
+ |                                                  |
+ | last modified:  28. Sept. 2001 by R. Engel       |
+ ====================================================
+
+
+ Table: J, sqs,  PT_cut,  SIG_tot,  SIG_inel,  B_el,  rho,  <n_s>,  <n_h>
+ ------------------------------------------------------------------------
+    1   1.000E+01    1.45   38.33   30.88   10.83  -0.185   1.964   0.003
+    1   1.259E+01    1.49   38.27   31.16   11.10  -0.127   1.949   0.006
+    1   1.585E+01    1.54   38.44   31.54   11.36  -0.078   1.934   0.012
+    1   1.995E+01    1.59   38.84   32.05   11.63  -0.037   1.918   0.019
+    1   2.512E+01    1.64   39.46   32.67   11.89  -0.004   1.901   0.029
+    1   3.162E+01    1.69   40.29   33.41   12.16   0.022   1.884   0.043
+    1   3.981E+01    1.75   41.28   34.24   12.42   0.044   1.864   0.060
+    1   5.012E+01    1.81   42.39   35.13   12.69   0.061   1.844   0.082
+    1   6.310E+01    1.87   43.59   36.05   12.95   0.075   1.821   0.109
+    1   7.943E+01    1.93   44.92   37.06   13.22   0.085   1.797   0.141
+    1   1.000E+02    2.00   46.35   38.14   13.48   0.094   1.771   0.180
+    1   1.259E+02    2.07   47.84   39.28   13.73   0.101   1.742   0.226
+    1   1.585E+02    2.14   49.39   40.48   13.98   0.106   1.711   0.279
+    1   1.995E+02    2.22   51.05   41.74   14.24   0.110   1.677   0.342
+    1   2.512E+02    2.30   52.84   43.09   14.50   0.113   1.641   0.413
+    1   3.162E+02    2.38   54.79   44.56   14.77   0.116   1.603   0.495
+    1   3.981E+02    2.46   56.96   46.16   15.05   0.118   1.563   0.587
+    1   5.012E+02    2.55   59.39   47.94   15.33   0.119   1.522   0.691
+    1   6.310E+02    2.65   62.15   49.92   15.63   0.120   1.479   0.808
+    1   7.943E+02    2.74   65.29   52.14   15.95   0.121   1.435   0.938
+    1   1.000E+03    2.84   68.88   54.63   16.28   0.122   1.391   1.081
+    1   1.259E+03    2.95   72.53   57.06   16.61   0.123   1.347   1.240
+    1   1.585E+03    3.06   76.46   59.64   16.96   0.123   1.303   1.416
+    1   1.995E+03    3.17   80.67   62.36   17.32   0.123   1.259   1.609
+    1   2.512E+03    3.29   85.17   65.23   17.70   0.124   1.216   1.822
+    1   3.162E+03    3.41   89.95   68.24   18.10   0.124   1.175   2.055
+    1   3.981E+03    3.54   95.02   71.38   18.51   0.124   1.135   2.312
+    1   5.012E+03    3.67  100.35   74.64   18.95   0.124   1.096   2.594
+    1   6.310E+03    3.81  105.95   78.03   19.40   0.124   1.059   2.903
+    1   7.943E+03    3.95  111.81   81.53   19.88   0.124   1.024   3.244
+    1   1.000E+04    4.09  117.90   85.14   20.38   0.125   0.990   3.618
+    1   1.259E+04    4.25  124.23   88.85   20.90   0.125   0.959   4.030
+    1   1.585E+04    4.40  130.77   92.65   21.44   0.125   0.928   4.481
+    1   1.995E+04    4.57  137.51   96.55   22.01   0.125   0.898   4.957
+    1   2.512E+04    4.74  144.44  100.54   22.60   0.125   0.865   5.394
+    1   3.162E+04    4.91  151.56  104.61   23.21   0.125   0.825   5.707
+    1   3.981E+04    5.09  158.85  108.75   23.85   0.125   0.779   5.909
+    1   5.012E+04    5.28  166.31  112.98   24.50   0.125   0.733   6.076
+    1   6.310E+04    5.47  173.92  117.27   25.17   0.125   0.690   6.243
+    1   7.943E+04    5.67  181.69  121.64   25.87   0.125   0.649   6.417
+    1   1.000E+05    5.88  189.60  126.07   26.58   0.125   0.611   6.601
+    1   1.259E+05    6.10  197.66  130.57   27.31   0.125   0.576   6.799
+    1   1.585E+05    6.32  205.84  135.12   28.06   0.125   0.543   7.013
+    1   1.995E+05    6.55  214.16  139.74   28.82   0.125   0.512   7.247
+    1   2.512E+05    6.78  222.60  144.41   29.60   0.125   0.484   7.501
+    1   3.162E+05    7.03  231.15  149.14   30.39   0.125   0.457   7.763
+    1   3.981E+05    7.28  239.82  153.93   31.21   0.125   0.432   7.999
+    1   5.012E+05    7.54  248.60  158.77   32.03   0.125   0.408   8.168
+    1   6.310E+05    7.81  257.49  163.65   32.87   0.125   0.385   8.278
+    1   7.943E+05    8.08  266.48  168.59   33.72   0.125   0.364   8.364
+    1   1.000E+06    8.37  275.56  173.57   34.59   0.125   0.343   8.442
+    1   1.259E+06    8.67  284.75  178.60   35.47   0.125   0.324   8.517
+    1   1.585E+06    8.97  294.03  183.68   36.36   0.125   0.307   8.590
+    1   1.995E+06    9.28  303.40  188.80   37.27   0.125   0.290   8.663
+    1   2.512E+06    9.61  312.87  193.97   38.18   0.125   0.274   8.735
+    1   3.162E+06    9.94  322.42  199.18   39.11   0.125   0.259   8.808
+    1   3.981E+06   10.29  332.06  204.43   40.05   0.125   0.246   8.882
+    1   5.012E+06   10.64  341.79  209.72   41.00   0.125   0.233   8.957
+    1   6.310E+06   11.00  351.59  215.05   41.95   0.125   0.220   9.032
+    1   7.943E+06   11.38  361.49  220.42   42.92   0.125   0.209   9.101
+    1   1.000E+07   11.77  371.46  225.83   43.90   0.125   0.198   9.158
+
+ Table: J, sqs,  PT_cut,  SIG_tot,  SIG_inel,  B_el,  rho,  <n_s>,  <n_h>
+ ------------------------------------------------------------------------
+    2   1.000E+01    1.45   23.12   19.94   10.12  -0.067   1.845   0.009
+    2   1.259E+01    1.49   23.22   20.21   10.27  -0.031   1.827   0.016
+    2   1.585E+01    1.54   23.49   20.57   10.42  -0.001   1.807   0.026
+    2   1.995E+01    1.59   23.87   21.00   10.56   0.024   1.783   0.040
+    2   2.512E+01    1.64   24.33   21.49   10.70   0.045   1.756   0.058
+    2   3.162E+01    1.69   24.89   21.95   10.84   0.061   1.727   0.081
+    2   3.981E+01    1.75   25.53   22.48   10.97   0.074   1.695   0.110
+    2   5.012E+01    1.81   26.25   23.06   11.11   0.085   1.659   0.144
+    2   6.310E+01    1.87   27.04   23.69   11.24   0.093   1.621   0.185
+    2   7.943E+01    1.93   27.89   24.36   11.37   0.100   1.580   0.233
+    2   1.000E+02    2.00   28.80   25.08   11.50   0.105   1.536   0.288
+    2   1.259E+02    2.07   29.72   25.72   11.44   0.109   1.490   0.351
+    2   1.585E+02    2.14   30.77   26.45   11.39   0.113   1.442   0.423
+    2   1.995E+02    2.22   31.96   27.30   11.38   0.115   1.392   0.503
+    2   2.512E+02    2.30   33.35   28.30   11.40   0.117   1.340   0.592
+    2   3.162E+02    2.38   34.99   29.47   11.45   0.119   1.288   0.690
+    2   3.981E+02    2.46   36.93   30.84   11.55   0.120   1.236   0.798
+    2   5.012E+02    2.55   39.22   32.47   11.69   0.121   1.184   0.916
+    2   6.310E+02    2.65   41.93   34.36   11.88   0.122   1.132   1.044
+    2   7.943E+02    2.74   45.10   36.57   12.12   0.123   1.082   1.183
+    2   1.000E+03    2.84   48.81   39.13   12.41   0.123   1.033   1.333
+    2   1.259E+03    2.95   52.13   41.43   12.78   0.123   0.987   1.497
+    2   1.585E+03    3.06   55.69   43.86   13.16   0.124   0.942   1.673
+    2   1.995E+03    3.17   59.50   46.43   13.56   0.124   0.899   1.865
+    2   2.512E+03    3.29   63.55   49.12   13.97   0.124   0.859   2.072
+    2   3.162E+03    3.41   67.83   51.93   14.39   0.124   0.821   2.298
+    2   3.981E+03    3.54   72.35   54.85   14.83   0.124   0.785   2.543
+    2   5.012E+03    3.67   77.09   57.87   15.28   0.124   0.752   2.809
+    2   6.310E+03    3.81   82.06   61.01   15.75   0.124   0.720   3.099
+    2   7.943E+03    3.95   87.24   64.23   16.24   0.125   0.691   3.416
+    2   1.000E+04    4.09   92.61   67.55   16.74   0.125   0.664   3.761
+    2   1.259E+04    4.25   98.18   70.96   17.26   0.125   0.638   4.138
+    2   1.585E+04    4.40  103.94   74.44   17.79   0.125   0.614   4.543
+    2   1.995E+04    4.57  109.87   78.01   18.34   0.125   0.591   4.953
+    2   2.512E+04    4.74  115.96   81.65   18.91   0.125   0.565   5.303
+    2   3.162E+04    4.91  122.21   85.36   19.50   0.125   0.535   5.552
+    2   3.981E+04    5.09  128.62   89.15   20.10   0.125   0.504   5.737
+    2   5.012E+04    5.28  135.18   92.99   20.72   0.125   0.474   5.904
+    2   6.310E+04    5.47  141.88   96.91   21.35   0.125   0.446   6.067
+    2   7.943E+04    5.67  148.71  100.88   22.00   0.125   0.419   6.233
+    2   1.000E+05    5.88  155.68  104.91   22.66   0.125   0.394   6.404
+    2   1.259E+05    6.10  162.78  109.01   23.34   0.125   0.371   6.584
+    2   1.585E+05    6.32  170.01  113.15   24.03   0.125   0.349   6.776
+    2   1.995E+05    6.55  177.35  117.36   24.73   0.125   0.329   6.981
+    2   2.512E+05    6.78  184.81  121.61   25.45   0.125   0.311   7.200
+    2   3.162E+05    7.03  192.39  125.92   26.18   0.125   0.293   7.426
+    2   3.981E+05    7.28  200.08  130.28   26.93   0.125   0.277   7.632
+    2   5.012E+05    7.54  207.88  134.69   27.68   0.125   0.261   7.792
+    2   6.310E+05    7.81  215.78  139.15   28.45   0.125   0.246   7.909
+    2   7.943E+05    8.08  223.78  143.66   29.23   0.125   0.232   8.004
+    2   1.000E+06    8.37  231.89  148.21   30.02   0.125   0.219   8.090
+    2   1.259E+06    8.67  240.09  152.81   30.83   0.125   0.207   8.172
+    2   1.585E+06    8.97  248.39  157.46   31.64   0.125   0.196   8.249
+    2   1.995E+06    9.28  256.79  162.15   32.47   0.125   0.185   8.324
+    2   2.512E+06    9.61  265.28  166.88   33.30   0.125   0.175   8.398
+    2   3.162E+06    9.94  273.86  171.66   34.15   0.125   0.165   8.470
+    2   3.981E+06   10.29  282.53  176.48   35.01   0.125   0.156   8.542
+    2   5.012E+06   10.64  291.29  181.34   35.88   0.125   0.148   8.615
+    2   6.310E+06   11.00  300.14  186.25   36.76   0.125   0.140   8.687
+    2   7.943E+06   11.38  309.08  191.19   37.64   0.125   0.133   8.756
+    2   1.000E+07   11.77  318.10  196.18   38.54   0.125   0.126   8.819
+
+ Table: J, sqs,  PT_cut,  SIG_tot,  SIG_inel,  B_el,  rho,  <n_s>,  <n_h>
+ ------------------------------------------------------------------------
+    3   1.000E+01    1.45   18.30   16.13    8.84  -0.025   1.845   0.009
+    3   1.259E+01    1.49   18.83   16.69    9.14   0.009   1.827   0.016
+    3   1.585E+01    1.54   19.41   17.26    9.40   0.036   1.807   0.026
+    3   1.995E+01    1.59   20.02   17.83    9.64   0.056   1.783   0.040
+    3   2.512E+01    1.64   20.63   18.41    9.85   0.072   1.756   0.058
+    3   3.162E+01    1.69   21.32   18.99   10.05   0.084   1.727   0.081
+    3   3.981E+01    1.75   22.04   19.60   10.23   0.093   1.695   0.110
+    3   5.012E+01    1.81   22.81   20.23   10.40   0.101   1.659   0.144
+    3   6.310E+01    1.87   23.61   20.89   10.56   0.106   1.621   0.185
+    3   7.943E+01    1.93   24.46   21.57   10.72   0.111   1.580   0.233
+    3   1.000E+02    2.00   25.34   22.28   10.87   0.114   1.536   0.288
+    3   1.259E+02    2.07   26.25   23.01   11.06   0.116   1.490   0.351
+    3   1.585E+02    2.14   27.21   23.75   11.21   0.118   1.442   0.423
+    3   1.995E+02    2.22   28.23   24.53   11.32   0.120   1.392   0.503
+    3   2.512E+02    2.30   29.37   25.37   11.42   0.121   1.340   0.592
+    3   3.162E+02    2.38   30.68   26.34   11.51   0.122   1.288   0.690
+    3   3.981E+02    2.46   32.25   27.48   11.61   0.123   1.236   0.798
+    3   5.012E+02    2.55   34.17   28.86   11.73   0.123   1.184   0.916
+    3   6.310E+02    2.65   36.58   30.57   11.90   0.123   1.132   1.044
+    3   7.943E+02    2.74   39.60   32.72   12.12   0.124   1.082   1.183
+    3   1.000E+03    2.84   43.42   35.40   12.41   0.124   1.033   1.333
+    3   1.259E+03    2.95   46.38   37.52   12.78   0.124   0.987   1.497
+    3   1.585E+03    3.06   49.56   39.77   13.16   0.124   0.942   1.673
+    3   1.995E+03    3.17   52.95   42.14   13.56   0.124   0.899   1.865
+    3   2.512E+03    3.29   56.56   44.63   13.97   0.124   0.859   2.072
+    3   3.162E+03    3.41   60.38   47.23   14.39   0.125   0.821   2.298
+    3   3.981E+03    3.54   64.41   49.94   14.83   0.125   0.785   2.543
+    3   5.012E+03    3.67   68.64   52.75   15.28   0.125   0.752   2.809
+    3   6.310E+03    3.81   73.06   55.67   15.75   0.125   0.720   3.099
+    3   7.943E+03    3.95   77.67   58.67   16.24   0.125   0.691   3.416
+    3   1.000E+04    4.09   82.46   61.77   16.74   0.125   0.664   3.761
+    3   1.259E+04    4.25   87.43   64.95   17.26   0.125   0.638   4.138
+    3   1.585E+04    4.40   92.55   68.21   17.79   0.125   0.614   4.543
+    3   1.995E+04    4.57   97.83   71.55   18.34   0.125   0.591   4.953
+    3   2.512E+04    4.74  103.26   74.97   18.91   0.125   0.565   5.303
+    3   3.162E+04    4.91  108.83   78.45   19.50   0.125   0.535   5.552
+    3   3.981E+04    5.09  114.54   82.01   20.10   0.125   0.504   5.737
+    3   5.012E+04    5.28  120.37   85.63   20.72   0.125   0.474   5.904
+    3   6.310E+04    5.47  126.34   89.31   21.35   0.125   0.446   6.067
+    3   7.943E+04    5.67  132.43   93.05   22.00   0.125   0.419   6.233
+    3   1.000E+05    5.88  138.64   96.85   22.66   0.125   0.394   6.404
+    3   1.259E+05    6.10  144.96  100.72   23.34   0.125   0.371   6.584
+    3   1.585E+05    6.32  151.39  104.63   24.03   0.125   0.349   6.776
+    3   1.995E+05    6.55  157.93  108.61   24.73   0.125   0.329   6.981
+    3   2.512E+05    6.78  164.58  112.63   25.45   0.125   0.311   7.200
+    3   3.162E+05    7.03  171.33  116.71   26.18   0.125   0.293   7.426
+    3   3.981E+05    7.28  178.17  120.84   26.93   0.125   0.277   7.632
+    3   5.012E+05    7.54  185.12  125.01   27.68   0.125   0.261   7.792
+    3   6.310E+05    7.81  192.15  129.24   28.45   0.125   0.246   7.909
+    3   7.943E+05    8.08  199.28  133.52   29.23   0.125   0.232   8.004
+    3   1.000E+06    8.37  206.50  137.84   30.02   0.125   0.219   8.090
+    3   1.259E+06    8.67  213.80  142.20   30.83   0.125   0.207   8.172
+    3   1.585E+06    8.97  221.19  146.62   31.64   0.125   0.196   8.249
+    3   1.995E+06    9.28  228.67  151.07   32.47   0.125   0.185   8.324
+    3   2.512E+06    9.61  236.23  155.58   33.30   0.125   0.175   8.398
+    3   3.162E+06    9.94  243.87  160.12   34.15   0.125   0.165   8.470
+    3   3.981E+06   10.29  251.60  164.71   35.01   0.125   0.156   8.542
+    3   5.012E+06   10.64  259.40  169.34   35.88   0.125   0.148   8.615
+    3   6.310E+06   11.00  267.28  174.01   36.76   0.125   0.140   8.687
+    3   7.943E+06   11.38  275.24  178.72   37.64   0.125   0.133   8.756
+    3   1.000E+07   11.77  283.28  183.48   38.54   0.125   0.126   8.819
+ ====================================================
+ |                                                  |
+ |                 S I B Y L L  2.3d                |
+ |                                                  |
+ |         HADRONIC INTERACTION MONTE CARLO         |
+ |                        BY                        |
+ |            Eun-Joo AHN, Felix RIEHN              |
+ |      R. ENGEL, A. FEDYNITCH, R.S. FLETCHER,      |
+ |       T.K. GAISSER, P. LIPARI, T. STANEV         |
+ |                                                  |
+ | Publication to be cited when using this program: |
+ | Eun-Joo AHN et al., Phys.Rev. D80 (2009) 094003  |
+ | F. RIEHN et al., hep-ph: 1912.03300              |
+ | last modifications: F. Riehn (05/20/2020)        |
+ ====================================================
+
+ SIG_AIR_INI: initializing target: (i,A)           1           0 air..
+ SIG_AIR_INI: initializing target: (i,A)           2          14 nit..
+ SIG_AIR_INI: initializing target: (i,A)           3          16 oxy..
+ ====================================================
+ |                                                  |
+ |         QUARK GLUON STRING JET -II MODEL         |
+ |                                                  |
+ |         HADRONIC INTERACTION MONTE CARLO         |
+ |                        BY                        |
+ |                 S. OSTAPCHENKO                   |
+ |                                                  |
+ |             e-mail: sergei@tf.phys.ntnu.no       |
+ |                                                  |
+ |                   Version II-04                  |
+ |                                                  |
+ | Publication to be cited when using this program: |
+ | S.Ostapchenko, PRD 83 (2011) 014018              |
+ |                                                  |
+ | last modification:  09.04.2013                   |
+ |                                                  |
+ | Any modification has to be approved by the author|
+ ====================================================
+
+ DATDIR/Users/hdembinski/Extern/impy/src/impy/iamdata/qgsjet/                                                                                                                                                                                                          
+  qgaini: cross sections readout from the file: qgsdat-II-04
+ done
+  qgaini: nuclear cross sections readout from the file sectnu-II-04
+impy.models.epos.EposLHC::init_generator(): First initialization
+ /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/          52
+initializations ...
+###################################################################
+#        EPOS LHC      K. WERNER, T. PIEROG                       #
+#                      Contact: tanguy.pierog@kit.edu             #
+###################################################################
+#        WARNING: This is a special retuned version !!!           #
+#     Do not publish results without contacting the authors.      #
+###################################################################
+read from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.iniev ...
+read from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.initl ...
+read from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.inirj.lhc ...
+read from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.inics.lhc ...
+ Compute Cross-section (can take a while...)
+seedj:   3862463     0.2000000000000000D+01
+ EPOS used with FUSION option
+1                                                                              
+ ******************************************************************************
+ ******************************************************************************
+ **                                                                          **
+ **                                                                          **
+ **              *......*                  Welcome to the Lund Monte Carlo!  **
+ **         *:::!!:::::::::::*                                               **
+ **      *::::::!!::::::::::::::*          PPP  Y   Y TTTTT H   H III   A    **
+ **    *::::::::!!::::::::::::::::*        P  P  Y Y    T   H   H  I   A A   **
+ **   *:::::::::!!:::::::::::::::::*       PPP    Y     T   HHHHH  I  AAAAA  **
+ **   *:::::::::!!:::::::::::::::::*       P      Y     T   H   H  I  A   A  **
+ **    *::::::::!!::::::::::::::::*!       P      Y     T   H   H III A   A  **
+ **      *::::::!!::::::::::::::* !!                                         **
+ **      !! *:::!!:::::::::::*    !!       This is PYTHIA version 6.428      **
+ **      !!     !* -><- *         !!       Last date of change:  5 Sep 2013  **
+ **      !!     !!                !!                                         **
+ **      !!     !!                !!       Now is  0 Jan 2000 at  0:00:00    **
+ **      !!                       !!                                         **
+ **      !!        lh             !!       Disclaimer: this program comes    **
+ **      !!                       !!       without any guarantees. Beware    **
+ **      !!                 hh    !!       of errors and use common sense    **
+ **      !!    ll                 !!       when interpreting results.        **
+ **      !!                       !!                                         **
+ **      !!                                Copyright T. Sjostrand (2011)     **
+ **                                                                          **
+ ** An archive of program versions and documentation is found on the web:    **
+ ** http://www.thep.lu.se/~torbjorn/Pythia.html                              **
+ **                                                                          **
+ ** When you cite this program, the official reference is to the 6.4 manual: **
+ ** T. Sjostrand, S. Mrenna and P. Skands, JHEP05 (2006) 026                 **
+ ** (LU TP 06-13, FERMILAB-PUB-06-052-CD-T) [hep-ph/0603175].                **
+ **                                                                          **
+ ** Also remember that the program, to a large extent, represents original   **
+ ** physics research. Other publications of special relevance to your        **
+ ** studies may therefore deserve separate mention.                          **
+ **                                                                          **
+ ** Main author: Torbjorn Sjostrand; Department of Theoretical Physics,      **
+ **   Lund University, Solvegatan 14A, S-223 62 Lund, Sweden;                **
+ **   phone: + 46 - 46 - 222 48 16; e-mail: torbjorn@thep.lu.se              **
+ ** Author: Stephen Mrenna; Computing Division, GDS Group,                   **
+ **   Fermi National Accelerator Laboratory, MS 234, Batavia, IL 60510, USA; **
+ **   phone: + 1 - 630 - 840 - 2556; e-mail: mrenna@fnal.gov                 **
+ ** Author: Peter Skands; CERN/PH-TH, CH-1211 Geneva, Switzerland            **
+ **   phone: + 41 - 22 - 767 24 47; e-mail: peter.skands@cern.ch             **
+ **                                                                          **
+ **                                                                          **
+ ******************************************************************************
+ ******************************************************************************
+1****************** PYINIT: initialization of PYTHIA routines *****************
+
+ ==============================================================================
+ I                                                                            I
+ I             PYTHIA will be initialized for a p+ on p+ collider             I
+ I                  at  13000.000 GeV center-of-mass energy                   I
+ I                                                                            I
+ ==============================================================================
+
+ ******** PYMAXI: summary of differential cross-section maximum search ********
+
+           ==========================================================
+           I                                      I                 I
+           I  ISUB  Subprocess name               I  Maximum value  I
+           I                                      I                 I
+           ==========================================================
+           I                                      I                 I
+           I   92   Single diffractive (XB)       I    7.1189D+00   I
+           I   93   Single diffractive (AX)       I    7.1189D+00   I
+           I   94   Double  diffractive           I    1.0176D+01   I
+           I   95   Low-pT scattering             I    5.4002D+01   I
+           I   96   Semihard QCD 2 -> 2           I    1.2350D+04   I
+           I                                      I                 I
+           ==========================================================
+
+ ****** PYMULT: initialization of multiple interactions for MSTP(82) = 4 ******
+        pT0 = 3.28 GeV gives sigma(parton-parton) = 5.65D+02 mb: accepted
+
+ ****** PYMIGN: initialization of multiple interactions for MSTP(82) = 4 ******
+        pT0 = 3.28 GeV gives sigma(parton-parton) = 2.38D+02 mb: accepted
+
+ ********************** PYINIT: initialization completed **********************
+
+
+
+
for Model, h in histograms.items():
+    plt.figure()
+    h2 = h[bh.loc(-10):bh.loc(10):bh.rebin(5), bh.loc(-10):bh.loc(10):bh.rebin(5)]
+    h3 = h2[:, sum]
+    nxy = h2.values() / nevent
+    nx = ny = h3.values() / nevent
+    corr = nxy / np.outer(nx, ny) ** 0.5
+    print(Model.__name__, np.mean(corr))
+    plt.pcolormesh(*h2.axes.edges.T, np.ma.masked_invalid(corr).T)
+    plt.xlabel("eta 1")
+    plt.ylabel("eta 2")
+    plt.colorbar()
+    plt.clim(0, 0.05)
+    plt.title(Model.__name__)
+
+
Sibyll21 0.020957722916460298
+Sibyll23d 0.020768885431597404
+QGSJetII04 0.02160297235716634
+EposLHC 0.021529706058337213
+Pythia6 0.022110986939247387
+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+ + + +
+ +
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zzKdTC%|G6oc&11?g&0=QD-c@)G$hi2?{s&U$8#Hk(b&#B}O|6tru_a5dK zs1k8Jt|UzmXkQT1)_xenri(MsN>EC0=nw+FQ&bei9ddqVxh`IM@m>ukIO`2hE-W_2 z{$z{_`@sJFdhpbex_COO(5*WaXGW4kw78N}jKKloIBh5m22F?^m3MQg8) KM#k===l= + + + + + + + + +Hans Dembinski’s blog - Inelastic cross-sections of proton-proton and pion-proton collisions + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+ +
+ +
+
+
+

Inelastic cross-sections of proton-proton and pion-proton collisions

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

According to a simple parton model, the cross-section for an inelastic proton-proton collision should be 3/2 of a pion-proton collision. In this picture, only the valence quarks matter. We know that at high energies most of the collisions involve sea quarks and gluons, so another theory says that the cross-sections might become equal at high energies.

+

I check this here by downloading cross-section data from the Particle Data Group, interpolating the data points and then plot the ratio. We get indeed a ratio of about 3/2=1.5. In reality, it is a bit larger, about 1.6. There is no visible trend that it does down to 1 with energy.

+
+
import matplotlib.pyplot as plt
+import numpy as np
+import requests
+
+
+
def read(url):
+    r = requests.get(url)
+    tab = r.text
+    xy = []
+    skip = 11
+    for line in tab.strip().split("\n"):
+        if skip:
+            skip -= 1
+            continue
+        items = line.split()
+        xi = float(items[3])
+        yi = float(items[4])
+        yei = (0.5 * sum(float(i) ** 2 for i in items[5:9])) ** 0.5
+        xy.append((xi, yi, yei))
+    xy.sort(key=lambda p: p[0])
+    return np.transpose(xy)
+
+
+
pim_p = read("https://pdg.lbl.gov/2020/hadronic-xsections/rpp2020-pimp_total.dat")
+pip_p = read("https://pdg.lbl.gov/2020/hadronic-xsections/rpp2020-pipp_total.dat")
+p_p = read("https://pdg.lbl.gov/2020/hadronic-xsections/rpp2020-pp_total.dat")
+
+
+
plt.figure(figsize=(16, 6))
+plt.errorbar(*pim_p, fmt="o", label="pi- p")
+plt.errorbar(*pip_p, fmt="s", label="pi+ p")
+plt.errorbar(*p_p, fmt="D", label="p p")
+plt.loglog()
+plt.legend()
+
+
<matplotlib.legend.Legend at 0x12b7185b0>
+
+
+

+
+
+
+
from scipy.interpolate import interp1d
+l_pip_p = interp1d(*pip_p)
+l_p_p = interp1d(*p_p)
+x = np.geomspace(1, 1e4)
+plt.plot(x, l_p_p(x) / l_pip_p(x))
+plt.axhline(3/2, ls="--", color="0.5")
+plt.semilogx()
+
+
[]
+
+
+

+
+
+ + + +
+ +
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+ +
+ +
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+

Fitting weighted histograms with SPD method

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+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

We test different fit methods of weighted binned data. The tested weight distributions are normal, exponential, and uniform. The toy case is a common HEP fit of a gaussian peak above a smooth background (here exponential).

+

Fits of weighted data are discussed in F. James, “Statistical Methods in Experimental Physics” (2nd edition), World Scientific, in section 8.5. Two approximate least-squares methods for binned data are presented in section 8.5.4. Here we test the method \(Q_1'\).

+

Bohm and Zech (2014) derived an approximate maximum-likelihood method for weighted histograms with on average positive weights. The authors show that their approximation yields less biased results than the aforementioned least-squares method.

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Still, a notable bias is present even when the SPD method is used. The largest bias is found for exponentially distributed weights with deviations of 4-5% in the fitted parameters.

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The remaining bias of the SPD method originates from the fact that the true weight variance in each bin is unknown and is replaced by the sample variance. This is inaccurate in bins with small number of samples and fails for empty bins, which must be skipped. The same is true for the alternative least-squares fit, however, and the biases are even larger there.

+

The study suggest that biases are larger if the weight distribution deviates more from the normal distribution.

+
+
#!pip install boost-histogram iminuit scipy matplotlib numpy numba progressbar2 pyik
+import boost_histogram as bh
+from iminuit import Minuit
+from scipy.stats import norm, expon, uniform, poisson
+import multiprocessing
+multiprocessing.set_start_method('fork') # needed on OSX, see https://github.com/ipython/ipython/issues/12396
+from multiprocessing.pool import Pool
+from matplotlib import pyplot as plt
+from pyik.mplext import plot_stacked
+import numpy as np
+import numba as nb
+from progressbar import progressbar
+import math
+
+
+
kwd = {"fastmath": True}
+
+
+@nb.njit(**kwd)
+def sum_log(fs, spdf, fb, bpdf):
+    return np.sum(np.log(fs * spdf + fb * bpdf))
+
+
+@nb.njit(**kwd)
+def norm_pdf(x, mu, sigma):
+    invs = 1.0 / sigma
+    z = (x - mu) * invs
+    invnorm = 1 / np.sqrt(2 * np.pi) * invs
+    return np.exp(-0.5 * z**2) * invnorm
+
+
+@nb.njit(**kwd)
+def nb_erf(x):
+    y = np.empty_like(x)
+    for i in nb.prange(len(x)):
+        y[i] = math.erf(x[i])
+    return y
+
+
+@nb.njit(**kwd)
+def norm_cdf(x, mu, sigma):
+    invs = 1.0 / (sigma * np.sqrt(2))
+    z = (x - mu) * invs
+    return 0.5 * (1 + nb_erf(z))
+
+
+@nb.njit(**kwd)
+def expon_pdf(x, lambd):
+    inv_lambd = 1.0 / lambd
+    return inv_lambd * np.exp(-inv_lambd * x)
+
+
+@nb.njit(**kwd)
+def expon_cdf(x, lambd):
+    inv_lambd = 1.0 / lambd
+    return 1.0 - np.exp(-inv_lambd * x)
+
+
+@nb.njit
+def cost_lsq(n, mu):
+    r = 0.0
+    for i in range(len(mu)):
+        if n[i] == 0:
+            continue
+        r += (n[i] - mu[i]) ** 2 / n[i]
+    return r
+
+
+@nb.njit
+def _cost_lsq_w(w, wvar, mu):
+    r = 0.0
+    n = len(mu)
+    for i in range(n):
+        if wvar[i] == 0:
+            continue
+        r += (w[i] - mu[i]) ** 2 / wvar[i]
+    return r
+
+
+def cost_lsq_w(w, mu):
+    return _cost_lsq_w(w.value, w.variance, mu)
+
+
+@nb.njit
+def cost_ml(n, mu):
+    r = 0.0
+    for i in range(len(mu)):
+        r += mu[i] - n[i] * np.log(mu[i])
+    return 2 * r
+
+
+@nb.njit
+def _cost_ml_w(w, wvar, mu):
+    # Bohm and Zech, NIMA 748 (2014) 1-6
+    r = 0.0
+    for i in range(len(mu)):
+        if wvar[i] > 0.0:
+            if w[i] > 0.0:
+                s_inv = w[i] / wvar[i]
+                mu_prime = mu[i] * s_inv
+                n_prime = w[i] * s_inv
+                r += mu_prime - n_prime * np.log(mu_prime)
+            else:  # extension
+                r += 0.5 * (w[i] - mu[i]) ** 2 / wvar[i]
+    return 2 * r
+
+
+def cost_ml_w(w, mu):
+    return _cost_ml_w(w.value, w.variance, mu)
+
+
+# these distributions are later shifted and scaled to give E[w] = 1 and Var(w) = 1
+weight_distributions = [None, np.ones, norm().rvs, uniform().rvs, expon().rvs]
+wd_labels = ("no weights", "w=1", "norm", "uniform", "expon")
+
+mu_npts = 1000
+signal_fraction = 0.2
+
+par_labels = ("bkg: fraction", "sig: fraction",
+              "bkg: lambda", "sig: mu", "sig: sigma")
+truth = ((1.0 - signal_fraction), signal_fraction, 1.0, 2.0, 0.2)
+
+
+def run(wd, wd_label):
+    sub_pars = []
+    for imc in progressbar(range(nmc), redirect_stdout=True):
+        rng = np.random.default_rng(imc)
+        all_valid = False
+        while not all_valid:
+            # sample total number of points from poisson distribution
+            npts = rng.poisson(mu_npts)
+
+            # generate background
+            b = rng.exponential(truth[2], size=int(truth[0] * npts))
+
+            # generate signal
+            s = rng.normal(truth[3], truth[4], size=int(truth[1] * npts))
+            x = np.append(b, s)
+
+            if wd is None:
+                h = bh.Histogram(bh.axis.Regular(20, 0, 3))
+                h.fill(x)
+                xe = h.axes[0].edges
+                w = h.view()
+
+            else:
+                # generate weights
+                wi = wd(len(x))
+                if np.std(wi) > 0:
+                    # wi have mean(wi) = 1 and std(wi) = 1
+                    wi = (wi - np.mean(wi)) / np.std(wi) + 1.0
+                h = bh.Histogram(bh.axis.Regular(20, 0, 3),
+                                 storage=bh.storage.Weight())
+                h.fill(x, weight=wi)
+                xe = h.axes[0].edges
+                w = h.view()
+
+            
+            sub_sub_pars = []
+            all_valid = True
+            for fit_mode in range(2):
+                if wd is None:
+                    if fit_mode == 0:
+                        cost = cost_lsq
+                    else:
+                        cost = cost_ml
+                else:
+                    if fit_mode == 0:
+                        cost = cost_lsq_w
+                    else:
+                        cost = cost_ml_w
+
+                def cost_model(par):
+                    bkg = expon_cdf(xe, par[2])
+                    sig = norm_cdf(xe, par[3], par[4])
+                    mu_bkg = par[0] * npts * np.diff(bkg)
+                    mu_sig = par[1] * npts * np.diff(sig)
+                    res = cost(w, mu_bkg + mu_sig)
+                    return res
+
+                m = Minuit.from_array_func(
+                    cost_model,
+                    truth,
+                    limit=((0, None), (0, None),
+                           (0, None), None, (0, None)),
+                    pedantic=False,
+                )
+
+                try:
+                    m.migrad()
+                    all_valid &= m.valid
+                    sub_sub_pars.append(m.np_values())
+                except RuntimeError:
+                    all_valid = False
+
+        if nmc == 1 and wd is not None:
+            plt.figure()
+            h_bkg = bh.Histogram(h.axes[0], storage=bh.storage.Weight())
+            h_bkg.fill(x[:len(b)], weight=wi[:len(b)])
+            h_sig = bh.Histogram(h.axes[0], storage=bh.storage.Weight())
+            h_sig.fill(x[len(b):], weight=wi[len(b):])
+            plot_stacked(xe, np.array((h_bkg.view().value, h_sig.view().value)),
+                         colors=("0.8", "r"), labels=("bkg", "sig"))
+            plt.errorbar(h.axes[0].centers, w.value,
+                         w.variance ** 0.5, fmt="ok")
+            plt.title(wd_label)
+            plt.ylim(0, None)
+            plt.xlim(xe[0], xe[-1])
+
+        sub_pars.append(sub_sub_pars)
+    return sub_pars
+
+
+# for drawing examples
+nmc = 1
+for wd, wd_label in zip(weight_distributions, wd_labels):
+    run(wd, wd_label)
+
+
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+
+

+
+
+

+
+
+
+
nmc = 10000
+with Pool() as p:
+    pars = p.starmap(run, zip(weight_distributions, wd_labels))
+pars = np.array(pars)
+
+
100% (10000 of 10000) |##################| Elapsed Time: 0:01:15 Time:  0:01:15
+100% (10000 of 10000) |##################| Elapsed Time: 0:01:31 Time:  0:01:31
+100% (10000 of 10000) |##################| Elapsed Time: 0:01:35 Time:  0:01:35
+100% (10000 of 10000) |##################| Elapsed Time: 0:01:35 Time:  0:01:35
+100% (10000 of 10000) |##################| Elapsed Time: 0:01:36 Time:  0:01:36
+
+
+
+
fit_labels = ("LSQ", "ML")
+pars_range = ((0.4, 1.0), (0.0, 0.4), (0.2, 1.6), (1.75, 2.15), (0.0, 0.4))
+for ipar, (par_label, t) in enumerate(zip(par_labels, truth)):
+    fig, ax = plt.subplots(1, 2, figsize=(14, 4), sharex=True, sharey=True)
+    plt.suptitle(par_label)
+    for iwd, wd_label in enumerate(wd_labels):
+        for ifit, fit_label in enumerate(fit_labels):
+            plt.sca(ax[ifit])
+            p = pars[iwd, :, ifit, ipar]
+            plt.hist(p, alpha=0.5, label=wd_label,
+                     bins=100, range=pars_range[ipar])
+    for label, axi in zip(fit_labels, ax):
+        axi.axvline(t, color="r")
+        axi.legend(loc="upper left", title=label)
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+
+
for ipar, (par_label, t) in enumerate(zip(par_labels, truth)):
+    fig, ax = plt.subplots(1, 2, figsize=(14, 3), sharex=True, sharey=True)
+    plt.suptitle(f"{par_label}", y=1.05)
+    for iwd, wd_label in enumerate(wd_labels):
+        for ifit in range(2):
+            plt.sca(ax[ifit])
+            p = pars[iwd, :, ifit, ipar]
+            mean = np.mean(p)
+            sigma = np.std(p) / len(p) ** 0.5
+            plt.errorbar(mean, iwd, 0, sigma, fmt="o", ms=10)
+    for fit_label, axi in zip(fit_labels, ax):
+        axi.set_title(fit_label, y=1.05)
+        axi.axvline(t, color="r")
+        for percent in (1, 2, 3, 4, 5):
+            dt = t * 1e-2 * percent
+            axi.axvline(t - dt, color="0.5", ls=":")
+            axi.axvline(t + dt, color="0.5", ls=":")
+            axi.text(t + dt, 1.01, f"+{percent}%", transform=axi.get_xaxis_transform(),
+                     ha="center", fontsize="x-small")
+            axi.text(t - dt, 1.01, f"-{percent}%", transform=axi.get_xaxis_transform(),
+                     ha="center", fontsize="x-small")
+    plt.sca(ax[0])
+    plt.yticks(np.arange(len(wd_labels)), wd_labels)
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Fit weighted histograms with SPD method_files/figure-html/cell-3-output-2.png b/posts/Fit weighted histograms with SPD method_files/figure-html/cell-3-output-2.png new file mode 100644 index 0000000000000000000000000000000000000000..47241d7cb03f9ca1418a252830882458c6a5f25c GIT binary patch literal 6238 zcmZu$3p|r;|KF^vgTu2*iX=mdip(*G)ng8eD8~>BF{eTflQ8OuMX1SPjy(|}=8#hk zTjdZoq60FvmIy0D4rA=dR5QJTMx~ z2X5+w7+(yr3GfQRxn1`JU3Ciy^bZK}_i;ZI=6OBXC*T_5g!&2fV~4y$LIO{zs{Si} zBH%h!HC^`JT@XmJ8I3w)8=muh6o1e2>y4qgSxA!_*yy74bE6eaRFUOihgMI*WyjbH z-nJdJBGx5$G~q*vQnf8UIYmSdY~jGQTIvm0=fk|pHV2K&NH_bCGizY;w(0Zwnm5X=u){b_er#}#p$$!!-12P2jm!qNZ@{BG_Ees}`NI<&Q30ep%o~Mu(?mlDa=_9zO|>;Pd!kHgZe@v zL1#y^e2a?fudC`A2tPj;DU8Zk1NTxnfc3f~6maO-a{=a`XL++H$oouE@w?X8GTyNKXnCp@U)4iDG+XqS$K$wE&Y zuL}UT=8!to7uKj?){7lq(iH3-C1?roB9Y}oty{v~hJ)i1s%=nh7AQ_4A&T6(>bup= zjQGh;nP=K|ta2Nze7Tm?h(OePPShoynD$BY!+-YOkn zFbavkWC`~m<6H;1mb{Rm`KcRsJJ=;@tVPSF>7T$ 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100644 index 0000000..8b21e19 --- /dev/null +++ b/posts/From fixed target to sqrt_s_nn and back.html @@ -0,0 +1,477 @@ + + + + + + + + + +Hans Dembinski’s blog - From a fixed-target to center-of-mass frame and back + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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From a fixed-target to center-of-mass frame and back

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I compute with Sympy how to transform a 4-vector from a fixed-target frame (aka laboratory frame) to the center-of-mass frame, where the colliding particles have equal but opposite momenta. It is fun to do this with Sympy, because it can compute the Taylor series expansion for the common case where the momenta are large compared to the particle masses.

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from sympy import *
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e1, p1, e2, p2, m = symbols("E_1 p_1 E_2 p_2 m",real=True, positive=True)
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We work in 1-dimensional coordinates with \(p = p_z\). We generally use \(m = m_p = m_n\), in other words we neglect the mass difference between protons and neutrons. We consider a proton projectile and a nuclear target with \(A\) nucleons.

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We first compute the cms energy \(\sqrt{s_{nn}}\) in the nucleon-nucleon system from the fixed target system.

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In this system, we have these 4-vectors: * projectile \((E, p)\) * target \((m, 0)\)

+
+
s2 = (e1 + m)**2 - (p1 + 0)**2; s2
+
+

\(\displaystyle - p_{1}^{2} + \left(E_{1} + m\right)^{2}\)

+
+
+
+
s2 = s2.subs(e1, sqrt(p1 ** 2 + m ** 2)); s2
+
+

\(\displaystyle - p_{1}^{2} + \left(m + \sqrt{m^{2} + p_{1}^{2}}\right)^{2}\)

+
+
+
+
s2.series(m, 0, 2)
+
+

\(\displaystyle 2 m p_{1} + O\left(m^{2}\right)\)

+
+
+
+
s = sqrt((e1 + m)**2 - (p1 + 0)**2); s
+
+

\(\displaystyle \sqrt{- p_{1}^{2} + \left(E_{1} + m\right)^{2}}\)

+
+
+
+
s = s.subs(e1, sqrt(p1 ** 2 + m ** 2)); s
+
+

\(\displaystyle \sqrt{- p_{1}^{2} + \left(m + \sqrt{m^{2} + p_{1}^{2}}\right)^{2}}\)

+
+
+
+
s.series(m, 0, 2)
+
+

\(\displaystyle \sqrt{2} \sqrt{m} \sqrt{p_{1}} + \frac{\sqrt{2} m^{\frac{3}{2}}}{2 \sqrt{p_{1}}} + O\left(m^{2}\right)\)

+
+
+
+
def sqrt_snn(energy):
+    mass = (938.27 + 939.57) * 0.5e-9 # nucleon mass in PeV
+    return sqrt(2 * mass * energy) * 1e3 # sqrt(sNN) in TeV
+
+
+
0.5 * (sqrt_snn(320000+90000) - sqrt_snn(320000-90000))
+
+

\(\displaystyle 110.127117815583\)

+
+
+
+
def beam_energy(sqrt_snn):
+    mass = (938.27 + 939.57) * 0.5e-3 # nucleon mass in GeV
+    return (sqrt_snn * 1e3) ** 2 / (2 * mass) / 1e6 # beam in PeV
+
+
+
beam_energy(14)
+
+
104.37523963702976
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/HESSE_vs_MINOS_TwoGauss.html b/posts/HESSE_vs_MINOS_TwoGauss.html new file mode 100644 index 0000000..f822de6 --- /dev/null +++ b/posts/HESSE_vs_MINOS_TwoGauss.html @@ -0,0 +1,1351 @@ + + + + + + + + + +Hans Dembinski’s blog - Coverage of HESSE and MINOS intervals + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Coverage of HESSE and MINOS intervals

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

The MINUIT package provides two ways to compute uncertainty intervals for a fitted parameter, the HESSE method and the MINOS method. Ideally, these intervals should have 68 % coverage probability: if the experiment is repeated with new random data each time, the intervals should cover the true value in 68 % of the cases.

+

This only asymptotically correct, and if the samples are small, the actual coverage probability can deviate from the expected one. It is often thought that MINOS provides better coverage (closer to the expected value) because its intervals can be asymmetric and seem to retain more information.

+

As a toy example, I fit the sum of two normal distribution which have equal width but different location parameters. The fitted parameters are the two locations and the shared width. Here is one example fit.

+
+
from iminuit import Minuit
+from iminuit.cost import UnbinnedNLL
+from numba_stats import norm
+import numpy as np
+import numba as nb
+from matplotlib import pyplot as plt
+from IPython.display import display
+
+
+
sigma = 1.0
+
+def gen_data(rng, n):
+    n0 = rng.binomial(n, 0.5)
+    n1 = n - n0
+    return np.append(rng.normal(0, sigma, size=n0), rng.normal(1, sigma, size=n1))
+
+@nb.njit(nogil=True)
+def model(x, mu_a, mu_b):
+    return 0.5 * (norm.pdf(x, mu_a, sigma) + norm.pdf(x, mu_b, sigma))
+
+rng = np.random.default_rng(0)
+
+data = gen_data(rng, 10000)
+m = Minuit(UnbinnedNLL(data, model), mu_a=0, mu_b=1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 3.065e+04Nfcn = 31
EDM = 9.53e-06 (Goal: 0.0002)time = 4.5 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0mu_a-0.0140.020
1mu_b1.0130.020
+ + + + + + + + + + + + + + + + + + +
mu_amu_b
mu_a0.000412-0.2e-3 (-0.386)
mu_b-0.2e-3 (-0.386)0.00041
+ + + + + + + + 2024-05-12T23:24:23.630891 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+

I now run this fit many times on newly drawn random samples. To speed things up, the calculation is done in parallel.

+
+
def compute(n):
+    n_try = 1000
+    rng = np.random.default_rng(n)
+    n_hesse = 0
+    n_minos = 0
+    h_pcov = []
+    m_pcov = []
+    cor = []
+    n_fail = 0
+    for i_try in range(n_try):
+        for sub in range(5):
+            data = gen_data(rng, n)
+            nll = UnbinnedNLL(data, model)            
+            m = Minuit(nll, mu_a=0, mu_b=1)
+            m.migrad()
+            m.hesse()
+            m.minos()
+            if m.valid and m.accurate:
+                break
+            n_fail += 1
+        mu = m.values["mu_a"]
+        h_mu = m.errors["mu_a"]
+        m_mu = m.merrors["mu_a"]
+        cov = m.covariance
+        c = cov[1,0] / np.sqrt(cov[0, 0] * cov[1, 1])
+        if abs(mu) < h_mu:
+            n_hesse += 1
+        if (mu + m_mu.lower) < 0 < (mu + m_mu.upper):
+            n_minos += 1
+        cor.append(c)
+    h_pcov.append(n_hesse / n_try)
+    m_pcov.append(n_minos / n_try)
+    print(f"{n} hesse {h_pcov[-1]:.2f} minos {m_pcov[-1]:.2f} <corr> {np.mean(cor):.3f} {n_fail=}", flush=True)
+    return h_pcov, m_pcov, cor
+
+if __name__ == "__main__":
+    from concurrent.futures import ThreadPoolExecutor as Pool
+
+    n_pts = (10, 20, 50, 70, 100, 200, 300, 400, 500, 1000, 2000, 3000)
+
+    h_pcov = []
+    m_pcov = []
+    corr = []
+    with Pool() as pool:
+        for (h, m, c) in pool.map(compute, n_pts):
+            h_pcov.append(h)
+            m_pcov.append(m)
+            corr.append(c)
+
+
10 hesse 0.76 minos 0.75 <corr> -0.388 n_fail=7
+20 hesse 0.77 minos 0.73 <corr> -0.450 n_fail=0
+50 hesse 0.75 minos 0.67 <corr> -0.458 n_fail=0
+100 hesse 0.77 minos 0.69 <corr> -0.461 n_fail=0
+70 hesse 0.75 minos 0.67 <corr> -0.472 n_fail=0
+200 hesse 0.69 minos 0.66 <corr> -0.435 n_fail=0
+1000 hesse 0.67 minos 0.67 <corr> -0.407 n_fail=0
+2000 hesse 0.69 minos 0.69 <corr> -0.401 n_fail=0
+500 hesse 0.70 minos 0.69 <corr> -0.413 n_fail=0
+3000 hesse 0.69 minos 0.69 <corr> -0.404 n_fail=0
+400 hesse 0.70 minos 0.69 <corr> -0.422 n_fail=0
+300 hesse 0.68 minos 0.67 <corr> -0.420 n_fail=0
+
+
+

I plot the actual coverage probability of the HESSE and MINOS intervals. For small samples, both methods overcover, but the MINOS intervals converge faster to the correct value. So in this case, MINOS did perform better. This is not always the case, however.

+
+
plt.plot(n_pts, h_pcov, label="HESSE")
+plt.plot(n_pts, m_pcov, label="MINOS")
+plt.axhline(0.68, ls=":", color="0.5", zorder=0)
+plt.semilogx()
+plt.legend()
+plt.ylabel("coverage")
+plt.xlabel("sample size")
+plt.ylim(0, 1);
+
+

+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/HESSE_vs_MINOS_TwoGauss_files/figure-html/cell-5-output-1.png b/posts/HESSE_vs_MINOS_TwoGauss_files/figure-html/cell-5-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..554b0c20a5e80aa80205cc059ed990406016b80c GIT binary patch literal 19163 zcmeIacT`hfw>BE}XG4LX1ymFeL8YTq0Ra^O0qIS;C><#R0j1gi0TBsJ>AeMmbO=pF zL_m6p^bV4cP(v?wZr}Hud&c*kGwwO}e&hc2IW%fW*4}%qx#oQ4GoQHv)m0T}X_#nG zC={*Yo!goy)Lt^bbVy=;JllK zvx|+RgAo5UenH;z)~>EjE)oI)_W%9?en)33f%BU?-(Zl#PIvTOP$+scd&UUMJBz^$V+%j7P^i?%bhQk+`KE^LoGV>DwC1nO2!u?p6W|l#`(s zh-akcP9BLWj3ZpvP&;-zOz7Yf&D&p`II99MH||`Ns#w~57*2bYE$z`S^CwcQq299; z_Zp$i-Ye$A>n-kW21+jPREPpoj{>n&zaDQ)z#H0DJgYSK3$6G@?E72 z;%5t*gxzNES(uyuetwXpja)j66R2KG3E|m~LNWWLR~cRFe31WpJXEG9O)*^c*Ys!r zYtL9+FtwImrmP=Vv*R&2J#E*3xtN)xghI6qtv{)FHg&JcqZkt&PAW=IOG8GzHYzwd zIms;`;4b4X5#QL{taJPJD_%xr!$?@~>p!j>Wnf_7;^$Y@wHudAkkBq`Xl&GUb#;v| z`Tg6Pfsrw%s7QOyo;}nkMP=pAIB^$Moj?>SiPpl>lADb!xm}u>d!Xps@=RxBObpS- 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zqw@aBO{CZiJ8`*QW=8{#QOV|*`B*xnX=0Lq_^(0dCO$mG`;!a|lVPhAIEV-b=Yb$! zi#Bo&1dN#rWvR(m+lf!c4mZb%Yk_w`94F_~&_lyFa##Tpu;Bz20#d_9 zYIRK4obFduBW@0YGX;*>f$BHj5xiCn2iL`ma>xWk@f+tL03IogAbsv&oKGjL_9Q)K ztj&`4KY{AYhHUmOod2K>4zW~uPurd=NFVWCKR8GLUI(E{@>%;dlGbM2LSW>7>veJW Zoui-a>}gFI$g)w2a;mp+x9-3AzW^D-u+0Df literal 0 HcmV?d00001 diff --git a/posts/Interactive plotting in Jupyter with matplotlib.html b/posts/Interactive plotting in Jupyter with matplotlib.html new file mode 100644 index 0000000..d9d7bc6 --- /dev/null +++ b/posts/Interactive plotting in Jupyter with matplotlib.html @@ -0,0 +1,452 @@ + + + + + + + + + +Hans Dembinski’s blog - Interactive plotting in Jupyter with matplotlib + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Interactive plotting in Jupyter with matplotlib

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

This is a little demo on how to make interactive plots with matplotlib and ipympl.

+
+
#!pip install ipywidgets matplotlib scipy numpy ipympl
+%matplotlib widget
+from matplotlib import pyplot as plt
+from scipy.stats import johnsonsu, norm, t
+from ipywidgets import interact, widgets
+import numpy as np
+
+
+
x = np.linspace(-5, 5, 1000)
+
+fig = plt.figure()
+plt.plot(x, norm.pdf(x), label="normal")
+line, = plt.plot(x, x, label="other")
+plt.legend()
+plt.ylim(0, 1)
+
+def update(a, b, which):
+    if which == "Johnson SU":
+        y = johnsonsu.pdf(x, a, np.exp(b))
+    elif which == "Student-t":
+        y = t.pdf(x, np.exp(b))
+    line.set_ydata(y)
+    fig.canvas.draw_idle()
+    
+interact(update,
+         a=(-10.0, 10.0),
+         b=widgets.FloatSlider(0.0, min=-10.0, max=5.0),
+         which=["Johnson SU", "Student-t"]);
+
+ +
+
+ +
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Leave-one-out cross-validation.html b/posts/Leave-one-out cross-validation.html new file mode 100644 index 0000000..fff67b7 --- /dev/null +++ b/posts/Leave-one-out cross-validation.html @@ -0,0 +1,499 @@ + + + + + + + + + +Hans Dembinski’s blog - Leave-one-out cross-validation + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Leave-one-out cross-validation

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

General steps for LOO cross-validation: - Remove i-th datum from input data set - Fit model to remaining data set - Use fitted model to predict the i-th datum and store difference to original i-th datum - Do this for all i and compute variance of the differences - Select model with smallest variance

+

I demonstrate below how to select the optimal order for a polynomial model with leave-one-out (LOO) cross validation.

+
+
#!pip install numpy matplotlib iminuit tabulate
+import numpy as np
+import matplotlib.pyplot as plt
+from iminuit import Minuit 
+from tabulate import tabulate
+
+
+
rng = np.random.default_rng(seed=1)
+
+
+
# create some toy data that follows polynomials of increasing degree
+x = np.linspace(0, 1, 100)
+y_set = np.empty((3, len(x)))
+for poly_degree in (0, 1, 2):
+    y_set[poly_degree] = np.polyval(np.ones(poly_degree + 1), x) + rng.normal(0, 0.1, len(x))
+    
+for y in y_set:
+    plt.plot(x, y, ".")
+
+

+
+
+
+
# apply leave-one-out cross-validation
+for true_poly_order, y in enumerate(y_set):
+
+    variances = []
+    tested_poly_orders = np.arange(5)
+    for tested_poly_order in tested_poly_orders:
+        deltas = []
+
+        # leave out the i-th (x, y) pair in the fit
+        for i in range(len(y)):
+            mask = np.ones(len(y), dtype=bool)
+            mask[i] = 0
+            x_without_i = x[mask]
+            y_without_i = y[mask]
+
+            # least-squares cost function to fit a polynomial to the toy data
+            def lsq(par):
+                ym = np.polyval(par, x_without_i)
+                return np.sum((y_without_i - ym) ** 2 / 0.1 ** 2)
+            
+            m = Minuit.from_array_func(lsq, np.zeros(tested_poly_order+1), pedantic=False)
+            m.strategy = 0 # faster and does not compute errors automatically
+            m.migrad()
+
+            # predict the i-th y value with the fitted model and 
+            # store the delta to the real value
+            assert m.migrad_ok()
+            
+            yi_predicted = np.polyval(m.np_values(), x[i])
+            deltas.append(y[i] - yi_predicted)
+
+        variances.append(np.var(deltas))
+
+    print(f"true polynomial order {true_poly_order}; selected order {np.argmin(variances)}")
+    print(tabulate(np.transpose((tested_poly_orders, variances)),
+                   headers=("model order", "LOO variance")) + "\n")
+
+
true polynomial order 0; selected order 0
+  model order    LOO variance
+-------------  --------------
+            0       0.0105525
+            1       0.0105614
+            2       0.0107262
+            3       0.0107479
+            4       0.0107608
+
+true polynomial order 1; selected order 1
+  model order    LOO variance
+-------------  --------------
+            0       0.0903401
+            1       0.0121111
+            2       0.0122902
+            3       0.0124952
+            4       0.0124547
+
+true polynomial order 2; selected order 2
+  model order    LOO variance
+-------------  --------------
+            0       0.347969
+            1       0.017877
+            2       0.0117249
+            3       0.0118761
+            4       0.011895
+
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Leave-one-out cross-validation_files/figure-html/cell-4-output-1.png b/posts/Leave-one-out cross-validation_files/figure-html/cell-4-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..1f0a44b30419c938d03ebdd5db8ddde0d9773663 GIT binary patch literal 10638 zcma)i1yEegv+u!O7AFK-+#P}kSOSFLF2OapJ1p)N+#zTdmtX-x0)Y?`vbakixCi&U zeBb}wSFi4?S8uEK?4C0{-90_szd6&>vFfS{c-T*{0RX^LQiNy%0Fo!-|0O0m;(BXC z4MF^%ddMniVA|5{Tl-KvvaFhi`UB!5K{&zdKi@Plk6yQS$q63s5GTOfR`vpNM z#*=;(4k%u+A7*BWXbmcxQX-Vy7Dm=rqpX)~fVYpXIU# z$W1X0bGPqBCXx@O)1zPEP@zVK6Ni)M&_M`rF+=4eIO%IK`I{(8#}t0O`gEhW6V-o3eyyxDajdMK)(O&UQc8>hy!i%cks8H<4}Mjw%m`aiA_Z*(yV zMO+G#5)!Pud}T4&7)wh_U0^%0szuh!TMP^gc7jg~s0#^Yv14QQJZ};*asIDt{s-f- zZ!=4LHH%$jmH~qn%RV-KTM>j~E(!b8eXyb$Sr1rjNhp0yhaKLV6NkqAA`n8kGRRsY zdajmA3;YYj0Xb1*mD*TIh9x2;y{bri5Z+6OZUuyA$V{lg20KGchvgwjXh*0}OR@Qf z=Q^Jp=Qk*T3qT*;j24<)y_2M`{1THMty1_?ln{BX41(_t%cSK8BS3)Q4ik!_eFlxe zMZWBaxz*PZF>LKX?+w;@%?Z{+xjY1qt|h zVg!TFs!ioNMCl{`WHXKvFY>-L(w<}kgg`lXD+=HyZ5`7V$5oQV`BLAiodVzh73$wD zwdR4C3lybFmQE1su)NMP@{s_LH6t~LXGRaiD3bQ_LQMzNr4TevA8CGbY=PY^Soou3 zejvNjLrrIdjR;o83ySKIkFsI>$Se`3Opj#d79E^`Yg-I%@gfkYrbA1XngOh6!YR_V%@ykrw|dw>gSy%^&DBn3=w})Yb0EWAy(Qt)N;W| zBuo*6R0e)48YRWVvN4G@5=T|8t@5&%ji5FRny<+O$Y=$GvQJ|iVuEvnCAi3CF_lsT z?|dIUkWukJPhzoH>Lq^Wdpo9wd)|1RmA;6?^E#T77_h#8a=!J#!f+O zI(A7sWihcr)4sm_ddsEDif}0+WR`gAbpsB8f|E#^M{2s{aTH!zgxe5W3CY)y|Ik48u+Yv~6WY?}9`usuF6euX2RCgGv{ z=bIP-O4O^z;k^u>qEQG#~WD7y#A@Ru1MvwuDDM&X~IzH$uqH z+X*QA0FXV#wpNoZQOT=`P^^PKLBVMZ)3;o}J2`JKMx6Cu`c(~|(!Y`z=a<6B!MaML zkzF_l<=#@gl6>Q!ZBnbPQLA0)MBMg*?w78T@#FY9zCe6yr+%cX{{GISXYu(M=cRcb zBs5GF;db5*8-Fj1m!@TxA|LN@ryMgSDhX^N>x<*R$s%IWmjy}&71ud)XmuJvIiiD_ zdf%VQ>2$L#y(5yn2Gw8v)GPeJzq`7QorW1i)QbnZu8GM9%(Nl*COW1~PHi$eJn^4m zB2t2xVy}J9%St5`I8c#?-9uRZA~m%4^@YYLcZ zOw!It1z3%AUN#D4_ zu?4&K09KoS?gP%r)#?b~30reOL_5mO>#~?RGWpf> z0{gs@h$u;+?Elail#hpU9F2f)EA&NBu155%xDYO*C`AdetPSDx&iQ%CO`lw!{zpHq z{j3OHnrm8-r`KwVAuB?6T=J67OynNy5%P%25r~>$TifsJk$RwG;0vNjs&N5``Y+sN~<@31OHkj1`&2YPUGW&IO{saUx# zBjxLFLlMivDW#IvbM&+RSB9@U=W64TBm4uS-@S()*){s=Nqu!+x8 ztquNV32}t>V;K7Tgw;Myr1S0XF4{Lj=|3aFRnvV>c7F7KvS|b$zam>FtE5G@@Mp4fiT>ah^ zl)Qe{5R*a#8tkPLf6A);6fwlNy!Jm0-cB?GZ(GEZMw3Db=!cC;kYW}4ZouXg{TU`sWV2_;r7Gefwi;2rE*3_2|?{fP^V2yn_? 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+
+ +
+ +
+
+
+

MCMC demo

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

This is a tiny demo of the MCMC algorithm from the emcee library, which computes the posterior for some parameters based on the likelihood function and priors on the parameters.

+
+
import numpy as np
+import emcee
+import matplotlib.pyplot as plt
+import numba as nb
+
+
+
rng = np.random.default_rng(1)
+
+y = 2.0
+
+@nb.njit
+def model(mu):
+    return mu + np.random.randn()
+
+@nb.njit
+def log_prob(mu, y):
+    diff = y - model(mu)
+    return -0.5 * np.sum(diff ** 2)
+
+nwalkers = 1000
+ndim = 1
+sampler = emcee.EnsembleSampler(nwalkers, ndim, log_prob, args=[y])
+
+start = rng.normal(size=(nwalkers, ndim))
+r = sampler.run_mcmc(start, 100, progress=True)
+
+plt.hist(r.coords)
+print(np.mean(r.coords), np.std(r.coords) / np.sqrt(len(r.coords)))
+
+
100%|██████████| 100/100 [00:00<00:00, 168.91it/s]
+
+
+
1.962706921253352 0.04570914440079373
+
+
+

+
+
+ + + +
+ +
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+
+ +
+ +
+
+
+

Introduction into Machine Learning

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

Hans Dembinski

+
+

Overview

+

+
    +
  • Caution: Field is riddled with buzzwords

    +
      +
    • Machine learning, artificial intelligence, deep learning, belief networks, attention-based networks, long short-term memory, features, hyperparameters …
    • +
  • +
  • Anthropomorphized and technical names hide what are usually basic mathematical algorithms

  • +
  • Machine learning: umbrella term for multivariate statistical methods for classification and regression

    +
      +
    • Multivariate: algorithms use more than one variable as input, e.g. (x, y, z)
    • +
    • Statistical: algorithms can handle noisy input and are founded on statistical theory
    • +
    • Classification: sort inputs based on input variables into categories
    • +
    • Regression: find a mapping of input vectors to output vectors (fitting)
    • +
  • +
+

+
    +
  • Rise of the machines: Why do we need them? +
      +
    • Humans good at working with 1D or 2D data, because we can visualize it
    • +
    • Humans bad at working with higer-dimensional data
    • +
    • Algorithms (“machines”) scale well to high-dimensional data
    • +
  • +
+
+

Notable software

+ +
+
+

Learning resources

+
    +
  • Internet provides excellent resources for autodidactic learning

    +
      +
    • Example code
    • +
    • Tutorials
    • +
    • Documentation

    • +
  • +
  • Scikit-Learn

  • +
  • Kaggle

  • +
  • All methods can do classification and regression and work with any input

  • +
  • What to use?

    +
      +
    • Structured data (i.e. tables of numbers): Ensembles of decision trees +
        +
      • Good results with default settings
      • +
      • Can process continuous and categorical input (m/w, red/green/blue, …)
      • +
      • No data preprocessing required
      • +
      • Compute feature importance
      • +
    • +
    • Unstructured data (i.e. images, waveforms, text…): Neural networks +
        +
      • Deep networks learn to extract features automatically
      • +
      • Requires data preprocessing
      • +
      • Networks difficult to optimize for non-experts
      • +
      • Often need very large data sets for training
      • +
    • +
  • +
+

Some definitions * sample: one or more numbers which describe outcome of stochastic process * dataset: consists of samples * features: subset of numbers from each sample which carry interesting information, also numbers computed from each sample * signal: samples which originate from stochastic process of interest, typically particle decay * background: samples which do not originate from process of interest

+

Discuss how the typical decay \(K^0_S \to \pi^+ \pi^-\) is reconstructed and why in general there is always some background.

+

Clarify how the previous terms would be used in the context of this example.

+
+
+
+

Classification

+
    +
  • Setting: dataset is mix of two or more components, typically signal + one or more backgrounds
  • +
  • Want only signal component, invert a cut to select signal candidates
  • +
  • Features are variables which allow to discriminate between signal and background, for example, reconstructed mass of decay candidate
  • +
+
+
#!pip install scipy matplotlib numpy scikit-learn xgboost
+
+
+
import scipy
+import matplotlib
+import numpy
+import sklearn
+import xgboost
+
+print(f"scipy {scipy.__version__}")
+print(f"matplotlib {matplotlib.__version__}")
+print(f"numpy {numpy.__version__}")
+print(f"sklearn {sklearn.__version__}")
+print(f"xgboost {xgboost.__version__}")
+
+
scipy 1.10.0
+matplotlib 3.7.1
+numpy 1.21.6
+sklearn 1.2.2
+xgboost 1.7.4
+
+
+
+
from scipy.stats import norm
+from matplotlib import pyplot as plt
+import numpy as np
+
+background = norm(0, 1).rvs(1000, random_state=1)
+signal = norm(1, 0.1).rvs(500, random_state=1)
+
+x = np.append(signal, background)
+y = np.append(np.ones(len(signal)), np.zeros(len(background)))
+
+
+
def plot(x, y):
+    signal = x[y == 1]
+    background = x[y == 0]
+    t, xe = np.histogram(x, bins=100)
+    s = np.histogram(signal, bins=xe)[0]
+    b = np.histogram(background, bins=xe)[0]
+    plt.stairs(b, xe, fill=True, label="background")
+    plt.stairs(t, xe, baseline=b, fill=True, label="signal")
+    plt.xlabel("x")
+    plt.legend();
+    return xe
+
+plot(x, y);
+
+

+
+
+

Exercise: * Propose simple selection (“cut”) that keeps most of the signal and removes most of the background * Is it possible to eliminate the background completely?

+
+

Efficiency and purity of a selection

+
    +
  • Efficiency: fraction of signal left after applying selection +
      +
    • Number of true signal samples after cut / Number of all true signal samples
    • +
  • +
  • Purity: fraction of background still contained in selected sample +
      +
    • Number of true signal samples after cut / Number of selected samples
    • +
  • +
  • Want to maximize efficiency and purity
  • +
+
+

Single-parameter cut

+
+
xmin = 0.7
+
+selected_signal = np.sum(x[y==1] > xmin)
+total_signal = np.sum(y==1)
+selected = np.sum(x > xmin)
+
+efficiency = selected_signal / total_signal
+purity = selected_signal / selected
+
+plot(x, y)
+plt.axvspan(-3.5, xmin, facecolor="k", alpha=0.5)
+plt.legend(title=f"efficiency={efficiency:.2f}\npurity={purity:.2f}");
+
+

+
+
+
+
xmin = np.linspace(-3, 3, 100)
+efficiency = []
+purity = []
+for xm in xmin:
+    selected_signal = np.sum(signal > xm)
+    total_signal = len(signal)
+    selected = np.sum(x > xm)
+    e = selected_signal / total_signal
+    p = selected_signal / selected
+    efficiency.append(e)
+    purity.append(p)
+plt.plot(xmin, efficiency, label="efficiency")
+plt.plot(xmin, purity, label="purity")
+plt.xlabel("xmin")
+plt.legend();
+
+

+
+
+
+
plt.scatter(efficiency, purity, c=xmin);
+plt.xlim(0, 1)
+plt.ylim(0, 1)
+plt.xlabel("efficiency")
+plt.ylabel("purity")
+plt.colorbar().set_label("xmin")
+
+

+
+
+
+
arg = np.argmax(purity)
+print("efficiency at max purity", efficiency[arg])
+
+
efficiency at max purity 0.904
+
+
+
    +
  • How to choose cut?

    +
      +
    • Want to maximize efficiency and purity, but what trade-off between the two?
    • +
  • +
  • Often subjective choice is acceptable, because exact answer is difficult

  • +
  • Objective answer given by figure-of-merit (FOM)

    +
      +
    • Popular is Punzi FOM
    • +
    • Most FOM (including Punzi FOM) only minimize statistical uncertainty
    • +
    • Correct FOM has to minimize total uncertainty = statistical ⊕ systematic
    • +
    • Systematic uncertainty often difficult to quantify: fallback to Punzi or subjective choice
    • +
  • +
+
+
+

BDT-based cut

+
    +
  • We can do better with two cuts, on xmin and xmax

  • +
  • Space of cut parameters then already two-dimensional

  • +
  • With 2D data

    +
      +
    • Four rectangular cuts possible (4 cut parameters)
    • +
    • Infinitely many non-rectangular cuts possible
    • +
  • +
  • Use machine instead, needs to be trained on labeled data

  • +
  • Use BDT classifier from XGBoost

  • +
  • Gradient boosting machines regularly win Kaggle competitions

  • +
  • Well documented, fast, parallizable, tunable

  • +
+
+
from xgboost.sklearn import XGBClassifier
+from sklearn.model_selection import train_test_split
+
+# generate training data, ideally we have more than actual data
+background = norm(0, 1).rvs(100000, random_state=2)
+signal = norm(1, 0.1).rvs(50000, random_state=2)
+x2 = np.append(signal, background)
+
+# features in general matrix [N, K] for N samples and K features
+X = x2.reshape((len(x2), 1))
+# labels: signal=1, background=0
+y = np.append(np.ones_like(signal), np.zeros_like(background))
+
+
+
# ALWAYS do this: split into train and test datasets
+X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=3)
+
+# actual magic happens here
+clf = XGBClassifier(random_state=4)
+clf.fit(X_train, y_train);
+
+
+
fig, ax = plt.subplots(2, sharex=True)
+
+plt.sca(ax[0])
+xe = plot(X_test[:, 0], y_test)
+plt.xlabel(None)  # unset label set by `plot`
+
+plt.sca(ax[1])
+score = clf.score(X_test, y_test)
+xm = np.linspace(xe[0], xe[-1], 200)
+ps, pb = clf.predict_proba(xm).T
+
+plt.xlabel("x")
+plt.plot(xm, ps, "C0-", label="background probability")
+plt.plot(xm, pb, "C1-", label="signal probability")
+plt.legend(title=f"score={score:.2f}", frameon=False);
+
+

+
+
+

Exercise: * Vary some hyperparameters in XGBClassifier to see how score, runtime, and predicted probability changes * n_estimators try e.g. 2, 5, 10 * max_depth try e.g. 1, 5, 10 * learning_rate try e.g. 0.001, 0.05, 0.5, 1

+
    +
  • BDT predicts signal probability
  • +
  • We use this to implement our cut: select regions with high signal probability
  • +
+
+
ps = clf.predict_proba(X_test[y_test==1])[:, 1]
+pb = clf.predict_proba(X_test[y_test==0])[:, 1]
+xe = plt.hist(ps, bins=20, alpha=0.5, label="true signal")[1]
+plt.hist(pb, bins=xe, alpha=0.5, label="true background")
+plt.legend()
+plt.xlabel("predicted signal probability")
+plt.ylim(0, 5e3);
+
+

+
+
+
+
xe = plot(X_test[:, 0], y_test)
+
+# chosen cut: signal probability > 0.6
+# visualize accepted region
+xm = np.linspace(xe[0], xe[-1], 200)
+mask = clf.predict_proba(xm)[:, 1] > 0.6
+plt.plot(xm, mask * 5000, "k-");
+
+

+
+
+
+
pmin = np.linspace(0, 1, 100)
+efficiency = []
+purity = []
+for pm in pmin:
+    mask = clf.predict_proba(X_test)[:, 1] > pm
+    selected = np.sum(mask)
+    selected_signal = np.sum(y_test[mask])
+    total_signal = np.sum(y_test)
+    efficiency.append(selected_signal / total_signal)
+    with np.errstate(divide="ignore", invalid="ignore"):
+        purity.append(selected_signal / selected)
+plt.plot(pmin, efficiency, label="efficiency")
+plt.plot(pmin, purity, label="purity")
+plt.xlabel("pmin")
+plt.legend();
+
+

+
+
+
+
plt.scatter(efficiency, purity, c=pmin);
+plt.xlim(0, 1)
+plt.ylim(0, 1)
+plt.xlabel("efficiency")
+plt.ylabel("purity")
+plt.colorbar().set_label("pmin")
+
+

+
+
+
    +
  • Physicists like efficiency and purity, but ROC curve is standard
  • +
  • Receiver-operator characteristic
  • +
  • ROC plots true positive rate (fraction of signal that passes cut) over false positive rate (fraction of background that passes cut)
  • +
  • Scikit-Learn provides a quick way to generate this plot
  • +
+
+
from sklearn.metrics import RocCurveDisplay
+
+RocCurveDisplay.from_estimator(clf, X_test, y_test);
+
+

+
+
+
    +
  • area under curve (AUC) is a common metric to compare the performance of two different classifiers
  • +
  • can also use Scikit-Learn to compute ROC points and AUC and plot these ourselves
  • +
+
+
from sklearn.metrics import roc_curve, auc
+
+# plot XGBoost curve again
+fpr, tpr, thresholds = roc_curve(y_test, clf.predict_proba(X_test)[:, 1])
+plt.plot(fpr, tpr, label=f"XGBoost AUC={auc(fpr, tpr):.2f}")
+
+# plot ROC curve for our simple x > 0.7 cut for comparison
+fpr2, tpr2, _ = roc_curve(y_test, X_test[:, 0] > 0.7)
+plt.plot(fpr2, tpr2, label=f"x > 0.7 AUC={auc(fpr2, tpr2):.2f}")
+
+plt.xlabel("False positive rate")
+plt.ylabel("True positive rate")
+plt.legend();
+
+

+
+
+
    +
  • Exercise: run scikit-learn example
  • +
  • Try adding XGBClassifier into the comparison
  • +
  • Play around with the hyperparameters of the classifiers
  • +
+
+
+
+
+

Regression

+
    +
  • Regression (aka fitting): “learn” function y = f(x) from points(x, y), where both x and y can be vectors

  • +
  • Machines can do this, but we rarely need this in HEP

  • +
  • We usually prefer to use theoretically motivated models or hand-crafted low-order polynoms

  • +
  • Let’s try regression on one-dimensional function

  • +
+
+
def model(x, a, b, c, d):
+    return a + b * np.sin(c * x) * np.exp(- d * x)
+
+truth = 0.2, 1, 2, 0.5
+
+x = np.linspace(0, 10, 1000)
+plt.plot(x, model(x, *truth))
+plt.xlabel("x")
+plt.ylabel("y");
+
+

+
+
+
    +
  • generate random points on this curve and add some noise to make it more realistic
  • +
+
+
rng = np.random.default_rng(1)
+
+x = rng.uniform(0, 10, 1000)
+y = model(x, 0.2, 1, 2, 0.5)
+y += rng.normal(0, 0.05, len(y))
+
+X = np.reshape(x, (-1, 1))
+
+
    +
  • (X, y) are inputs for regression similar to before, but y is a real number now, not a label (integer)
  • +
+
+

Neural network

+
    +
  • We use MLPRegressor from Scikit-Learn, a basic neural network
  • +
  • Neural networks always need preprocessing
  • +
  • Scikit-Learn offers pipelines to automate this, use them
  • +
+
+
from sklearn.neural_network import MLPRegressor
+from sklearn.preprocessing import StandardScaler
+from sklearn.pipeline import make_pipeline
+
+# network with 5 hidden layers with 100 nodes each
+regressor = MLPRegressor(
+    hidden_layer_sizes=(100,)*5,
+    random_state=1)
+
+# we use basic StandardScaler for preprocessing
+pipeline = make_pipeline(StandardScaler(), regressor)
+pipeline.fit(X, y);
+
+
+
# compute fitted curve
+X2 = np.reshape(np.linspace(0, 10, 1000), (-1, 1))
+y2 = pipeline.predict(X2)
+
+plt.plot(x, y, ".")
+plt.plot(X2[:, 0], y2, color="r", label="Neural net")
+plt.plot(X2[:, 0], model(X2[:, 0], *truth), color="k", label="truth")
+plt.legend()
+plt.xlabel("x")
+plt.ylabel("y");
+
+

+
+
+
    +
  • Exercise: play around with hyperparameters of MLPRegressor
  • +
  • What happens if you remove the StandardScaler?
  • +
  • What performs better, deep or wide networks?
  • +
+
+
+

Gaussian Process

+
    +
  • For low-dimensional data, Gaussian Process Regression can work well
  • +
  • Not easy to set up: need to select suitable kernels and configure them
  • +
  • Advantage: Predicts uncertainty and automatically provides smooth solution
  • +
+
+
from sklearn.gaussian_process import GaussianProcessRegressor
+from sklearn.gaussian_process.kernels import RBF, WhiteKernel
+
+kernel = RBF() + WhiteKernel()
+
+gpr = GaussianProcessRegressor(kernel=kernel)
+gpr.fit(X, y)
+y3, y3err = gpr.predict(X2, return_std=True)
+
+
+
plt.plot(x, y, ".")
+plt.plot(X2[:, 0], model(X2[:, 0], *truth), color="k", label="truth")
+plt.plot(X2[:, 0], y2, color="r", label="Neural net")
+l = plt.plot(X2[:, 0], y3, label="Gaussian Process")[0]
+plt.fill_between(X2[:, 0], y3 - y3err, y3 + y3err, alpha=0.8, color=l.get_color())
+plt.legend();
+
+

+
+
+ + +
+
+ +
+ +
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+ + + + + + + + + + + + + + + +

+
+ +
+ +
+
+
+

New Fit Result Display

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

The next release of iminuit will feature a new Fit Result display.

+
    +
  • The new fit result display is less cluttered.
  • +
  • It has a new field that warns if a parameter is fitted at a limit.
  • +
  • ASCII tables were replaced with beautiful unicode tables.
  • +
  • Errors are now easier to spot in text tables.
  • +
+

You can see old vs. new below. Please commment!

+
+

Old displays

+
+
!pip install iminuit==1.4.8
+import iminuit
+
+
Collecting iminuit==1.4.8
+  Using cached iminuit-1.4.8-cp38-cp38-macosx_10_9_x86_64.whl (390 kB)
+Requirement already satisfied: numpy>=1.11.3 in /usr/local/lib/python3.8/site-packages (from iminuit==1.4.8) (1.19.0)
+Installing collected packages: iminuit
+  Attempting uninstall: iminuit
+    Found existing installation: iminuit 1.4.9
+    Uninstalling iminuit-1.4.9:
+      Successfully uninstalled iminuit-1.4.9
+Successfully installed iminuit-1.4.8
+
+
+
+

Good fit

+
+
m = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
FCN = 1Ncalls = 19 (19 total)
EDM = 0 (Goal: 0.0002)up = 1.0
Valid Min.Valid Param.Above EDMReached call limit
TrueTrueFalseFalse
Hesse failedHas cov.AccuratePos. def.Forced
FalseTrueTrueTrueFalse
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0a01
1b01
+
+
+
+
m = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)
+print(m.migrad())
+
+
------------------------------------------------------------------
+| FCN = 1                       |      Ncalls=19 (19 total)      |
+| EDM = 0 (Goal: 0.0002)        |            up = 1.0            |
+------------------------------------------------------------------
+|  Valid Min.   | Valid Param.  | Above EDM | Reached call limit |
+------------------------------------------------------------------
+|     True      |     True      |   False   |       False        |
+------------------------------------------------------------------
+| Hesse failed  |   Has cov.    | Accurate  | Pos. def. | Forced |
+------------------------------------------------------------------
+|     False     |     True      |   True    |   True    | False  |
+------------------------------------------------------------------
+------------------------------------------------------------------------------------------
+|   | Name |   Value   | Hesse Err | Minos Err- | Minos Err+ | Limit-  | Limit+  | Fixed |
+------------------------------------------------------------------------------------------
+| 0 | a    |     0     |     1     |            |            |         |         |       |
+| 1 | b    |     0     |     1     |            |            |         |         |       |
+------------------------------------------------------------------------------------------
+
+
+
+
+

Bad fits

+
+
m = iminuit.Minuit(lambda x, y : 0, pedantic=False)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
FCN = 0Ncalls = 20 (20 total)
EDM = 0 (Goal: 0.0002)up = 1.0
Valid Min.Valid Param.Above EDMReached call limit
FalseTrueFalseFalse
Hesse failedHas cov.AccuratePos. def.Forced
TrueTrueFalseFalseFalse
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x00
1y00
+
+
+
+
m = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
FCN = 8.015e-09Ncalls = 31 (31 total)
EDM = 8.01e-09 (Goal: 0.0002)up = 1.0
Valid Min.Valid Param.Above EDMReached call limit
TrueTrueFalseFalse
Hesse failedHas cov.AccuratePos. def.Forced
FalseTrueFalseFalseTrue
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0.30.801
1y016
+
+
+
+
m = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)
+print(m.migrad())
+
+
------------------------------------------------------------------
+| FCN = 8.015e-09               |      Ncalls=31 (31 total)      |
+| EDM = 8.01e-09 (Goal: 0.0002) |            up = 1.0            |
+------------------------------------------------------------------
+|  Valid Min.   | Valid Param.  | Above EDM | Reached call limit |
+------------------------------------------------------------------
+|     True      |     True      |   False   |       False        |
+------------------------------------------------------------------
+| Hesse failed  |   Has cov.    | Accurate  | Pos. def. | Forced |
+------------------------------------------------------------------
+|     False     |     True      |   False   |   False   |  True  |
+------------------------------------------------------------------
+------------------------------------------------------------------------------------------
+|   | Name |   Value   | Hesse Err | Minos Err- | Minos Err+ | Limit-  | Limit+  | Fixed |
+------------------------------------------------------------------------------------------
+| 0 | x    |    0.3    |    0.8    |            |            |    0    |    1    |       |
+| 1 | y    |     0     |    16     |            |            |         |         |       |
+------------------------------------------------------------------------------------------
+
+
+
+
+
+

New displays

+

This part requires a restart of the Jupyter kernel.

+
+
!pip install --upgrade ../iminuit
+import iminuit
+
+
Processing /Users/hdembinski/Extern/iminuit
+Requirement already satisfied, skipping upgrade: numpy>=1.11.3 in /usr/local/lib/python3.8/site-packages (from iminuit==1.4.9) (1.19.0)
+Building wheels for collected packages: iminuit
+  Building wheel for iminuit (setup.py) ... done
+  Created wheel for iminuit: filename=iminuit-1.4.9-cp38-cp38-macosx_10_15_x86_64.whl size=384673 sha256=9ba56dcf93e21dc8844737e5caf597fa22c78cc05bb431ac1b8bc3ddeca07510
+  Stored in directory: /private/var/folders/tl/pv6mt7z17tz0stm1fjfg01cc0000gn/T/pip-ephem-wheel-cache-2v6nwkbn/wheels/e7/61/b4/0dd6b8fc20e6d6b170f3e0bd185533497f2fe08c4ae74cc799
+Successfully built iminuit
+Installing collected packages: iminuit
+  Attempting uninstall: iminuit
+    Found existing installation: iminuit 1.4.9
+    Uninstalling iminuit-1.4.9:
+      Successfully uninstalled iminuit-1.4.9
+Successfully installed iminuit-1.4.9
+
+
+
+

Good fit

+
+
import iminuit
+m = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
FCN = 1Ncalls = 19 (19 total)
EDM = 0 (Goal: 0.0002)up = 1.0
Valid MinimumValid ParametersNo Parameters at limit
Below EDM goalBelow call limit
Hesse okHas CovarianceAccuratePos. def.Not forced
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0a01
1b01
+
+
+
+
m = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)
+print(m.migrad())
+
+
┌──────────────────────────────────┬──────────────────────────────────────┐
+│ FCN = 1                          │        Ncalls = 19 (19 total)        │
+│ EDM = 0 (Goal: 0.0002)           │               up = 1.0               │
+├───────────────┬──────────────────┼──────────────────────────────────────┤
+│ Valid Minimum │ Valid Parameters │        No Parameters at limit        │
+├───────────────┴──────────────────┼──────────────────────────────────────┤
+│          Below EDM goal          │           Below call limit           │
+├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤
+│   Hesse ok    │  Has Covariance  │ Accurate  │  Pos. def.  │ Not forced │
+└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘
+┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐
+│   │ Name │   Value   │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit-  │ Limit+  │ Fixed │
+├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤
+│ 0 │ a    │     0     │     1     │            │            │         │         │       │
+│ 1 │ b    │     0     │     1     │            │            │         │         │       │
+└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘
+
+
+
+
+

Bad fits

+
+
m = iminuit.Minuit(lambda x, y : 0, pedantic=False)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
FCN = 0Ncalls = 20 (20 total)
EDM = 0 (Goal: 0.0002)up = 1.0
INVALID MinimumValid ParametersNo Parameters at limit
Below EDM goalBelow call limit
Hesse FAILEDHas CovarianceAPPROXIMATENOT pos. def.Not forced
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x00
1y00
+
+
+
+
m = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
FCN = 8.015e-09Ncalls = 31 (31 total)
EDM = 8.01e-09 (Goal: 0.0002)up = 1.0
Valid MinimumValid ParametersSOME Parameters at limit
Below EDM goalBelow call limit
Hesse okHas CovarianceAPPROXIMATENOT pos. def.FORCED
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0x0.30.801
1y016
+
+
+
+
m = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)
+print(m.migrad())
+
+
┌──────────────────────────────────┬──────────────────────────────────────┐
+│ FCN = 8.015e-09                  │        Ncalls = 31 (31 total)        │
+│ EDM = 8.01e-09 (Goal: 0.0002)    │               up = 1.0               │
+├───────────────┬──────────────────┼──────────────────────────────────────┤
+│ Valid Minimum │ Valid Parameters │       SOME Parameters at limit       │
+├───────────────┴──────────────────┼──────────────────────────────────────┤
+│          Below EDM goal          │           Below call limit           │
+├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤
+│   Hesse ok    │  Has Covariance  │APPROXIMATE│NOT pos. def.│   FORCED   │
+└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘
+┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐
+│   │ Name │   Value   │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit-  │ Limit+  │ Fixed │
+├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤
+│ 0 │ x    │    0.3    │    0.8    │            │            │    0    │    1    │       │
+│ 1 │ y    │     0     │    16     │            │            │         │         │       │
+└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘
+
+
+ + +
+
+ +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Numerically stable calculation of invariant mass.html b/posts/Numerically stable calculation of invariant mass.html new file mode 100644 index 0000000..20f67cf --- /dev/null +++ b/posts/Numerically stable calculation of invariant mass.html @@ -0,0 +1,646 @@ + + + + + + + + + +Hans Dembinski’s blog - Numerically stable calculation of invariant mass + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Numerically stable calculation of invariant mass

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

The art of writing numerical stable formulas is to find a mathematically equivalent computation for a formula that produces accurate results even though computation with floating point numbers suffers from finite precision and round-off. We derive here numerically stable formulas for the calculation of the invariant mass of an ancestor particle in particle physics from the momenta and (assumed) masses of its products.

+
+

Two-body decay

+

We start with a derivation of a numerically stable formula for the 1D case, and then move to the general 3D case. The 1D solution is not applicable in practice, but it is easier to understand one of the two main numerical issues with the naive implementation when we start in 1D.

+
+

Solution in 1D

+

We start with the formula for the invariant mass squared of an ancestor particle which decays into two children,

+

\[ +M^2 = m_1^2 + m_2^2 + 2 (E_1 E_2 - \vec{p}_1 \vec{p}_2). +\]

+

For small \(m_i\), we have a catastrophic cancelation in the third term. In floating point precision, the third term can be zero, even if the masses are small but positive. Due to round-off errors, it is even possible that the third term gets negative.

+

To fix this, we rewrite the product of the energies

+

\[ +E_1 E_2 = \sqrt{|\vec{p}_1|^2 + m_1^2}\sqrt{|\vec{p}_2| + m_2^2} = |\vec{p}_1| |\vec{p}_2| \sqrt{\left(1 + \frac{m_1^2}{|\vec{p}_1|^2}\right)\left(1 + \frac{m_2^2}{|\vec{p}_2|^2}\right)} = |\vec{p}_1| |\vec{p}_2| \sqrt{1 + x} +\]

+

with

+

\[ +x = \frac{m_1^2}{|\vec{p}_1|^2} + \frac{m_2^2}{|\vec{p}_2|^2} + \frac{m_1^2}{|\vec{p}_1|^2} \frac{m_2^2}{|\vec{p}_2|^2} +\]

+

In 1D, we have (assuming parallel vectors)

+

\[ +E_1 E_2 - \vec{p}_1 \vec{p}_2 = E_1 E_2 - |\vec{p}_1||\vec{p}_2| = |\vec{p}_1||\vec{p}_2| (\sqrt{x + 1} - 1) +\]

+

The cancellation is now reduced to the expression \(\sqrt{x + 1} - 1\). With a brilliant trick from Stackoverflow, we can rewrite this as

+

\[ +\sqrt{x + 1} - 1 = \frac{(\sqrt{x + 1} - 1)(\sqrt{x + 1} + 1)}{\sqrt{x + 1} + 1} = \frac{x}{\sqrt{x + 1} + 1} +\]

+

which is numerically stable, as the following plot demonstrates.

+
+
import numpy as np
+import matplotlib.pyplot as plt
+
+x = np.geomspace(1e-8, 1e2, 1000).astype(np.float32)
+
+plt.plot(x, np.sqrt(x ** 2 + 1) - 1, label="sqrt(x^2 + 1) - 1")
+plt.plot(x, x ** 2 / (np.sqrt(x ** 2 + 1) + 1), label="x^2 / (sqrt(x^2 + 1) + 1)")
+plt.loglog()
+plt.legend();
+
+

+
+
+

The rewritten formula for \(M^2\) produces a value that is always non-negative even in the presence of round-off errors. This is a common issue with naive implementations even in double precision.

+

Below, we compare a naive implementation msq1 with the numerically stable implementation msq2 on random values. The reference is computed with the naive implementation using mpmath on floating point numbers with 100 digits of precision.

+
+
import numpy as np
+import numba as nb
+import matplotlib.pyplot as plt
+import mpmath as mp
+
+# naive implementation
+def msq1(p1, p2, m1, m2):
+    m1_sq = m1 ** 2
+    m2_sq = m2 ** 2
+    
+    # energies of particles 1 and 2
+    e1 = np.sqrt(p1 ** 2 + m1_sq)
+    e2 = np.sqrt(p2 ** 2 + m2_sq)
+
+    # dangerous cancelation in third term
+    return m1_sq + m2_sq + 2 * (e1 * e2 - p1 * p2)
+
+
+# numerically stable implementation
+@nb.vectorize
+def msq2(p1, p2, m1, m2):
+    x1 = (m1 / p1) ** 2
+    x2 = (m2 / p2) ** 2
+    x = x1 + x2 + x1 * x2
+    y = 2 * p1 * p2 * x / (np.sqrt(1 + x) + 1)
+    return m1 ** 2 + m2 ** 2 + y
+
+
+rng = np.random.default_rng(1)
+
+p1 = 10 ** rng.uniform(0, 5, size=10000).astype(np.float32)
+p2 = 10 ** rng.uniform(0, 5, size=10000).astype(np.float32)
+m1 = 10 ** rng.uniform(-5, 5, size=10000).astype(np.float32)
+m2 = 10 ** rng.uniform(-5, 5, size=10000).astype(np.float32)
+
+# as reference we use naive formula computed with mpmath
+M0 = []
+with mp.workdps(100):
+    for p1i, p2i, m1i, m2i in zip(p1, p2, m1, m2):        
+        p1i = mp.mpf(float(p1i))
+        p2i = mp.mpf(float(p2i))
+        
+        m1_sq = mp.mpf(float(m1i)) ** 2
+        m2_sq = mp.mpf(float(m2i)) ** 2
+
+        e1 = mp.sqrt(p1i ** 2 + m1_sq)
+        e2 = mp.sqrt(p2i ** 2 + m2_sq)
+
+        M0.append(float(m1_sq + m2_sq + 2 * (e1 * e2 - p1i * p2i)))
+
+M0 = np.array(M0)
+
+M = {f.__name__: f(p1, p2, m1, m2) for f in (msq1, msq2)}
+
+fig, ax = plt.subplots(1, len(M), figsize=(len(M) * 4.5, 4), sharey=True)
+for (fname, Mi), axi in zip(M.items(), ax):    
+    axi.hist((Mi - M0) / M0, bins=100)
+    axi.set_yscale("log")
+    axi.set_title(fname)
+fig.supxlabel(r"$(M'^2 - M^2) / M^2$");
+
+

+
+
+

The numerically stable formula retains full accuracy, while the naive formula suffers a catastrophic loss of accuracy for certain input values.

+
+
+

Solution in 3D

+

In 3D, the solution is more complex than in the 1D case. We have now

+

\[ +E_1 E_2 - \vec{p}_1 \vec{p}_2 = |\vec{p}_1| |\vec{p}_2| \left(\sqrt{x + 1} - \frac{\vec{p}_1 \vec{p}_2}{|\vec{p}_1| |\vec{p}_2|}\right) = |\vec{p}_1| |\vec{p}_2| \left(\sqrt{x + 1} - \cos{\alpha}\right) +\]

+

where \(\alpha\) is the angle between \(\vec{p}_1\) and \(\vec{p}_2\). We have again a numerical instability for small \(x\) and small \(\alpha\), when \(\vec{p}_1\) and \(\vec{p}_2\) are nearly parallel, which happens often in practice.

+

To avoid the cancelation in the subtraction of two numbers close to 1, we use the same trick as before, but get a different outcome

+

\[ +\sqrt{x + 1} - \cos{\alpha} = \frac{x + 1 - \cos^2{\alpha}}{\sqrt{x + 1} + \cos{\alpha}} = \frac{x + \sin^2{\alpha}}{\sqrt{x + 1} + \cos{\alpha}}. +\]

+

This time we only replace the left-hand side with the right-hand side if \(\cos{\alpha} > 0\), since for other values of the cosine, there is no subtraction happening. The expression is again guaranteed to produce a non-negative value even in the presence of round-off errors.

+

So far so good, but we face an additional issue. The naive computation of \(\alpha\) is also not numerically stable for small \(\alpha\) and we need another trick from Stackoverflow to compute it accurately,

+

\[ +\alpha = \rm{arctan2}(|\vec{p}_1 \times \vec{p}_2|, \vec{p}_1 \vec{p}_2 ) +\]

+

where \(\rm{arctan2}\) refers to the computer function with the same name.

+

We again compare a naive implementation msq1 with the numerically stable version msq2 on random inputs.

+
+
import numpy as np
+import numba as nb
+import matplotlib.pyplot as plt
+import mpmath as mp
+
+# naive implementation
+def msq1(px1, py1, pz1, px2, py2, pz2, m1, m2):
+    p1_sq = px1 ** 2 + py1 ** 2 + pz1 ** 2
+    p2_sq = px2 ** 2 + py2 ** 2 + pz2 ** 2
+    m1_sq = m1 ** 2
+    m2_sq = m2 ** 2
+    
+    # energies of particles 1 and 2
+    e1 = np.sqrt(p1_sq + m1_sq)
+    e2 = np.sqrt(p2_sq + m2_sq)
+
+    # dangerous cancelation in third term
+    return m1_sq + m2_sq + 2 * (e1 * e2 - (px1 * px2 + py1 * py2 + pz1 * pz2))
+
+
+# numerically stable implementation
+@nb.vectorize
+def msq2(px1, py1, pz1, px2, py2, pz2, m1, m2):
+    p1_sq = px1 ** 2 + py1 ** 2 + pz1 ** 2
+    p2_sq = px2 ** 2 + py2 ** 2 + pz2 ** 2
+    m1_sq = m1 ** 2
+    m2_sq = m2 ** 2
+    x1 = m1_sq / p1_sq
+    x2 = m2_sq / p2_sq
+    x = x1 + x2 + x1 * x2
+    a = angle(px1, py1, pz1, px2, py2, pz2)
+    cos_a = np.cos(a)
+    if cos_a >= 0:
+        y1 = (x + np.sin(a) ** 2) / (np.sqrt(x + 1) + cos_a) 
+    else:
+        y1 = -cos_a + np.sqrt(x + 1) 
+    y2 = 2 * np.sqrt(p1_sq * p2_sq)
+    return m1_sq + m2_sq + y1 * y2
+
+
+# numerically stable calculation of angle
+@nb.njit
+def angle(x1, y1, z1, x2, y2, z2):
+    # cross product
+    cx = y1 * z2 - y2 * z1
+    cy = x1 * z2 - x2 * z1
+    cz = x1 * y2 - x2 * y1
+    
+    # norm of cross product
+    c = np.sqrt(cx * cx + cy * cy + cz * cz)
+    
+    # dot product
+    d = x1 * x2 + y1 * y2 + z1 * z2
+    
+    return np.arctan2(c, d)
+
+
+rng = np.random.default_rng(1)
+
+dtype = np.float32
+size = 10000
+px1 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)
+py1 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)
+pz1 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)
+px2 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)
+py2 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)
+pz2 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)
+m1 = 10 ** rng.uniform(-5, 5, size=size).astype(dtype)
+m2 = 10 ** rng.uniform(-5, 5, size=size).astype(dtype)
+
+# as reference we use naive formula computed with mpmath
+M0 = []
+with mp.workdps(100):
+    for px1i, py1i, pz1i, px2i, py2i, pz2i, m1i, m2i in zip(px1, py1, pz1, px2, py2, pz2, m1, m2):
+        px1i = mp.mpf(float(px1i))
+        py1i = mp.mpf(float(py1i))
+        pz1i = mp.mpf(float(pz1i))
+        px2i = mp.mpf(float(px2i))
+        py2i = mp.mpf(float(py2i))
+        pz2i = mp.mpf(float(pz2i))
+        
+        m1_sq = mp.mpf(float(m1i)) ** 2
+        m2_sq = mp.mpf(float(m2i)) ** 2
+
+        e1 = mp.sqrt(px1i ** 2 + py1i ** 2 + pz1i ** 2 + m1_sq)
+        e2 = mp.sqrt(px2i ** 2 + py2i ** 2 + pz2i ** 2 + m2_sq)
+
+        M0.append(float(m1_sq + m2_sq + 2 * (e1 * e2 - (px1i * px2i + py1i * py2i + pz1i * pz2i))))
+
+M0 = np.array(M0)
+
+M = {f.__name__: f(px1, py1, pz1, px2, py2, pz2, m1, m2) for f in (msq1, msq2)}
+
+fig, ax = plt.subplots(1, len(M), figsize=(len(M) * 4.5, 4), sharey=True)
+for (fname, Mi), axi in zip(M.items(), ax):
+    d = (Mi - M0) / M0
+    axi.hist(d, bins=100)
+    axi.set_yscale("log")
+    axi.set_title(f"{fname}")
+fig.supxlabel(r"$(M'^2 - M^2) / M^2$", y=-0.05);
+
+

+
+
+

We again see great loss of accuracy for the naive implementation, while the numerically stable formula produces accurate results.

+ + +
+
+ +
+ +
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literal 0 HcmV?d00001 diff --git a/posts/RooFit.html b/posts/RooFit.html new file mode 100644 index 0000000..97649c8 --- /dev/null +++ b/posts/RooFit.html @@ -0,0 +1,518 @@ + + + + + + + + + +Hans Dembinski’s blog - RooFit with Chebyshev and Bernstein polynomials + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

RooFit with Chebyshev and Bernstein polynomials

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

This is a little demonstration of how to fit a peak plus smooth background with either an empirical Chebyshev polynomial for the background or a Bernstein polynomial. The Bernstein polynomial is technically superior, because it allows us to ensure that the background density is always positive with a simple box constrain on its parameters. That is not possible with Chebyshev.

+

This example is based on the RooFit Tutorial. I generate a toy model which is an additive mixture of a normal distribution for a peak and a polynomial background. I draw a sample, and fit two variations of the toy model, one with either polynomial. The fitted lines look identical after the fit, as expected. The result of the maximum likelihood method is invariant to reparametrisations.

+
+
import ROOT as R
+
+
Welcome to JupyROOT 6.30/02
+
+
+
+
x = R.RooRealVar("x", "x", -10, 10)
+
+mu1 = R.RooRealVar("mu1", "mu", 0, -10, 10)
+sigma1 = R.RooRealVar("sigma1", "sigma", 1, 0.01, 1e2)
+gauss1 = R.RooGaussian("gauss1", "Gauss", x, mu1, sigma1)
+
+p0 = R.RooRealVar("p0", "p_0", 1, 0, 1e2)
+p1 = R.RooRealVar("p1", "p_1", 0.2, 0, 1e2)
+cheb = R.RooChebychev("cheb", "Chebyshev", x, R.RooArgList(p0, p1))
+
+mu2 = R.RooRealVar("mu2", "mu", 0, -10, 10)
+sigma2 = R.RooRealVar("sigma2", "sigma", 1, 0.01, 1e2)
+gauss2 = R.RooGaussian("gauss2", "Gauss", x, mu2, sigma2)
+
+fix = R.RooRealVar("one", "one", 0.1)
+q0 = R.RooRealVar("q0", "q_0", 1, 0, 1e2)
+q1 = R.RooRealVar("q1", "q_1", 0.2, 0, 1e2)
+bern = R.RooBernstein("bern", "Bernstein", x, R.RooArgList(fix, q0, q1))
+
+f1 = R.RooRealVar("f1", "signal fraction", 0.3, 0, 1)
+model1 = R.RooAddPdf("sum1", "S+B model", R.RooArgList(gauss1, cheb), f1)
+
+data = model1.generate(R.RooArgSet(x), 1000)
+
+model1.fitTo(data)
+
+f2 = R.RooRealVar("f2", "signal fraction", 0.3, 0, 1)
+model2 = R.RooAddPdf("sum2", "S+B model", R.RooArgList(gauss2, bern), f2)
+model2.fitTo(data, R.RooFit.PrintLevel(-1))
+
+c = R.TCanvas()
+frame = x.frame()
+data.plotOn(frame)
+model1.plotOn(frame)
+model2.plotOn(frame, R.RooFit.LineStyle(R.kDashed), R.RooFit.LineColor(R.kRed))
+model2.paramOn(frame)
+frame.Draw()
+
+
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
+[#1] INFO:Minimization --  The following expressions will be evaluated in cache-and-track mode: (gauss1,cheb)
+Minuit2Minimizer: Minimize with max-calls 2500 convergence for edm < 1 strategy 1
+Minuit2Minimizer : Valid minimum - status = 0
+FVAL  = 2737.75116242620834
+Edm   = 0.000269023969144123421
+Nfcn  = 100
+f1    = 0.275394     +/-  0.0242084 (limited)
+mu1   = -0.0175247   +/-  0.091251  (limited)
+p0    = 0.990493     +/-  0.0455302 (limited)
+p1    = 0.103726     +/-  0.0587728 (limited)
+sigma1    = 1.01262  +/-  0.0914058 (limited)
+[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
+[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
+[#1] INFO:Minimization --  The following expressions will be evaluated in cache-and-track mode: (gauss2,bern)
+[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
+
+
+
Info in <Minuit2>: MnSeedGenerator Computing seed using NumericalGradient calculator
+Info in <Minuit2>: MnSeedGenerator Initial state: FCN =       2739.667437 Edm =       1.767793129 NCalls =     21
+Info in <Minuit2>: MnSeedGenerator Initial state  
+  Minimum value : 2739.667437
+  Edm           : 1.767793129
+  Internal parameters:  [    -0.4115168461                0     -1.370461484     -1.481323767     -1.371459022] 
+  Internal gradient  :  [      15.50719084      34.67327852      -68.9043889      127.4588967     -139.7445175] 
+  Internal covariance matrix:
+[[   0.0035221149              0              0              0              0]
+ [              0   0.0001480147              0              0              0]
+ [              0              0  4.3539642e-05              0              0]
+ [              0              0              0  0.00023511592              0]
+ [              0              0              0              0  0.00010343293]]]
+Info in <Minuit2>: VariableMetricBuilder Start iterating until Edm is < 0.001 with call limit = 2500
+Info in <Minuit2>: VariableMetricBuilder    0 - FCN =       2739.667437 Edm =       1.767793129 NCalls =     21
+Info in <Minuit2>: VariableMetricBuilder    1 - FCN =       2738.073461 Edm =      0.1061417478 NCalls =     33
+Info in <Minuit2>: VariableMetricBuilder    2 - FCN =       2737.800519 Edm =     0.03895269349 NCalls =     46
+Info in <Minuit2>: VariableMetricBuilder    3 - FCN =       2737.753549 Edm =    0.002339429787 NCalls =     58
+Info in <Minuit2>: VariableMetricBuilder    4 - FCN =       2737.751162 Edm =   0.0002368589638 NCalls =     69
+Info in <Minuit2>: VariableMetricBuilder After Hessian
+Info in <Minuit2>: VariableMetricBuilder    5 - FCN =       2737.751162 Edm =   0.0002690239691 NCalls =    100
+Info in <Minuit2>: Minuit2Minimizer::Hesse Using max-calls 2500
+Info in <Minuit2>: Minuit2Minimizer::Hesse Hesse is valid - matrix is accurate
+
+
+
+
model1.getParameters(data).Print("v")
+
+
  1) 0x7f7e8f6c9310 RooRealVar::     f1 = 0.275394 +/- 0.0241924  L(0 - 1)  "signal fraction"
+  2) 0x7f7e8eff2700 RooRealVar::    mu1 = -0.0175247 +/- 0.0912425  L(-10 - 10)  "mu"
+  3) 0x7f7e8f2b7960 RooRealVar::     p0 = 0.990493 +/- 0.0454892  L(0 - 100)  "p_0"
+  4) 0x7f7e8f692e30 RooRealVar::     p1 = 0.103726 +/- 0.058683  L(0 - 100)  "p_1"
+  5) 0x7f7e8eff2a90 RooRealVar:: sigma1 = 1.01262 +/- 0.0913868  L(0.01 - 100)  "sigma"
+
+
+
+
model2.getParameters(data).Print("v")
+
+
  1) 0x7f7e8fda7940 RooRealVar::     f2 = 0.275478 +/- 0.0241593  L(0 - 1)  "signal fraction"
+  2) 0x7f7e8fd050e0 RooRealVar::    mu2 = -0.0183764 +/- 0.0911997  L(-10 - 10)  "mu"
+  3) 0x7f7e8fd08f60 RooRealVar::    one = 0.1 C  L(-INF - +INF)  "one"
+  4) 0x7f7e8fd092f0 RooRealVar::     q0 = 0.604314 +/- 0.403307  L(0 - 100)  "q_0"
+  5) 0x7f7e8fd15aa0 RooRealVar::     q1 = 1.84239 +/- 0.845585  L(0 - 100)  "q_1"
+  6) 0x7f7e8fd08bd0 RooRealVar:: sigma2 = 1.01227 +/- 0.0912609  L(0.01 - 100)  "sigma"
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/SPD with negative weights.html b/posts/SPD with negative weights.html new file mode 100644 index 0000000..79dec5f --- /dev/null +++ b/posts/SPD with negative weights.html @@ -0,0 +1,486 @@ + + + + + + + + + +Hans Dembinski’s blog - Applying the SPD approximation to negative weights + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Applying the SPD approximation to negative weights

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

In Bohm and Zech, NIMA 748 (2014) 1-6, the scaled Poisson distribution is discussed, which is an approximate distribution for sums of weights. This distribution is used in fits to histograms which are filled with weighted samples.

+

I demonstrate here that a fit based on the SPD performs better than a least-squares fit, which effectively uses a normal distribution instead of the SPD to describe random fluctuations. I run repeated simulations in which weighted samples are generated. They are filled into a generalized histogram that keeps track of the variance of the weights. The cost function for the fit is simple: all bins in the histogram have the same constant expectation. The cost is computed with the least-squares method and the SPD method, respectively.

+

I measure the bias and the variance of the results in repeated toy experiments. The SPD-based cost function yields a smaller bias and the estimated uncertainty better reflects the true variance of the fit results.

+
+
import numpy as np
+import numba as nb
+import matplotlib.pyplot as plt
+from iminuit import Minuit
+from scipy.stats import norm
+import boost_histogram as bh 
+
+
+
@nb.njit
+def cost_lsq(w, wvar, mu):
+    r = 0.0
+    for i in range(len(w)):
+        if wvar[i] > 0:
+            r += (w[i] - mu) ** 2 / wvar[i]
+    return r
+
+
+@nb.njit
+def cost_ml(w, wvar, mu):
+    # Bohm and Zech, NIMA 748 (2014) 1-6
+    r = 0.0
+    for i in range(len(w)):
+        if wvar[i] > 0 and w[i] * mu > 0:
+            s_inv = w[i] / wvar[i]
+            mu_prime = mu * s_inv
+            n_prime = w[i] * s_inv      
+            r += mu_prime - n_prime * np.log(mu_prime)
+    return 2 * r
+
+
+
nmc = 1000
+
+def run(cost, w_mu, w_sigma_rel, mu_pts, bins):
+    expected_mu = w_mu * mu_pts / bins
+    rng = np.random.default_rng(seed=1)
+    fitted_mu = []
+    fitted_mu_var = []
+    h = bh.Histogram(bh.axis.Regular(bins, 0, 1), storage=bh.storage.Weight())
+    for imc in range(nmc):
+        h.reset()
+        n = rng.poisson(mu_pts)
+        x = rng.uniform(size=n)
+        wi = rng.normal(w_mu, w_sigma_rel * w_mu, size=n)
+        h.fill(x, weight=wi)
+        m = Minuit(lambda mu: cost(h.values(), h.variances(), mu), mu=expected_mu)
+        m.limits["mu"] = (0, None)
+        m.migrad()
+        if m.valid:
+            fitted_mu.append(m.values[0])
+            fitted_mu_var.append(m.errors[0] ** 2)
+        else:
+            print("error", imc)
+    
+    bias = np.mean(fitted_mu) - expected_mu
+    sample_var = np.var(fitted_mu, ddof=1)
+    mean_fitted_var = np.mean(fitted_mu_var)
+    return expected_mu, bias, mean_fitted_var, sample_var
+
+results = tuple(map(lambda cost: run(cost, 1.0, 0.1, 100, 5), (cost_lsq, cost_ml)))
+results = np.array(results)
+
+labels = "LSQ", "ML"
+for label, (expected_mu, bias, mean_fitted_var, sample_var ) in zip(labels, results):
+    print(label)
+    print(f"  relative bias = {bias / expected_mu:.3f}")
+    print(f"  average estimated variance {mean_fitted_var ** 0.5:.3f}, true variance {sample_var ** 0.5:.3f}")
+
+
LSQ
+  relative bias = -0.042
+  average estimated variance 1.966, true variance 2.158
+ML
+  relative bias = -0.001
+  average estimated variance 2.008, true variance 2.035
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Simple parallelization in Jupyter Notebooks.html b/posts/Simple parallelization in Jupyter Notebooks.html new file mode 100644 index 0000000..cc2e1ea --- /dev/null +++ b/posts/Simple parallelization in Jupyter Notebooks.html @@ -0,0 +1,517 @@ + + + + + + + + + +Hans Dembinski’s blog - Simple parallelization in Jupyter Notebooks + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Simple parallelization in Jupyter Notebooks

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

The usual modules concurrent.futures and multiprocessing do not work correctly in notebooks on all platforms (notably on OSX there are issues). What does work is joblib, which also offers a simple way to run embarrassingly parallel jobs.

+
+
from joblib import Parallel, delayed
+import numpy as np
+
+
+
# silly function that simulates numerical workload
+def run(n):
+    r = 0
+    rng = np.random.default_rng(1)
+    for itry in range(1000):
+        x = rng.normal(size=n)
+        r += np.sum(x)
+    return r
+
+n = np.full((4,), 100000)
+y = Parallel(8)(delayed(run)(ni) for ni in n)
+
+
+
import time
+import matplotlib.pyplot as plt
+
+
+
njob = np.arange(1, 9)
+n = (10 ** np.arange(2, 6, 0.5)).astype(int)
+data = np.empty((len(njob), len(n)))
+for i, njobi in enumerate(njob):
+    for j, nj in enumerate(n):
+        t = time.perf_counter()
+        n_per_job = nj // njobi
+        Parallel(njobi)(delayed(run)(ni) for ni in np.full(njobi, n_per_job))
+        t = time.perf_counter() - t
+        data[i, j] = t
+        print(njobi, nj, t)
+
+
1 100
+1 316
+1 1000
+1 3162
+1 10000
+1 31622
+1 100000
+1 316227
+2 100
+2 316
+2 1000
+2 3162
+2 10000
+2 31622
+2 100000
+2 316227
+3 100
+3 316
+3 1000
+3 3162
+3 10000
+3 31622
+3 100000
+3 316227
+4 100
+4 316
+4 1000
+4 3162
+4 10000
+4 31622
+4 100000
+4 316227
+5 100
+5 316
+5 1000
+5 3162
+5 10000
+5 31622
+5 100000
+5 316227
+6 100
+6 316
+6 1000
+6 3162
+6 10000
+6 31622
+6 100000
+6 316227
+7 100
+7 316
+7 1000
+7 3162
+7 10000
+7 31622
+7 100000
+7 316227
+8 100
+8 316
+8 1000
+8 3162
+8 10000
+8 31622
+8 100000
+8 316227
+
+
+
+
for i, njobi in enumerate(njob):
+    plt.plot(n, data[0] / data[i], label=f"{njobi}")
+plt.loglog()
+plt.legend();
+
+

+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Simple parallelization in Jupyter Notebooks_files/figure-html/cell-6-output-1.png b/posts/Simple parallelization in Jupyter Notebooks_files/figure-html/cell-6-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..11934e373ce693f352147d1245048bc49610c9f2 GIT binary patch literal 22880 zcmafaWmH^Ivn=lJP9}H~+@0Vq3GVI=!QGt%cL*Mw;O_1a+#lP3c??e;u$G7dDXKWHCvnf`X6` zZq|w>_$D4+n1yiy>EaYOTlFg4$ISOqS;IFZ1dM>gL(ALKRT&u>WK2vnXssQ%c3h6(oM*iZH+` ziccRV;VsE?(ja+ZU~AMZ{2^o z=!ZGz6}&bj$gGA{T1-@-x!Giapy{YJ+Yn|h_%G0UK*4-fQ5nzV9ccG{0v{fJq^1s` z$#M(mZsmizx#3!K$G4kR>{L_11Rs<5R0?pwZQlGwMbZ9V2;y+zdXBdEGf#qX?efG< zqZo`~UeRgq24B%m?orlMaa1%<2R<=Habevf;eC`uhHD}8&L?bJ4FGjV8MeA6iwr68 z#8Jwbmy|f(9&6m5EK4XTpfmseJ=(A+X<^y#`pUPqq!~1Uzp`N7n~@n!#fa|YBarmG z(FS5+QCHO?Z|@iPLPfnNCcc>E+_19lT%VN8$;0l$@;8hWHj5UwQxGLjj}Z7m)`t^= zfj3!!YSvfWg0(ikl0GG&IT@O&=zQnWSV&Sp0X+GS%DApNq(#y)+}e5*eubC^_EkIo 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zcZ*~i`>~1*Ip8w(G}msW)I*216$ZPKT_Qu0Dw3!Rq0qb$`=h>wt}L}P!H|`+@%tOl zSI#q2X&mcjJS`?9tgr*oAKpg)XjTzy>#I^isYn$z;@$IyzrII>BGpu*xyhki1poWe zs?m(0jP2Fz2FXmbiJK+OhP)9YWX@Lz{^L#to4PHwS9d>R45HHG@7`S>F5~@c34eqQ zB>sa38?v@iD3UvKEXr$Bdep~4s@>XS$E!@hx_%PsAcU&jNzWA0MrGCsMNpGGCY_`b z4u#V4^t`fD66C0UOuW@f16Xu?d>+w-2?}`AJqG)(Cg}fC0sg + + + + + + + + +Hans Dembinski’s blog - Power consumption in sleep mode + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Power consumption in sleep mode

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

Some people make a big fuss about machines consuming power in standby. While it is very important to conserve energy, it makes no sense to turn off all devices but then run the air conditioner during the full night or run the car for a few minutes before you leave to heat it up. It is important to save energy where it really matters.

+

Here I compute that the CO2 emissions produced by a MacBook in sleep mode over a year are equal to burning 900 g of wood. So if you are worried about this, put one log less into the fireplace on Christmas, and it is compensated.

+
+
kg_of_co2_for_burning_kg_of_wood = 1.9
+kg_of_co2_per_wh = 0.94e-3 
+macbook_power_consumption_off_in_watt = 0.07
+macbook_power_consumption_sleep_mode_in_watt = 0.34
+
+
+
sleep_hours_per_year = 52 * (5 * 10 + 2 * 24)
+
+
+
macbook_power_consumption_sleep_mode_in_wh_per_year = sleep_hours_per_year * macbook_power_consumption_sleep_mode_in_watt
+
+
+
macbook_power_consumption_sleep_mode_in_wh_per_year * kg_of_co2_per_wh / kg_of_co2_for_burning_kg_of_wood
+
+
0.8572008421052631
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/UnbiasedEML.html b/posts/UnbiasedEML.html new file mode 100644 index 0000000..7e09364 --- /dev/null +++ b/posts/UnbiasedEML.html @@ -0,0 +1,471 @@ + + + + + + + + + +Hans Dembinski’s blog - Unbiasedness of EML fit for a mixture model with fixed component pdfs + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Unbiasedness of EML fit for a mixture model with fixed component pdfs

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

The log-likelihood for one observed Poisson-distributed count is (without constants) \[ +\ell_i(\lambda) := \ln \mathcal{L}_i(\lambda) = -\lambda + k_i \ln \lambda. +\]

+

The ML estimate is obtained by solving the score function \(\sum_i \partial \ell_i / \partial \lambda \overset!= 0\). The solution \(\hat\lambda = \frac1N \sum_i k_i\) is unbiased, \[ +E[\hat \lambda] = \frac1N \sum_i E[k_i] = \lambda. +\]

+

This can be generalised: if the estimator is a linear function of the observations, the estimator is unbiased.

+

We now consider a two component model, where the number density is \(\rho(x) = S f_s(x) + B f_b(x)\).

+

For one bin \(i\), the expectation is \(\lambda_i = S s_i + B b_i\) with constants \(s_i = \int_{a_i}^{b_i} f_s(x) dx\) and \(b_i = \int_{a_i}^{b_i} f_b(x) dx\).

+

The score functions are \[ +\ell_i(S, B) = -(S s_i + B b_i)/N + k_i \ln(S s_i + B b_i) +\] \[ +\frac{\partial \sum_i \ell_i}{\partial S} = -1 + \sum_i k_i \frac{s_i}{S s_i + B b_i} \overset!= 0 \quad\text{and}\quad +\frac{\partial \sum_i \ell_i}{\partial B} = -1 + \sum_i k_i \frac{b_i}{S s_i + B b_i} \overset!= 0. +\]

+

We calculate the solution explicitly for two bins. The calculation is fairly involved, although the solution is simple.

+
+
from sympy import *
+
+S, B, s1, s2, b1, b2, k1, k2 = symbols("S B s_1 s_2 b_1 b_2 k_1 k_2", real=True, nonnegative=True)
+
+
+
eq1 = Eq(k1 * s1 / (S * s1 + B * b1) + k2 * s2 / (S * s2 + B * b2), 1)
+eq1
+
+

\(\displaystyle \frac{k_{1} s_{1}}{B b_{1} + S s_{1}} + \frac{k_{2} s_{2}}{B b_{2} + S s_{2}} = 1\)

+
+
+
+
eq2 = Eq(k1 * b1 / (S * s1 + B * b1) + k2 * b2 / (S * s2 + B * b2), 1)
+eq2
+
+

\(\displaystyle \frac{b_{1} k_{1}}{B b_{1} + S s_{1}} + \frac{b_{2} k_{2}}{B b_{2} + S s_{2}} = 1\)

+
+
+
+
sols = solve((eq1, eq2), (S, B))
+sols
+
+
[((b_1*b_2*k_1 + b_1*b_2*k_2 - b_1*k_2*s_2 - b_2*k_1*s_1)/((b_1 - s_1)*(b_2 - s_2)),
+  (-b_1*k_1*s_2 - b_2*k_2*s_1 + k_1*s_1*s_2 + k_2*s_1*s_2)/((b_1 - s_1)*(b_2 - s_2)))]
+
+
+
+
for sol in sols:
+    for k, s in zip((S, B), sol):
+        print(f"Estimate for {k}:")
+        display(s)
+
+
Estimate for S:
+Estimate for B:
+
+
+

\(\displaystyle \frac{b_{1} b_{2} k_{1} + b_{1} b_{2} k_{2} - b_{1} k_{2} s_{2} - b_{2} k_{1} s_{1}}{\left(b_{1} - s_{1}\right) \left(b_{2} - s_{2}\right)}\)

+
+
+

\(\displaystyle \frac{- b_{1} k_{1} s_{2} - b_{2} k_{2} s_{1} + k_{1} s_{1} s_{2} + k_{2} s_{1} s_{2}}{\left(b_{1} - s_{1}\right) \left(b_{2} - s_{2}\right)}\)

+
+
+

These estimates are linear functions of the counts \(k_1\) and \(k_2\) and therefore unbiased.

+

What is left to show: that we also get a linear (and thus unbiased) solution for more than two bins.

+

Unbiasedness for an unbinned EML fit follows, since the unbinned EML fit is a limiting case of the binned EML fit as the bin width goes to zero.

+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/UncertaintiesLecture.html b/posts/UncertaintiesLecture.html new file mode 100644 index 0000000..7411f6d --- /dev/null +++ b/posts/UncertaintiesLecture.html @@ -0,0 +1,1423 @@ + + + + + + + + + +Hans Dembinski’s blog - LHCb Startertalk on Uncertainties + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

LHCb Startertalk on Uncertainties

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

by Hans Dembinski, TU Dortmund

+
+
+

+
image.png
+
+
+
+

What is an uncertainty, really?

+

We write \(a = 3 \pm 0.5\), where 0.5 is the standard error (jargon: “error” = uncertainty).

+

What does this truely mean?

+
    +
  1. The true value given by nature is 3, but when we measure it, it fluctuates (e.g. Heisenberg uncertainty principle).

  2. +
  3. I made an experiment and this expresses my belief that the value is most likely 3 and is located within the interval 2.5 to 3.5 with 68 % probability. This is based on the data that I recorded and my initial assumption that the value of \(a\) is positive.

  4. +
  5. Someone made an experiment and their estimate for the true value of \(a\) based on the data is 3. The interval (2.5, 3.5) contains the true value with 68 % probability.

  6. +
+
    +
    1. +
    2. Is nonsense, but b) and c) are valid interpretations
    3. +
  • +
  • In HEP, we generally mean c), because we compute uncertainties based on the Frequentist school of statistics
  • +
    1. +
    2. follows from the Bayesian school of statistics
    3. +
  • +
  • In b) or c), what does “68% probability” really mean in each context?
  • +
+
+
+

Statistical and systematic uncertainties

+
    +
  • Either type can be correlated or uncorrelated
  • +
+
+

Statistical uncertainty

+
    +
  • Origin: We use finite sample instead of infinite distribution

  • +
  • Goes down as simple size increases

  • +
  • Example

    +
      +
    • Arithmetic mean \(\bar x = \frac 1N\sum_i x_i\)
    • +
    • Variance of mean $ = {N(N - 1)}_i (x_i - x)^2$
    • +
  • +
  • We have reliable standard recipes to calculate these

  • +
  • Variance or standard deviation?

    +
      +
    • Standard deviation \(\sigma = \sqrt{V}\)
    • +
    • We communicate standard deviations, but compute variances
    • +
    • Variance is more fundamental +
        +
      • Second central moment of distribution
      • +
      • \(V = V_1 + V_2\), but \(\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}\)
      • +
    • +
  • +
+
+
+

Systematic uncertainty

+
    +
  • Origin: Imperfect appartus or technique, deviations of reality from simplified models
  • +
  • Quantifies potential mistakes in analysis or lack of knowledge
  • +
  • Does not go down as sample size increases
  • +
  • Correlated examples +
      +
    • Real-life: Measured the length of ten shoes with ruler than has factory tolerance of 0.1 mm
    • +
    • LHCb: Measured cross-sections with value of luminosity that has uncertainty of 1-2%
    • +
  • +
  • Uncorrelated examples +
      +
    • Real-life: ???
    • +
    • LHCb: Measured momentum of particle in LHCb differs from true momentum \(\sigma_p/p \approx 0.5\!-\!1\%\)
    • +
  • +
  • Conceptually different, requires different math
  • +
  • Not well covered in statistics books, but some theoretical results available
  • +
  • Many misconceptions
  • +
+
+
+

Statistical errors can be systematic errors

+
    +
  • Depends on context
  • +
  • Statistical error of parameter in control study is systematic error in main study
  • +
  • Example: LHCb performs calibration measurements to determine tracking and PID efficiencies +
      +
    • Statistical errors in calibration measurements are systematic errors in main measurement
    • +
  • +
+
+
+
+

Statistical uncertainties: Basic concepts

+

We need to distinguish between true value, estimator, and estimate (or estimated value).

+
    +
  • Sample \(\{ x_i \}\) drawn from distribution \(F(x)\) consists of random values \(x_i\) that are independent and identically distributed (iid)
  • +
  • Sample values occur with probability +
      +
    • \(P(k) = F(k)\) if \(k\) discrete
    • +
    • \(d P(x) = d F(x) = f(x) dx\) if \(x\) continuous
    • +
  • +
  • \(f(x)\) probability density function (pdf)
  • +
  • \(F(x)\) cumulative density function (cdf)
  • +
  • Will refer to \(f(x)\) and \(F(x)\) as the distribution
  • +
  • Beware lax jargon: distribution != sample
  • +
+
+
# Example: standard normal distribution ("Gaussian")
+from matplotlib import pyplot as plt
+import numpy as np
+from scipy.stats import norm
+x = np.linspace(-5, 5, 1000)
+fx = norm.pdf(x)
+Fx = norm.cdf(x)
+sample = norm.rvs(size=50, random_state=1)
+
+
+
def plot():
+    fig, ax = plt.subplots(1, 2, figsize=(14, 5), sharex=True)
+    for axi, y, label in zip(ax, (fx, Fx), ("f(x), pdf", "F(x), cdf")):
+        axi.plot(x, y, label=label)
+        axi.set_xlabel("x")
+    ax[0].plot(sample, np.zeros_like(sample), "|", label="sample")
+    for axi in ax: axi.legend()
+
+
+
plot()
+
+

+
+
+
    +
  • Distribution usually depends on parameters that we are interested in +
      +
    • Example: Normal distribution depends on parameters \(\mu, \sigma\) (location, width)
    • +
  • +
  • Notation: \(f(x; a, b, \dots)\), semicolon separates random variable \(x\) from parameters \(a, b, \dots\)
  • +
  • In general, \(x\) can be a vector and parameters can be written as vector, too: \(f(\vec x; \vec a)\)
  • +
+
+

Estimation

+
    +
  • Estimation is statistician speak for “measurement”
  • +
  • We have sample \(\{ x_i \}\) which is distributed according to \(f(x; a)\), \(a\) is unknown
  • +
  • Estimator \(h[x]\) maps data sample to estimate \(\hat a\) of \(a\)
  • +
  • Notation +
      +
    • \(h[x]\) estimator maps sample \(\{ x_i \}\) to value
    • +
    • \(f(x)\) is function that maps value to value
    • +
  • +
  • Examples of estimators of location of distribution +
      +
    • Arithmetic mean \(h_1[x] = \frac 1N \sum_i x_i\)
    • +
    • Sample median \(h_2[x]\): Sort \(x_i\), pick value in the middle
    • +
  • +
  • Estimator can be formula in closed form or an algorithm (recipe)
  • +
+
+
# Example: normal distribution with true value mu=1, sigma=0.5
+# mu is of interest; sigma is not of interest, a nuisance parameter
+
+from matplotlib import pyplot as plt
+import numpy as np
+from scipy.stats import norm
+
+mu, sigma = 1, 0.5
+dist = norm(mu, sigma)
+x = np.linspace(-1, 3, 1000)
+sample = dist.rvs(size=20, random_state=1)
+
+est1 = np.mean(sample)
+est2 = np.median(sample)
+
+
+
def plot():
+    plt.plot(x, dist.pdf(x))
+    plt.xlabel("x")
+    plt.plot(sample, np.zeros_like(sample), "|")
+    plt.axvline(dist.stats()[0], label="$\mu$", color="C3")
+    plt.axvline(est1, ls="--", label="mean", color="C2")
+    plt.axvline(est2, ls=":", label="median", color="C4")
+    plt.legend();
+
+
+
plot()
+
+

+
+
+
+
+

Optimal estimators

+
    +
  • Many kinds of estimators, some better than others
  • +
  • We want optimal estimators
  • +
  • Some desirable properties +
      +
    • As close to true value as possible (we come back to that)
    • +
    • Easy to compute and/or easy to apply to any problem
    • +
    • Robustness against outliers
    • +
  • +
  • In HEP, we mainly use +
      +
    • Maximum-likelihood estimator (MLE)
    • +
    • Plug-in estimator
    • +
    • Bootstrap estimator (meta estimator)
    • +
  • +
+
+
+

How far away is estimate from true value?

+
    +
  • Depends on estimator, sample size, distribution
  • +
  • Almost magic: accuracy of estimator can be also estimated from sample!
  • +
  • Define “close to true value” +
      +
    • In experiment, we have sample \(\{ x_i \}\) with size \(N\) and obtained estimate \(\hat a = h[x]\)
    • +
    • Sample \(\{ x_i \}\) is random and so is estimate \(\hat a\)
    • +
    • If experiment is identically repeated \(K\) times, we get sample \(\{ x_i \}_k\) and estimate \(\hat a_k\) each time
    • +
    • Quantify properties of sample of estimates \(\{ \hat a_1, \hat a_2, \dots, \hat a_K \}\) to describe accuracy of estimator with bias and variance estimates +
        +
      • Bias: \(\hat B[\hat a] = \frac1K \sum_i (\hat a_i - a)\)
      • +
      • Variance: \(\hat V[\hat a] = \frac1K\sum_i (\hat a_i - a)^2\)
      • +
    • +
  • +
  • Example for a biased estimator: if true mean \(\mu\) is unknown, variance estimate \(\hat V[x] = \frac1N\sum_i (x_i - \hat\mu)^2\) is biased
  • +
+
+
from matplotlib import pyplot as plt
+import numpy as np
+from scipy.stats import norm
+
+# true values and distribution
+mu, sigma = 1, 1
+dist = norm(mu, sigma)
+
+N, K = 10, 1000
+var_1, var_2 = [], []
+for k in range(K):
+    sample = dist.rvs(size=N, random_state=k+1)
+    hat_mu = np.mean(sample)
+    v1 = np.mean((sample - np.mean(sample)) ** 2)
+    v2 = np.mean((sample - mu) ** 2)
+    var_1.append(v1)
+    var_2.append(v2)
+
+
+
# hidden plotting code used in next cell
+def plot():
+    mu, sigma = dist.stats()
+    fig, ax = plt.subplots(1, 2, sharex=True, sharey=True, figsize=(14, 5))
+    ax[0].hist(var_1, bins=50, label="$\hat V[x; \hat \mu]$");
+    ax[1].hist(var_2, bins=50, label="$\hat V[x; \mu]$");
+    for axi in ax:
+        axi.axvline(sigma ** 2, color="C3", label="true V[x]")
+    ax[0].axvline(np.mean(var_1), color="C1", ls="--", label="mean of $\hat V[x; \hat \mu]$")
+    ax[1].axvline(np.mean(var_2), color="C1", ls="--", label="mean of $\hat V[x; \mu]$")
+    for axi in ax:
+        axi.legend()
+
+
+
plot()
+
+

+
+
+
    +
  • Perhaps surprising that \(\hat V[x; \hat \mu]\) is biased although \(\hat \mu\) is an unbiased estimate of \(\mu\)
  • +
  • In this case, simple well-known bias correction: \(\frac N{N-1}\hat V[x; \hat \mu]\) is unbiased (for derivation see e.g. F. James book (references at end))
  • +
+
+
# Example for variance of estimator: variance of mean and median
+from matplotlib import pyplot as plt
+import numpy as np
+from scipy.stats import norm, t
+
+mu, sigma = 0, 1
+
+dist = {
+    "norm": norm(mu, sigma),
+    "t": t(2, mu, sigma),
+} 
+
+N, K = 10, 1000
+data = {}
+for dkey, d in dist.items():
+    est = {"mean": [], "median": []}
+    all_samples = []
+    for k in range(K):
+        sample = d.rvs(size=N, random_state=k+1)
+        est["mean"].append(np.mean(sample))
+        est["median"].append(np.median(sample))
+        all_samples.append(sample)
+    data[dkey] = est, all_samples
+
+
+
def plot():
+    for dkey, (est, all_samples) in data.items():
+        fig, ax = plt.subplots(1, 3, figsize=(14, 4))
+        ax[0].hist(np.concatenate(all_samples), bins=50, range=(-10, 10))
+        ax[0].set_title(dkey)
+        ax[0].set_xlabel("x")
+        for axi, (key, esti) in zip(ax[1:], est.items()):
+            axi.hist(esti, bins=50)
+            axi.set_title(f"$\\langle \mathrm{{{key}}} "
+                          f"\\rangle = {np.mean(esti):.3f}$ "
+                          f"$\hat V[\mathrm{{{key}}}] = {np.var(esti):.3f}$""")
+            axi.set_xlabel(f"{key}")
+
+
+
plot()
+
+

+
+
+

+
+
+
    +
  • For normal distribution +
      +
    • Median has larger variance than arithmetic mean
    • +
  • +
  • For Student’s t distribution +
      +
    • Median has smaller variance than arithmetic mean
    • +
  • +
  • Which estimator is least biased and has smallest variance depends on distribution
  • +
  • Good estimator for any distribution: maximum-likelihood estimator
  • +
+
+
+

Plug-in estimator

+
    +
  • Plug-in principle: replace true values in some formula with estimates, e.g.
  • +
+

\[ +c = g(a, b, \dots) \to \hat c = g(\hat a, \hat b, \dots) +\]

+
    +
  • Justification (1D, likewise for ND) +
      +
    • For \(N\to \infty\), \(\Delta a = \hat a - a \to 0\) for any reasonable estimator
    • +
    • \(g(\hat a) = g(a + \Delta a) = g(a) + O(\Delta a) \to g(a)\) for smooth \(g\)
    • +
  • +
  • Example: \(\sqrt{N}\) uncertainty estimator for Poisson-distributed count \(N\) +
      +
    • Variance of Poisson distribution of counts \(N\): \(V_N = \lambda\)
    • +
    • Plug-in principle: Replace expected count \(\lambda\) with observed count \(N\): \(\hat V_N = N\)
    • +
    • Standard deviation is square-root of variance: \(\hat \sigma_N = \sqrt{\hat V_N} = \sqrt{N}\)
    • +
  • +
  • Plug-in estimators are usually biased (unless g is linear) and may fail to give reasonable answers for very small samples +
      +
    • Extreme case: \(\lambda = 0.01\), sample \(N = 0 \to \sqrt{0} = 0 \neq 0.1\)
    • +
  • +
  • Be aware of potentially bias when you use plug-in estimator
  • +
+
+
+

Bootstrapping estimator bias and variance

+
    +
  • Bootstrap method is generic way to compute bias and variance of any estimator
  • +
  • Looks like a magic trick, but well-founded in theory (excellent introductory book by inventor Brad Efron at the end)
  • +
+
+

Parametric bootstrap

+
    +
  • LHCb jargon: “running toy simulations”
  • +
  • Situation: +
      +
    • You have sample \(\{ x_i \}\) distributed along \(f(x; a)\) with unknown parameter \(a\)
    • +
    • You have estimator \(\hat a = h[x]\) (can be a formula, fit, anything)
    • +
    • You want to estimate bias and variance of \(h[x]\)
    • +
  • +
  • If we knew true \(a\), could simply simulate the experiment many times +
      +
    • Generate sample \(\{ x_i \}_k\) from \(f(x; a)\) with same size as \(\{ x_i \}\), compute \(\hat a_k\)
    • +
    • Compute bias and variance estimates from sample \(\{ \hat a_k \}\) as shown before
    • +
  • +
  • Don’t know \(f(x; a)\), but its plug-in estimate \(\hat f(x) = f(x; \hat a)\)!
  • +
  • So do the same but sample from \(\hat f(x)\)
  • +
  • This actually works, and very well!
  • +
+
+
# parametric bootstrap to estimate variance of mean
+import numpy as np
+from scipy.stats import norm
+
+mu, sigma = 1, 1
+N = 10
+sample = norm(mu, sigma).rvs(N, random_state=1)
+
+# in general, we use a fit here to get parameters of distribution
+mu_est = np.mean(sample)
+sigma_est = np.std(sample, ddof=1)
+dist = norm(mu_est, sigma_est)
+
+mu_est_b = []
+for k in range(1000):
+    b = dist.rvs(N, random_state=k)
+    mu_est_b.append(np.mean(b))
+    
+print(f"Bias = {np.mean(mu_est_b) - mu_est:.2f} "
+      f"std.dev. = {np.std(mu_est_b, ddof=1):.2f} "
+      f"(expected: {sigma / N ** 0.5:.2f})")
+
+
Bias = -0.01 std.dev. = 0.39 (expected: 0.32)
+
+
+
+
+

Non-parametric bootstrap

+
    +
  • Taking previous idea one step further
  • +
  • Sampling does not require \(\hat f(x)\), but \(\hat F(x)\) (the cdf)
  • +
  • \(\hat F(x)\) can be estimated directly from sample \(\{ x_i \}\) (empirical cdf)
  • +
  • Accuracy a bit worse than parametric bootstrap, but even easier to use
  • +
+
+
# bootstrap bias and variance of arithmetic mean
+import numpy as np
+from resample.bootstrap import resample
+from scipy.stats import norm
+
+mu, sigma = 1, 1
+N = 10
+sample = norm(mu, sigma).rvs(N, random_state=1)
+
+mu_est_b = []
+for replicas in resample(sample, size=1000):
+    mu_est_b.append(np.mean(replicas))
+    
+print(f"Bias = {np.mean(mu_est_b) - mu_est:.2f} "
+      f"std.dev. = {np.std(mu_est_b, ddof=1):.2f} "
+      f"(expected: {sigma / N ** 0.5:.2f})")
+
+
Bias = 0.00 std.dev. = 0.38 (expected: 0.32)
+
+
+
+
# Bonus: bootstrap bias and variance of np.var
+import numpy as np
+from resample.bootstrap import bias, variance
+from scipy.stats import norm
+
+mu, sigma = 1, 1
+N = 10
+sample = norm(mu, sigma).rvs(N, random_state=1)
+
+for ddof in (0, 1):
+    def fn(s):
+        return np.var(s, ddof=ddof)
+    
+    print(f"Bias = {bias(fn, sample, random_state=1):.2f} "
+          f"std.dev. = {np.sqrt(variance(fn, sample, random_state=1)):.2f}")
+
+
Bias = -0.13 std.dev. = 0.53
+Bias = 0.01 std.dev. = 0.59
+
+
+
    +
  • Use np.var(s, ddof=1) if you want unbiased estimate
  • +
+
+
+
+

Standard intervals, coverage and coverage probability

+
    +
  • Standard interval for parameter \(a\): \(\hat a \pm \sqrt{\hat V[\hat a]}\)

  • +
  • Not a statement about true value of \(a\), just a quantification of the properties of the estimate!

  • +
  • But what does it mean then?

  • +
  • Quadrature of circle: How to make a statement about the true value without using the true value?

  • +
  • Frequentist school of statistics: construct intervals such that true value is covered by interval in fraction \(P_\text{cov}\) of identically repeated experiments

  • +
  • Formal way to do this: Neyman construction

    +
      +
    • Works perfectly for continuous distributions, but not for discrete (Poisson, Binomial, …)
    • +
    • Rarely used in practice, too cumbersome
    • +
  • +
  • Coverage important and useful property of intervals

  • +
  • Jargon: “This interval has (proper) coverage”

    +
      +
    • Meaning: actual coverage probability in identically repeated experiments is equal to expected \(P_\text{cov}\)
    • +
  • +
  • For \(N \to \infty\), \(\hat a \pm \sqrt{\hat V[\hat a]}\) interval has coverage probability of 68%

    +
      +
    • And if well-constructed, also for finite \(N\)
    • +
  • +
  • Standard errors should form interval with 68 % coverage probability

    +
      +
    • If coverage is too small/large, adjust interval estimator
    • +
  • +
  • Example: Coverage of the commonly used Poisson interval \(N \pm \sqrt{N}\)?

  • +
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import poisson
+
+N = [0.5, 10., 45.]
+K = 100000
+
+data = []
+for Ni in N:
+    n = poisson(Ni).rvs(size=K, random_state=int(Ni))
+    sigma_est = np.sqrt(n)
+    inside = np.abs(Ni - n) < sigma_est
+    cov_prob = np.mean(inside)
+    data.append((Ni, n, sigma_est, inside, cov_prob))
+
+
+
def plot():
+    fig, ax = plt.subplots(1, len(N), sharex=True, figsize=(14, 4))
+    for (Ni, n, sigma_est, inside, cov_prob), axi in zip(data, ax):
+        axi.axhline(Ni, color="0.5", zorder=0)
+        i = np.arange(50)
+        n = n[:50]
+        sigma_est = sigma_est[:50]
+        inside = inside[:50]
+        for m, c in zip((~inside, inside), ("C0", "C1")):
+            axi.errorbar(i[m], n[m], sigma_est[m], fmt="o", color=c)
+        axi.set_title(f"P_cov = {cov_prob:.2f}")
+    fig.supxlabel("repeated experiments");
+
+
+
plot()
+
+

+
+
+
    +
  • Poisson interval \(N \pm \sqrt{N}\) only has proper coverage for \(N \gtrsim 45\) +
      +
    • Reason: if \(N\) fluctuates downwards, interval shrinks
    • +
  • +
  • Intervals can be empty in some experiments; this is not ideal, but not a defect
  • +
  • If you have an estimate \(\mu\) for the true value of \(N_i\), this is better: \(N_i \pm \sqrt{\mu}\)
  • +
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import poisson
+
+N = [0.5, 10, 20]
+K = 100000
+
+data = []
+for Ni in N:
+    n = poisson(Ni).rvs(size=K, random_state=int(Ni))
+    mu = np.mean(n)
+    sigma_est = np.sqrt(mu)
+    inside = np.abs(Ni - n) < sigma_est
+    cov_prob = np.mean(inside)
+    data.append((Ni, n, sigma_est * np.ones(len(n)), inside, cov_prob))
+
+
+
plot()
+
+

+
+
+
+
+

Maximum-likelihood estimator (MLE)

+
    +
  • MLE has attractive properties

    +
      +
    • For \(N \to \infty\), as close to true value as possible (proven) +
        +
      • Unbiased and minimum variance
      • +
    • +
    • In finite samples +
        +
      • Biased in general, but bias zero or small for many important distributions
      • +
      • Probably as close to true value as possible (conjecture)
      • +
    • +
    • Easy to compute MLE and its variance estimate (with suitable software)
    • +
    • But: Not robust against outliers or model misspecification
    • +
  • +
  • Maximum-likelihood estimator for parameter \(a\) \[ +\begin{aligned} +h_\text{MLE}[x] &= \text{argmax}_{a}\, \ln \mathcal L(a) \\ +\ln \mathcal L(a) &:= \ln \prod_i P(x_i; a) = \sum_i \ln f(x_i; a) + \text{const} +\end{aligned} +\]

  • +
  • MLE example: arithmetic mean is MLE for parameter \(\mu\) of normal distribution \[ +\begin{aligned} +\frac\partial{\partial \mu} \sum_i \ln\left( \frac1{\sqrt{2\pi} \sigma} \exp\left(-\frac{(x_i - \mu)^2}{2\sigma^2} \right)\right) &\overset{!}{=} 0 \\ +\underbrace{\frac\partial{\partial \mu} \sum_i \ln\left( \frac1{\sqrt{2\pi} \sigma} \right)}_{0} - \frac\partial{\partial \mu} \sum_i \frac{(x_i - \mu)^2}{2\sigma^2} &= 0\\ +\sum_i \frac{x_i - \mu}{\sigma^2} &= 0 \\ +\sum_i x_i - \sum_i\mu &= 0 \\ +\Rightarrow \hat\mu = \frac 1 N \sum_i x_i & +\end{aligned} +\]

  • +
  • In general: Use numerical optimiser (e.g. MINUIT) to find MLE for any distribution

  • +
+
+
+

Uncertainty of maximum likelihood estimate

+
    +
  • Two ways to compute uncertainty estimates for MLE based on theorems by Fisher and Wilks
  • +
  • Recommended: lecture on MLE by Geyer (2013) https://www.stat.umn.edu/geyer/5601/notes/sand.pdf
  • +
+
+

Variance estimate

+
    +
  • Variance estimate for MLE estimator \(\hat a\) of parameter \(a\) \[ +\hat V[\hat a] = -\left(\frac{\partial^2 \ln \mathcal L(a)}{\partial a^2}\Bigg|_{a=\hat a}\right)^{-1} +\]

  • +
  • Covariance estimate for multivariate MLE of parameter \(\vec a\) \[ +\hat C_{ik}[\hat{\vec a}] = -\left(\frac{\partial^2 \ln \mathcal L(\vec a)}{\partial a_i \partial a_k}\Bigg|_{\vec a=\hat {\vec a}}\right)^{-1} +\]

  • +
  • MINUIT computes this numerically (HESSE method)

  • +
  • Example: variance estimate for \(\hat\mu\) for normally distributed data with known width \(\sigma\) \[ +-\left(\frac{\partial^2 \ln \mathcal L(\mu)}{\partial \mu^2}\right)^{-1} = \dots = \left(\frac{\partial^2}{\partial \mu^2} \sum_i \frac{(x_i - \mu)^2}{2\sigma^2}\right)^{-1} +\] \[ = -\left(\frac{\partial}{\partial \mu} \sum_i \frac{(x_i - \mu)}{\sigma^2}\right)^{-1} = \left(\sum_i \frac1{\sigma^2}\right)^{-1} = \frac{\sigma^2}{N} +\]

  • +
  • Well-known result for error on the mean

  • +
+
+
+

Profile likelihood scan

+
    +
  • Produces asymmetric uncertainty intervals in general

  • +
  • For desired confidence level \(CL\) (a probability) solve \(\chi^2(\lambda;M) = CL\) for \(\lambda\), where \(M\) is number of parameters of interest (subset of all parameters)

  • +
  • Solve for contour in \(\vec a\)-space (parameters of interest) $$

  • +
  • 2 = $$ where \(\hat b(\vec a)\) are maximum-likelihood estimates of nuisance parameters \(\vec b\) for fixed values of \(\vec a\)

  • +
  • Too complicated? No problem, MINUIT computes all this numerically for you (MINOS method)

  • +
  • Commonly used to draw uncertainty contours

  • +
  • Sometimes advertised as superior to variance estimate (HESSE method), but isn’t

    +
  • +
  • If you really need this, publish full likelihood function with tools like pyhf or HistFactory

  • +
+
+
# Example: fit parameters of shifted Student's t distribution
+import numpy as np
+import matplotlib.pyplot as plt
+from iminuit import Minuit
+from iminuit.cost import UnbinnedNLL
+from scipy.stats import t
+
+nd = 3
+mu = 1.0
+sample = t(nd, mu).rvs(size=50, random_state=1)
+
+
+
def model(x, nd, mu):
+    return t(nd, mu).pdf(x)
+
+cost = UnbinnedNLL(sample, model)
+
+m = Minuit(cost, nd=2, mu=0)
+m.limits["nd"] = (1, None)
+m.migrad()  # also runs equivalent of m.hesse() by default
+m.minos()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 165Nfcn = 131
EDM = 2.85e-05 (Goal: 0.0002)
Valid MinimumNo Parameters at limit
Below EDM threshold (goal x 10)Below call limit
CovarianceHesse okAccuratePos. def.Not forced
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0nd3.61.5-1.22.11
1mu0.970.16-0.160.16
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
ndmu
Error-1.22.1-0.160.16
ValidTrueTrueTrueTrue
At LimitFalseFalseFalseFalse
Max FCNFalseFalseFalseFalse
New MinFalseFalseFalseFalse
+ + + + + + + + + + + + + + + + + + +
ndmu
nd2.360.00743 (0.030)
mu0.00743 (0.030)0.0264
+
+
+
+
xm = np.linspace(-10, 10, 1000)
+plt.hist(sample, bins=20, density=True, label="sample")
+plt.plot(xm, t(*m.values).pdf(xm), label="fit")
+plt.plot(xm, t(nd, mu).pdf(xm), label="truth")
+plt.legend();
+
+

+
+
+
+
fig, ax = plt.subplots(1, 2, figsize=(14, 4))
+for axi, par in zip(ax, ("nd", "mu")):
+    plt.sca(axi)
+    m.draw_mnprofile(par)
+
+

+
+
+
+
+
+

Misspecified models

+
    +
  • Maximum-likelihood method assumes that model is correctly specified, all proven properties rely on that

  • +
  • Examples of misspecified models

    +
      +
    • Assumed: signal only, actually: signal + background
    • +
    • Assumed: normal distribution, actually: distorted normal (wider tails, asymmetric, etc)
    • +
  • +
  • If model is misspecified

    +
      +
    • MLE is biased in general, but may not be
    • +
    • MLE uncertainties (from either method) no longer have coverage
    • +
  • +
  • Fundamentalist: models are essentially always misspecified, since they are always approximations

  • +
  • Pragmatist: Small deviations have negligible effect

  • +
  • Worse at HL-LHC: higher statistics, so more sensitive to small deviations

    +
      +
    • Need goodness-of-fit criterion (not discussed here) to check whether model is reliable
    • +
  • +
  • How to fix this: make model more general (by adding parameters) so it is no longer misspecified

    +
      +
    • Use Student’s t (or double-sided Crystal Ball) instead of normal distribution to fit peak
    • +
    • Use polynomial of higher order for smooth background
    • +
    • +
  • +
+
+
+
+

Systematic uncertainties: Known unknowns

+

image-3.png * Donald Rumsfeld: “There are known knowns, things we know that we know; and there are known unknowns, things that we know we don’t know. But there are also unknown unknowns, things we do not know we don’t know.”

+
    +
  • Why data analyses take so long: Study data, detector, and methods to turn unknown unknowns into known knowns or known unknowns

  • +
  • No formal way, but generally…

    +
      +
    • Apply checks +
        +
      • Split data by magnet polarity, fill, etc., analyse splits separately, check for agreement
      • +
    • +
    • Worry +
        +
      • What could go wrong?
      • +
      • What are my assumptions?
      • +
      • Example: You take an efficiency or correction factor from simulation, but simulation differs from real experiment; perform control measurement or use standard LHCb control measurements
      • +
    • +
    • Methods +
        +
      • If feasible, start from first principles or use methods that are known to be optimal/robust
      • +
      • Examples +
          +
        • Use Maximum-likelihood instead of least-squares fit
        • +
        • Use bootstrap estimate of error
        • +
      • +
    • +
  • +
  • Guidelines for handling systematic errors

    +
  • +
+
+
+

+
image.png
+
+
+
+
+

Error propagation

+
    +
  • You have vector of values \(\vec a\) with covariance matrix \(C_a\) and a smooth function \(\vec b = f(\vec a)\), you want to know \(C_b\)
  • +
  • \(\vec b\) can have different dimensionality from \(\vec a\)
  • +
+
+

Option 1: Standard error propagation

+
    +
  • Based on first order Taylor expansion \[ +C_b = J \, C_a \, J^T +\] with Jacobi matrix \(J\) \[ +J_{ik} = \frac{\partial f_i}{\partial a_k} +\]

  • +
  • For derivation, see F. James book (references at end)

  • +
  • Can be used for statistical and systematic uncertainties!

  • +
  • Useful special cases

    +
      +
    • Sum of scaled independent random values \(c = \alpha a + \beta b\) (\(\alpha, \beta\) are constant) \[ + \sigma^2_c = \alpha^2 \sigma^2_a + \beta^2 \sigma^2_b + \]
    • +
    • Product of powers of independent random values \(c = a^\alpha \, b^\beta\) \[ + \left(\frac{\sigma_c}{c}\right)^2 = \alpha^2 \left(\frac{\sigma_a}{a}\right)^2 + \beta^2 \left(\frac{\sigma_b}{b}\right)^2 + \]
    • +
    • Dominated by component with largest (relative) uncertainty
    • +
    • Put your effort in estimating largest component and be more lax on small components
    • +
  • +
  • Example

    +
      +
    • Arithmetic mean \[ + \hat \mu = \frac 1N\sum_i x_i + \]

    • +
    • Variance of x (unbiased formula for unkown mean) \[ + \hat V[x] = \frac 1{N-1} \sum_i (x_i - \hat\mu)^2 + \]

    • +
    • Variance of the mean \[ + \hat{V}[\hat \mu] = \frac1{N^2} \sum_i \hat{V}[x_i] = \frac 1 {N(N - 1)}\sum_i (x_i - \hat \mu)^2 + \]

    • +
  • +
  • Error-prone to do by hand, use software to compute numerical derivative

  • +
+
+
+

Option 2: Monte-Carlo propagation

+
    +
  • To apply, you need to know distribution of \(\vec a\)

  • +
  • Randomly sample \(K\) times

  • +
  • Compute sample \(\{\vec b_k \} = \{ f(\vec a_k) \}\)

  • +
  • Compute sample covariance of \(\{\vec b_k \}\)

  • +
+
+
import numpy as np
+from jacobi import propagate  # uses Option 1
+from scipy.stats import multivariate_normal
+
+a0, sigma_a0 = 2, 0.2
+a1, sigma_a1 = 3, 0.5
+
+a = [a0, a1]
+cov_a = np.zeros((2, 2))
+cov_a[0, 0] = sigma_a0 ** 2
+cov_a[1, 1] = sigma_a1 ** 2
+
+# option 1
+b, cov_b = propagate(np.mean, a, cov_a)
+print(f"option 1: b = {b:.2f} +/- {np.sqrt(cov_b):.2f}")
+
+# option 2
+a_sample = multivariate_normal(a, cov_a).rvs(size=10000)
+b_sample = np.mean(a_sample, axis=1) # sum over second axis
+cov_b = np.cov(b_sample, ddof=1)
+print(f"option 2: b = {b:.2f} +/- {np.sqrt(cov_b):.2f}")
+
+
option 1: b = 2.50 +/- 0.27
+option 2: b = 2.50 +/- 0.27
+
+
+
    +
  • Results identical here because function was linear; they differ in general
  • +
  • Which is better? Hard to say, depends on particular case +
      +
    • Option 2 is conceptually better, but you may need very large samples (>10000) to beat option 1 in accuracy
    • +
  • +
+
+
+
+

LHCb Statistics Guidelines

+
    +
  • Living document on GitLab to collect information about best practices +
      +
    • Everyone can edit this and propose new items!
    • +
    • LHCb StatML conveners are editors, liaisons check new content
    • +
  • +
  • Index instead of a book +
      +
    • Brief items with advice
    • +
    • Links to literature
    • +
  • +
  • In conversation, you can directly link to items, point to title and copy link (right-click menu)
  • +
+
+
+

Introductory statistics books

+ +
+

The END

+ + +
+
+ +
+ +
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z8v(uM!qMwq9e((#!_{zvr-h>e&Q1|}wI3mo(e;Oma1;*gG%oo05oTt3!=QnkzkWIQ zkrro&^2uwgOxscy+l&(~-2 zqD*#pDgY0&C&+G5wYYf1iC*!FOPmy;mPUfh%pWWm16*O@kOnXc1$bwMwNQU`j(a`K zi2r$iUBCO^q~M)?;_kc(Ea*Ia%IgMO;GMm|sMqvyZFuXtaEqH#^>o5G>< zSLy;4fyjJQ`xN0uGdPh{0QeK&W`5n-)mj)P4B?|>Cc!5F+=Xr2gh|WMTD9@;FO9;& zSzP?!<6bJz*0%#!&Odj(v^ z3rBE-A@H@+g%d|rY+n(-EW?8mm7U|ZES(`iro`=_N=N4kPebOmTc%=_*+!O?E z=#b{RE9?pidvPcL_`#i-JAXVdV*&@9bkmse9L?-6`^vMwPKUo?nSJQ3nO|XwhZ|M; zn5BKAD%{KbHp2dYwKtCHQ#{;Wv;ZYy3dme{0hw#x2scGsGZ@SX2S;WID}$=|v5zou zBmiL86$UjNVPV!U=HBi=yFzdYQz4=x-Zsq}uxx5*+_1=@=cD6-GgBqQn zYXt~!`}pEfySurW8zp^O1T>0YI_(7#UYd*txHo++5@QCBSNs%+X + + + + + + + + +Hans Dembinski’s blog - Uncertainty of efficiency from fitted decay yields + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+ +
+ +
+
+
+

Uncertainty of efficiency from fitted decay yields

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

We derive an approximate formula to compute the uncertainty of the efficiency computed from fitted yields of some decays. The formula is suitable to draw error bars in plots.

+
+
#!pip install numpy iminuit numba_stats matplotlib scipy sympy joblib
+import numpy as np
+from iminuit import Minuit
+from iminuit.cost import ExtendedBinnedNLL, NormalConstraint
+from numba_stats import bernstein, norm
+from matplotlib import pyplot as plt
+import joblib
+from sympy import *
+
+
+

Introduction

+

The efficiency of a selection is

+

\[ +\epsilon = \frac{k}{n}, +\]

+

where \(k\) is the number of events which pass and \(n\) is the total number of events.

+

For the sake of computing confidence intervals for an efficiency computed from a sample, \(\epsilon\) is typically assumed to be a binomial proportion. This means that \(k\) is binomially distributed for fixed \(n\) in repeated experiments.

+

In practice, this assumption is usually incorrect. The efficiency is usually computed from a total sample of events whose number \(n\) is Poisson distributed, since experiments that involve random processes are typically run for a fixed time rather than until a fixed number of events has occured. This produces a larger variance for \(\epsilon\) compared to what is expected from the binomial model.

+

We demonstrate this numerically.

+
+
rng = np.random.default_rng(1)
+
+x = np.linspace(0, 5, 100)
+
+# sigmoid
+p = 1.0 / (1.0 + np.exp(-x))
+
+plt.plot(x, p, color="k", label=r"$\epsilon(x) = 1/(1 + e^{-x})$")
+
+ntry = 10000
+for i, n in enumerate((5, 10, 100)):
+    eff1 = []
+    eff2 = []
+    for pi in p:
+        y1 = rng.binomial(n, pi, size=ntry) / n
+
+        ni = rng.poisson(n, size=ntry)
+        with np.errstate(invalid="ignore"):
+            y2 = rng.binomial(ni, pi) / ni
+
+        for y, eff in [(y1, eff1), (y2, eff2)]:
+#             q = np.nanquantile(y, [0.16, 1-0.16])
+            m = np.nanmean(y)
+            s = np.nanstd(y)
+            eff.append((m - s, m + s))
+    eff1 = np.array(eff1)
+    eff2 = np.array(eff2)
+
+    col = f"C{i}"
+    for eff, ls, label in ((eff1, ":", f"n={n} fixed"), 
+                           (eff2, "--", f"poisson n with $\\langle n \\rangle = {n}$ ")):
+        plt.plot(x, eff[:, 0], color=col, ls=ls, label=label)
+        plt.plot(x, eff[:, 1], color=col, ls=ls)
+
+    print(np.mean(np.diff(eff2, axis=1) / np.diff(eff1, axis=1)))
+
+plt.legend()
+plt.xlabel(r"$x$")
+plt.ylabel(r"$\epsilon$");
+
+
1.1395442804276614
+1.0638370186185993
+1.0046267505795776
+
+
+

+
+
+

Shown is an sigmoidal efficiency curve and a normal approximation interval computed from the standard deviation of the sample estimate \(\hat \epsilon\). For small \(n\), the interval is wider if \(n\) is Poisson distributed.

+

We compute the variance of \(\epsilon\) for both cases explicitly. In the following, it will be important to clearly distinguish sample values from their population values. We will use \(\hat x\) for the sample value, which has a population value \(x\). The sample value of the efficiency then is \(\hat \epsilon = \hat k / \hat n\).

+
+
+

No background, n fixed

+

If \(\hat n = n\) is fixed, the variance of \(\hat \epsilon\) can be computed from the variance of the binomial distribution \(\text{var}(\hat k) = n \epsilon (1 - \epsilon)\). We expand the formula for \(\epsilon\) in a Taylor series in the variable \(k = \epsilon n\) around the sample value \(\hat k\) and obtain

+

\[ +\epsilon = \sum_{i=0}^{\infty} \frac {(k - \hat k) ^i}{i!} \frac{\partial^i \epsilon}{\partial {k}^i}(\hat k) = \hat \epsilon + \frac {k - \hat k} n. +\]

+

The series naturally break off after the first order. Thanks to that, we can compute the variance of \(\hat \epsilon\) exactly,

+

\[ +\text{var}(\hat\epsilon) = \text{E}\big((\hat \epsilon - \epsilon)^2\big) = \text{E}\left(\frac{(\hat k - k)^2}{n^2}\right) = \frac{\text{E}\big((\hat k - k)^2\big)}{n^2} = \frac{\text{var}(\hat k)}{n^2} = \frac{\epsilon (1 - \epsilon)}{n} +\]

+

The same result is obtain with the standard error propagation formula, which is exact in this case, since \(\epsilon\) is a linear function of \(k\). We performed the calculation from first principles instead of using this formula, because it is instructive for the next step, where \(\epsilon\) becomes a non-linear function.

+
+
+

No background, n Poisson distributed

+

To simplify the calculation, we use that the efficiency can be written in terms of two independently Poisson-distributed numbers, the number of successes \(n_1\) and failures \(n_2\),

+

\[ +\epsilon = \frac{n_1}{n_1 + n_2} = \frac{n_1}{n}, +\]

+

with \(n = n_1 + n_2\).

+

In a particular sample, the sample values \(\hat n_k\) will deviate from the expectation values \(n_k\) by random offsets \(d_k = \hat n_k - n_k\). This has an effect on the estimator \(\hat \epsilon(\hat n_1, \hat n_2) = \hat n_1 / (\hat n_1 + \hat n_2)\), that we can evaluate by expanding the formula around the true values in a two-dimensional Taylor series with \(\hat n_k = n_k\), treating the integer values \(\hat n_k\) like real numbers,

+

\[ +\hat \epsilon = \sum_{i=0}^\infty\sum_{j=0}^\infty \frac {d_1^i \,d_2^j}{i! \, j!} \frac{\partial^{i+j} \hat \epsilon}{\partial \hat n_1^i \partial \hat n_2^j} (n_1, n_2) +\]

+
+
def bivariate_series(expr, x, y, dx, dy, order_max):
+    return sum(
+        sum(
+            dx ** i * dy ** j / 
+            (factorial(i) * factorial(j)) *
+            diff(expr, x, i, y, j)
+            for i in range(order_max + 1)
+            if (i + j) <= order_max
+        ) for j in range(order_max + 1)
+    )
+
+
+
n1, n2, n, epsilon = symbols("n_1 n_2 n epsilon", real=True, nonnegative=True)
+d1, d2 = symbols("d_1 d_2", real=True)
+
+eps = n1 / (n1 + n2)
+
+delta_eps = (bivariate_series(eps, n1, n2, d1, d2, 7) - eps)
+delta_eps = delta_eps.subs([(n1, epsilon * n), (n2, (1-epsilon) * n)])
+delta_eps = delta_eps.as_poly([d1, d2])
+delta_eps
+
+

\(\displaystyle \operatorname{Poly}{\left( \frac{1 - \epsilon}{n^{7}} d_{1}^{7} + \frac{6 - 7 \epsilon}{n^{7}} d_{1}^{6}d_{2} + \frac{\epsilon - 1}{n^{6}} d_{1}^{6} + \frac{15 - 21 \epsilon}{n^{7}} d_{1}^{5}d_{2}^{2} + \frac{6 \epsilon - 5}{n^{6}} d_{1}^{5}d_{2} + \frac{1 - \epsilon}{n^{5}} d_{1}^{5} + \frac{20 - 35 \epsilon}{n^{7}} d_{1}^{4}d_{2}^{3} + \frac{15 \epsilon - 10}{n^{6}} d_{1}^{4}d_{2}^{2} + \frac{4 - 5 \epsilon}{n^{5}} d_{1}^{4}d_{2} + \frac{\epsilon - 1}{n^{4}} d_{1}^{4} + \frac{15 - 35 \epsilon}{n^{7}} d_{1}^{3}d_{2}^{4} + \frac{20 \epsilon - 10}{n^{6}} d_{1}^{3}d_{2}^{3} + \frac{6 - 10 \epsilon}{n^{5}} d_{1}^{3}d_{2}^{2} + \frac{4 \epsilon - 3}{n^{4}} d_{1}^{3}d_{2} + \frac{1 - \epsilon}{n^{3}} d_{1}^{3} + \frac{6 - 21 \epsilon}{n^{7}} d_{1}^{2}d_{2}^{5} + \frac{15 \epsilon - 5}{n^{6}} d_{1}^{2}d_{2}^{4} + \frac{4 - 10 \epsilon}{n^{5}} d_{1}^{2}d_{2}^{3} + \frac{6 \epsilon - 3}{n^{4}} d_{1}^{2}d_{2}^{2} + \frac{2 - 3 \epsilon}{n^{3}} d_{1}^{2}d_{2} + \frac{\epsilon - 1}{n^{2}} d_{1}^{2} + \frac{1 - 7 \epsilon}{n^{7}} d_{1}d_{2}^{6} + \frac{6 \epsilon - 1}{n^{6}} d_{1}d_{2}^{5} + \frac{1 - 5 \epsilon}{n^{5}} d_{1}d_{2}^{4} + \frac{4 \epsilon - 1}{n^{4}} d_{1}d_{2}^{3} + \frac{1 - 3 \epsilon}{n^{3}} d_{1}d_{2}^{2} + \frac{2 \epsilon - 1}{n^{2}} d_{1}d_{2} + \frac{1 - \epsilon}{n} d_{1} - \frac{\epsilon}{n^{7}} d_{2}^{7} + \frac{\epsilon}{n^{6}} d_{2}^{6} - \frac{\epsilon}{n^{5}} d_{2}^{5} + \frac{\epsilon}{n^{4}} d_{2}^{4} - \frac{\epsilon}{n^{3}} d_{2}^{3} + \frac{\epsilon}{n^{2}} d_{2}^{2} - \frac{\epsilon}{n} d_{2}, d_{1}, d_{2}, domain=\mathbb{Z}\left(n, \epsilon\right) \right)}\)

+
+
+

To compute the variance \(\text{var}(\hat \epsilon)\) we need to square this and compute the expection on both sides. We use that \(\text{E}(d_1^i d_2^j) = \text{E}(d_1^i) \, \text{E}(d_2^j)\), since \(\hat n_1\) and \(\hat n_2\) are independently sampled. The expectation value \(\text{E}(d_k^j)\) is the \(j\)-th central moment of the Poisson distribution with \(\lambda = n_k\). These moments are Touchard polynomials, which are conveniently computed with a recursion formula.

+
+
# compute mapping d_k ** i -> E[d_k ** i]
+
+def central_moments(n):
+    x = symbols("x")
+    r = [1]
+    for ni in range(n):
+        # recursion for Touchard polynomial
+        m = x * sum(binomial(ni, k) * r[k] for k in range(ni))
+        r.append(m.expand())
+
+    return [lambdify([x], ri) for ri in r]
+
+replacements = central_moments(14)
+
+

Now we compute \(\text{var}(\hat \epsilon) = \text{E}\big((\hat \epsilon - \epsilon)^2\big)\). Since SymPy does not know how to compute an expectation and that it is a linear operator, we need to resort to advanced expression manipulation in the following steps. We use that the expectation is linear and \(\text{E}(n_i n_j) = \text{E}(n_i) \, \text{E}(n_j)\) for independent \(n_i\) and \(n_j\).

+
+
def expectation(expr, args, args2, replacements):
+    poly = expr.as_poly(args)
+    expr2 = []
+    for powers, coef in poly.terms():
+        x = prod(
+            1 if pi == 0 else replacements[pi](argi)
+            for pi, argi in zip(powers, args2)
+        )
+        expr2.append(coef * x)
+    return sum(expr2)
+
+
+
# replace powers of d_k with the corresponding central moments
+expr2 = expectation(delta_eps ** 2, (d1, d2), (epsilon * n, (1-epsilon)*n), replacements)
+expr2 = expr2.as_poly(1/n)
+expr2
+
+

\(\displaystyle \operatorname{Poly}{\left( \left(- \epsilon^{2} + \epsilon\right) \frac{1}{n^{13}} + \left(- 4081 \epsilon^{2} + 4081 \epsilon\right) \frac{1}{n^{12}} + \left(- 233058 \epsilon^{2} + 233058 \epsilon\right) \frac{1}{n^{11}} + \left(- 1413895 \epsilon^{2} + 1413895 \epsilon\right) \frac{1}{n^{10}} + \left(- 1356075 \epsilon^{2} + 1356075 \epsilon\right) \frac{1}{n^{9}} + \left(- 130095 \epsilon^{2} + 130095 \epsilon\right) \frac{1}{n^{8}} + \left(720 \epsilon^{2} - 720 \epsilon\right) \frac{1}{n^{7}} + \left(118 \epsilon^{2} - 118 \epsilon\right) \frac{1}{n^{6}} + \left(- 24 \epsilon^{2} + 24 \epsilon\right) \frac{1}{n^{5}} + \left(- 6 \epsilon^{2} + 6 \epsilon\right) \frac{1}{n^{4}} + \left(- 2 \epsilon^{2} + 2 \epsilon\right) \frac{1}{n^{3}} + \left(- \epsilon^{2} + \epsilon\right) \frac{1}{n^{2}} + \left(- \epsilon^{2} + \epsilon\right) \frac{1}{n}, \frac{1}{n}, domain=\mathbb{Z}\left[\epsilon\right] \right)}\)

+
+
+

We keep only terms up to \(O(1/n^2)\).

+
+
expr3 = []
+for (p, ), coef in expr2.terms():
+    if p <= 2:
+        expr3.append(coef * n ** -p)
+eps_var = sum(expr3).subs(epsilon, n1/n).subs(n, n1 + n2).simplify()
+eps_var
+
+

\(\displaystyle \frac{n_{1} n_{2} \left(n_{1} + n_{2} + 1\right)}{\left(n_{1} + n_{2}\right)^{4}}\)

+
+
+

Up to the extra factor \((n + 1)/n\), this is the variance of the binomial distribution we derived previously.

+

This formula is accurate for \(n > 10\). To get accurate results at even smaller \(n\), we need to include to higher orders.

+
+
expr3 = []
+for (p, ), coef in expr2.terms():
+    if p <= 4:
+        expr3.append(coef * n ** -p)
+eps_var = sum(expr3).subs(epsilon, n1/n).subs(n, n1 + n2).simplify()
+eps_var
+
+

\(\displaystyle \frac{n_{1} n_{2} \left(2 n_{1} + 2 n_{2} + \left(n_{1} + n_{2}\right)^{3} + \left(n_{1} + n_{2}\right)^{2} + 6\right)}{\left(n_{1} + n_{2}\right)^{6}}\)

+
+
+

This formula is accurate down to \(n \ge 5\).

+

The values \(n_k\) that appear in the formula are the unknown expectation values and need to be replaced by their sample estimates \(\hat n_k\) in practice. This is an application of the plug-in principle on which also bootstrap theory is based. There is a small bias involved in doing this, but it is negligible here.

+
+
eps_var_py = lambdify((n1, n2), eps_var)
+
+
+
rng = np.random.default_rng(1)
+eps = np.linspace(0, 1, 100)
+data = {}
+for n in (5, 10, 20, 50, 100, 1000):
+    s1 = []
+    s2 = []
+    for epsi in eps:
+        n1 = n * epsi
+        n2 = n * (1 - epsi)
+        k1 = rng.poisson(n1, size=10000)
+        k2 = rng.poisson(n2, size=10000)
+        ma = k1 + k2 > 0
+        k1 = k1[ma]
+        k2 = k2[ma]
+        s1.append(np.std(k1 / (k1 + k2)))
+        # it is important to average the variance instead of the standard deviation
+        s2.append(np.mean(eps_var_py(k1, k2)) ** 0.5)
+    data[n] = np.array((eps, s1, s2))
+
+
+
plt.figure(figsize=(10, 4))
+for n, (eps, s1, s2) in data.items():
+    l, = plt.plot(eps, s1, ":", label=f"MC, {n=}")
+    plt.plot(eps, s2, "--", color=l.get_color(), label=f"formula, {n=}")
+    print(f"{n=} median deviation {np.median(s1[s2 > 0] / s2[s2 > 0]) - 1:.2f}")
+plt.subplots_adjust(right=0.6)
+plt.xlabel(r"$\epsilon$")
+plt.ylabel(r"$\sigma(\epsilon)$")
+plt.figlegend(loc=(0.69, 0.2));
+
+
n=5 median deviation 0.00
+n=10 median deviation -0.01
+n=20 median deviation -0.00
+n=50 median deviation -0.00
+n=100 median deviation 0.00
+n=1000 median deviation 0.00
+
+
+

+
+
+
+

Wilson interval

+

To-do.

+
+
+
+

With background

+

We now assume that the estimates \(\hat n_1\) and \(\hat n_2\) cannot obtained by direct counting, but need to be estimated from a fit that inclues a signal and background component. If the fit is unbiased, the average contribution from the background is zero, which we will assume, but it enlargens the uncertainty of the estimate.

+

We can model this by adding a random contribution from the background to our estimate,

+

\[ +\hat n = n + z_1 \sigma_\text{poisson} + z_2 \sigma_\text{bkg} = n + z_1 \sqrt{n} + z_2 \sigma_\text{bkg}, +\]

+

where \(z_1\) and \(z_2\) are independent random variables with zero mean and unit variance.

+

The expected variance of \(\hat n\) then is

+

\[ +\sigma^2(\hat n) = n + \sigma^2_\text{bkg}. +\]

+

In practice, estimating \(\sigma^2_\text{bkg}\) directly is difficult, but fortunately, the left-hand side \(\sigma^2(\hat n)\) can be estimated from the fit (MINUITs Hesse error). Thus we can compute an estimate for the unknown \(\sigma^2_\text{bkg}\) with the formula,

+

\[ +\sigma^2_\text{bkg} = \sigma^2(\hat n) - n. +\]

+

In practice, we have to replace the expectation values \(\sigma^2(\hat n)\) and \(n\) with the corresponding estimates from the fit.

+

We insert this into the formula for \(\hat \epsilon\) and get:

+

\[ +\hat \epsilon = \frac{\hat n_1}{\hat n_1 + \hat n_2} = \frac{n_1 + z_1 \sqrt{n_1} + z_2 \sigma_\text{bkg,1}}{n_1 + n_2 + z_1 \sqrt{n_1} + z_2 \sigma_\text{bkg,1} + z_3 \sqrt{n_2} + z_4 \sigma_\text{bkg,2}} +\]

+

As before, the variance of \(\hat \epsilon\) is computed via standard error propagation, using that the \(z_i\) are independent and standard normal. This reflects the assumption that the background in both fits is independently sampled, which in practice is usually not exactly true. If the correlation can be estimated, it can be trivially included in the calculation, but I will not do this here.

+

\[ +\sigma^2(\hat \epsilon) = \left[\left(\frac{\text{d}\hat \epsilon}{\text{d}z_1}\right)^2 + \left(\frac{\text{d}\hat \epsilon}{\text{d}z_2}\right)^2 + +\left(\frac{\text{d}\hat \epsilon}{\text{d}z_3}\right)^2 + +\left(\frac{\text{d}\hat \epsilon}{\text{d}z_4}\right)^2 +\right]_{z_1 = 0,\, z_2=0,\,z_3=0,\,z_4=0}. +\]

+
+
(n1, n2, sigma1, sigma2,
+ sigma_bkg1, sigma_bkg2,
+ z1, z2, z3, z4) = symbols(
+    "n_1 n_2 sigma_1 sigma_2 "
+    "sigma_bkg_1 sigma_bkg_2 "
+    "z_1 z_2 z_3 z_4", real=True)
+
+g = (n1 + z1 * sqrt(n1) + z2 * sigma_bkg1) / \
+    (n1 + n2
+     + z1 * sqrt(n1) + z2 * sigma_bkg1 
+     + z3 * sqrt(n2) + z4 * sigma_bkg2); g
+
+

\(\displaystyle \frac{\sqrt{n_{1}} z_{1} + n_{1} + \sigma_{bkg 1} z_{2}}{\sqrt{n_{1}} z_{1} + n_{1} + \sqrt{n_{2}} z_{3} + n_{2} + \sigma_{bkg 1} z_{2} + \sigma_{bkg 2} z_{4}}\)

+
+
+
+
g = g.subs([(sigma_bkg1, sqrt(sigma1 ** 2 - n1)), 
+            (sigma_bkg2, sqrt(sigma2 ** 2 - n2))]); g
+
+

\(\displaystyle \frac{\sqrt{n_{1}} z_{1} + n_{1} + z_{2} \sqrt{- n_{1} + \sigma_{1}^{2}}}{\sqrt{n_{1}} z_{1} + n_{1} + \sqrt{n_{2}} z_{3} + n_{2} + z_{2} \sqrt{- n_{1} + \sigma_{1}^{2}} + z_{4} \sqrt{- n_{2} + \sigma_{2}^{2}}}\)

+
+
+
+
g_var = sum(diff(g, z) ** 2 for z in (z1, z2, z3, z4))
+g_var = g_var.subs([(z1, 0), (z2, 0), (z3, 0), (z4, 0)])
+g_var = g_var.simplify(); g_var
+
+

\(\displaystyle \frac{n_{1}^{2} \sigma_{2}^{2} + n_{2}^{2} \sigma_{1}^{2}}{\left(n_{1} + n_{2}\right)^{4}}\)

+
+
+

We thus obtain

+

\[ +\sigma(\hat \epsilon) = \frac{\sqrt{ +n_1^2 \, \sigma^2(\hat n_2) + +n_2^2 \, \sigma^2(\hat n_1) +}}{n^2}, +\]

+

with \(n = n_1 + n_2\). In practice, \(n_1\), \(\sigma^2(\hat n_1)\), … need to be replaced with their fit estimates.

+

As a check, I show how one can recover the previous formula, which is a special case, by setting \(\sigma^2(\hat n_k) = n_k\) (which means no background contribution),

+

\[ +\sigma^2(\hat \epsilon) = \frac{ +n_1^2 \, \sigma^2(\hat n_2) + +n_2^2 \, \sigma^2(\hat n_1) +}{n^4} = +\frac{ +n_1^2 \, n_2 + +n_2^2 \, n_1 +}{n^4} = +\frac{ +n_1 \, n_2 \, (n_1 + n_2) +}{n^4} = +\frac{ +n_1 \, n_2 +}{n^3}. +\]

+

We again check the formula against Monte-Carlo simulations.

+
+
g_var_py = lambdify([n1, sigma1, n2, sigma2], g_var)
+
+
+
def model(x, b0, b1, b2, s, mu, sigma):
+    nb = bernstein.scaled_cdf(x, (b0, b1, b2), -1, 1)
+    ns = s * norm.cdf(x, mu, sigma)
+    return ns + nb
+
+@joblib.delayed
+def run(eps, n):
+    rng = np.random.default_rng(1)
+
+    t1 = []
+    t2 = []
+    for itry in range(100):
+        sigma = 0.1
+        results = []
+        for p in (eps, 1 - eps):
+            s = rng.normal(0, sigma, size=rng.poisson(p * n))
+            b = rng.uniform(-1, 1, size=n // 5)
+            x = np.append(s, b)
+            w, xe = np.histogram(x, bins=50, range=(-1, 1))
+            cost = ExtendedBinnedNLL(w, xe, model)
+            cost += NormalConstraint(("mu", "sigma"), (0, sigma), (1, 1))
+            s_start = eps * n
+            b_start = n
+            m = Minuit(cost, 
+                       b0=b_start, b1=b_start, b2=b_start,
+                       s=s_start + 1, mu=0, sigma=sigma)
+            m.limits[:4] = (0, None)
+            m.limits["mu"] = (-0.5, 0.5)
+            m.limits["sigma"] = (0.02, 1)
+            m.migrad()
+            if m.valid and m.accurate:
+                results.append((m.values["s"], m.errors["s"]))
+            else:
+                print(n, f"{eps:.2f}", itry, "signal" if p == eps else "background",
+                      "fit failed")
+        if len(results) < 2:
+            continue
+
+        (n_pass, sigma_pass), (n_fail, sigma_fail) = results
+
+        t1i = n_pass / (n_pass + n_fail)
+        t2i = g_var_py(n_pass, sigma_pass, n_fail, sigma_fail)
+        t1.append(t1i)
+        t2.append(t2i)
+
+    s1 = np.std(t1)
+    s2 = np.mean(t2) ** 0.5
+    return s1, s2
+
+data = {}
+eps = np.linspace(0, 1, 20)
+with joblib.Parallel(-1) as pool:
+    for n in (50, 100, 1000):
+        s1, s2 = np.transpose(pool(run(epsi, n) for epsi in eps))
+        data[n] = (eps, s1, s2)
+
+
+
plt.figure(figsize=(10, 4))
+for n, (eps, s1, s2) in data.items():
+    l, = plt.plot(eps, s1, ":", label=f"MC, {n=}")
+    plt.plot(eps, s2, "--", color=l.get_color(), label=f"formula, {n=}")
+    print(f"{n=} median deviation {np.median(s1[s2 > 0] / s2[s2 > 0]) - 1:.2f}")
+plt.subplots_adjust(right=0.6)
+plt.xlabel(r"$\epsilon$")
+plt.ylabel(r"$\sigma(\epsilon)$")
+plt.figlegend(loc=(0.69, 0.2));
+
+
n=50 median deviation 0.02
+n=100 median deviation -0.03
+n=1000 median deviation 0.04
+
+
+

+
+
+

Again, the derived formula does reasonably well with deviations around 5 % for \(n > 50\). This is usually sufficient to calculate error bars for efficiency plots. There should be some bias for small \(n\) as before, but it is difficult to see here due to other larger random scatter, since I generated less toys to keep the computation time on my laptop reasonable.

+

Finally, I compare the uncertainty estimates for various levels of background. The estimate with zero background corresponds to the estimate which one obtains with the naive formula.

+
+
plt.figure(figsize=(14, 4))
+eps = np.linspace(0, 1, 100)
+for n in (10, 50, 100):
+    n_pass = n * eps
+    n_fail = n * (1 - eps)
+    n_bkg = 0
+    col = None
+    for i, n_bkg in enumerate((0, 1, 5, 10)):
+        s2 = g_var_py(n_pass, (n_pass + n_bkg) ** 0.5, n_fail, (n_fail + n_bkg) ** 0.5) ** 0.5
+        col = plt.plot(eps, s2, "--" if i > 0 else ":", color=col,
+                       lw=0.5 * i + 1.5, label=f"{n=} {n_bkg=}")[0].get_color()
+plt.subplots_adjust(right=0.5)
+plt.figlegend(loc=(0.64, 0.2))
+plt.xlabel(r"$\epsilon$")
+plt.ylabel(r"$\sigma(\hat \epsilon)$")
+plt.semilogy();
+
+

+
+
+

The error of using a formula that neglects the background contribution is largest when the efficiency is close to zero or one.

+

These are frequently the interesting cases. For example, the tracking efficency of the LHCb experiment is estimated from fits to decays in calibration samples and the efficiency is around 96 %, so that the difference in the formulas may impact the estimated uncertainty.

+
+

Formula for fits to selected and total events

+

Sometimes it is not easy to split the two samples into failed and passed cases, one has instead fits to the passed and total cases.

+

In this case, we need to replace \(n_2\) and \(\sigma^2(\hat n_2)\) in the main formula for \(\sigma(\hat \epsilon)\), since we then only have \(n = n_1 + n_2\), \(\sigma^2(\hat n)\), \(n_1\), and \(\sigma^2(\hat n_1)\) available.

+

Since both estimates \(\hat n_1\) and \(\hat n_2\) are completely independent (again assuming independent background), their sum \(\hat n = \hat n_1 + \hat n_1\) must have the variance \(\sigma^2(\hat n) = \sigma^2(\hat n_1) + \sigma^2(\hat n_2)\).

+

Thus the formula for this case becomes

+

\[ +\sigma(\hat \epsilon) = \frac{\sqrt{ +n_1^2 \, (\sigma^2(\hat n) - \sigma^2(\hat n_1)) + +(n - n_1)^2 \, \sigma^2(\hat n_1) +}}{n^2}. +\]

+ + +
+
+ +
+ +
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HcmV?d00001 diff --git a/posts/Wilson Score Interval with Weighted Histograms.html b/posts/Wilson Score Interval with Weighted Histograms.html new file mode 100644 index 0000000..71ccc39 --- /dev/null +++ b/posts/Wilson Score Interval with Weighted Histograms.html @@ -0,0 +1,503 @@ + + + + + + + + + +Hans Dembinski’s blog - The Wilson score interval and weighted histograms + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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The Wilson score interval and weighted histograms

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The Wilson Score interval (WSI) was won the LHCb internal challenge for the most suitable interval for efficiencies. The question what to do when events are weighted was left open at the time. It is addressed here, the WSI can be used also with weighted histograms if ordinary counts in unweighted histograms are replaced with effective counts in weighted histograms.

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import numpy as np
+import boost_histogram as bh
+from matplotlib import pyplot as plt
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https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval

+

Wilson score interval: \[ +p_\text{min} = \frac{n_S + \frac 12 z^2}{n_S + n_F + z^2} - \frac{z}{n_S + n_F + z^2} \sqrt{\frac{n_S n_F}{n_S + n_F} + \frac{z^2}{4}} +\] \[ +p_\text{max} = \frac{n_S + \frac 12 z^2}{n_S + n_F + z^2} + \frac{z}{n_S + n_F + z^2} \sqrt{\frac{n_S n_F}{n_S + n_F} + \frac{z^2}{4}} +\] with \(n_S\) counts of selected events, \(n_F\) counts of rejected events, \(z\) number of sigmas (\(z = 1\) for standard intervals).

+

In case of weighted events, the \(n_S,n_F\) are the effective counts, computed as \[ +n_\text{effective} = \frac{\big(\sum_i w_i\big)^2}{\sum_i w^2_i}. +\] In boost-histogram, this is computed as n_eff = h.values() ** 2 / h.variances() for a histogram h with WeightedSum storage.

+
+
def wilson_score_interval(n_s, n_f, z=1):
+    n = n_s + n_f
+    z2 = z * z
+    center = (n_s + 0.5 * z2) / (n + z2)
+    delta = z / (n + z2) * np.sqrt(n_s * n_f / n + 0.25 * z2)
+    return center - delta, center + delta
+
+
+
p_truth = 0.2
+
+rng = np.random.default_rng(1)
+
+w_s = bh.accumulators.WeightedSum()
+w_f = bh.accumulators.WeightedSum()
+
+w = rng.exponential(10, size=100)
+m = rng.uniform(size=len(w)) < p_truth
+
+w_s.fill(w[m])
+w_f.fill(w[~m])
+
+
WeightedSum(value=944.245, variance=24939.8)
+
+
+
+
n_s = w_s.value ** 2 / w_s.variance if w_s.variance > 0 else 0
+n_f = w_f.value ** 2 / w_f.variance if w_f.variance > 0 else 0
+
+p = n_s / (n_s + n_f)
+p_min, p_max = wilson_score_interval(n_s, n_f)
+
+plt.errorbar([1], [p], [[p-p_min], [p_max-p]], fmt="ok")
+
+print(f"efficiency = {p:.2f} - {p-p_min:.2f} + {p_max-p:.2f}")
+
+
efficiency = 0.16 - 0.05 + 0.06
+
+
+

+
+
+

Note that the error bar is asymmetric in general. This is especially noticable when p is close to 0 or 1.

+

Let’s check with a parametric bootstrap simulation whether this uncertainty is correct. It is parametric because I put in we know that the weights are exponentially distributed.

+
+
rng = np.random.default_rng(1)
+
+w_mean = np.mean(w)
+
+p_boot = []
+for iboot in range(1000):
+    wi = rng.exponential(w_mean, size=len(w))
+    mi = rng.uniform(size=len(w)) < p
+    t_s = bh.accumulators.WeightedSum()
+    t_f = bh.accumulators.WeightedSum()
+    t_s.fill(wi[mi])
+    t_f.fill(wi[~mi])
+    p_boot.append(t_s.value / (t_s.value + t_f.value))
+    
+center = np.mean(p_boot)
+delta = np.std(p_boot)
+p_min_boot = center - delta
+p_max_boot = center + delta
+
+plt.errorbar([1], [p], [[p-p_min], [p_max-p]], fmt="ok")
+plt.errorbar([2], [p], [[p-p_min_boot], [p_max_boot-p]], fmt="sr")
+plt.axhline(p_truth, ls="--", color="0.5")
+
+print(f"efficiency = {p:.2f} - {p-p_min:.2f} + {p_max-p:.2f}")
+print(f"efficiency(boot) = {p:.2f} - {p-p_min_boot:.2f} + {p_max_boot-p:.2f}")
+
+
efficiency = 0.16 - 0.05 + 0.06
+efficiency(boot) = 0.16 - 0.05 + 0.05
+
+
+

+
+
+

Ok, check passed.

+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/Wilson Score Interval with Weighted Histograms_files/figure-html/cell-5-output-2.png b/posts/Wilson Score Interval with Weighted Histograms_files/figure-html/cell-5-output-2.png new file mode 100644 index 0000000000000000000000000000000000000000..e1a3a45693e166bbde5230fd77c50802865f9277 GIT binary patch literal 4403 zcmcgwc~lekzMmwLsDRp97Z$Uw>|8i=yf@(ctii0tcaz1}O21d_GPuP){LEI`dQa*2Pesnr$H)o` zS6{FzA0O1c=f9imy33WbNiFZ+uhR}ipZ^^68 z(sn>567pr5X+lZb!zzWBEcaB?c_ibxm*=a32i?;@hs;wOm19@7!@7EKKS5*Io=xVl zovU|O%RmMw81dGEM(MTq&x_z|LEV-+>3PrOW|5ai;$0AV?>0+g=g?nIkR{`~bApb? zM@#;69gd4xj;A#;gp64S4}7-TPnz~eyb&Wr^dozf38tvL>Ybwnu=nhTB4gX}G(?3P z!1dyOCCmU>2V|o@-CK;7#}VjGkLQnb_0bV+uDnnTk+O5UR3PKos$5suu_BZP&+QIU z4r-X9qmd8MZI|;6>4VPVK-S38#3Gf;;&zgEX+jw;Id@ED!y|^HfA%v(0x{c36st0M z2Yk&T237^ibu>udV>x}6S&QlDlY_)ttgP{(V%w7=2fIm>LH9Xv}R{&f+>kEmmbR z9QjTyqy0SZxNf=cR#?Z!o9H=MD$K`)*jIvu#Xj+M=r)`H*G)I(uEG_Dyk}rK^(jS) zv?+!uDg-A@K8fDN0UQJsrl}(4QP->B+wUXCf3@e^xsB#kW+o(S9{p=g6g}$X!PAJ2 zys4b(@2vRmqxBy3KZW#P71drb;bnlWDz7p{&kiovXNg{ctzZPmlRHim9EhlkA{(YR zSR(iJ1KNm=*d9C?{{YK|%6wmx5aZsV1`6Q-=U%4x6QLp370Zg71dn=6{E`k@=eO=# z^>qt`$64qjx|pt-Ksd+Gg)bKv_F~i~lj}Dbk>O&PuN;>cryFwiF<6*lViqeuHQl!K zjUWu{$0{)g$#n?S?IGnBmy*OMfSsx>uy*o4L+#Lm0ZR!NbE^uIr zCj|~b(RZl4vVy{>Q#DwaXku2s^N1?%Hk5|3$2lbsz$LXH!}uGqmkaz9f_#v(kM~^A z9Pa`lY0s{rOw@oWio|1TH*w0*FygDtV<@W5E#h9|rZ9l9Vzh}hQUUgRO?hHUN>P{0 z^i&Jv{!71gj7nMLQW14JtP|ce@1iTb8LGFR18rS7{%y0 za$;Qy%C-nT!b@c=Q=l?KUN&fIfmD6vg6P!x#GCLw73vAK+;#<7BQPUjoX~+;AZ3V8 z2#e|EarHopl!juJFJv2N8TF&%Z>j~L2^<^9rcsZvztMpvZZaB~i}nFZn_4m*uOWG7 zX8(g!U(76vQKkPG^3 ziqW$B&13&ZYWtUTC`@e5)GzmF+!nLc zgulI)?oc07d!aZt!A#TI<7tAdqCx1eaX{j>eeRNn>*sFjXya>jl2*kFtp(1@cMcEV z^VzV&ljAQ4Ns>1P-ed;8`EDmE+@^aAk;!e=@2|1g1!;SZK6)2&J=Nr^dX54k5&O#kvga)q<)F=HK zYlaXJMN;r!UfnK8=~Nk!R}R^=oEb;*lq5)imh^6V6CLL2LUz{jmC6o;9>WHe2`r@u zwZXCC?9C54AUfK#N{SsWish?(43Rho-MfF^X;?FiHYsB>mc1j*j{mfmqyOf9sKr_fLze2v_N3}B1mL^%QotqM z9m3=##~(nSuWC&W!LUw6{wpB` zs>6&2&@>sTg(%FQ*ROy!;8CXNmxIeTr?hU05<|{B1PkjwZXHU0)9ROs(G_c{K;`_Q z^vo3@t+XkB zfY&Mk&$xNvdZ_^8#PWT;iih8K>(bH)4G2kAmw40BKRm)nxB?0=L~IjMa|Vu`zQP85 zV&1!i=Z-0RT&!T|HC!>@I{=Q-FCBM5c6U5<(uXhaeJ#i&V%( z>KR>Nu1-c?+T;{vw$sxSTZoS<<)D9LvlIQxU^Oyk5I!Ze(o}~cm8GZM{gD(L71#d& zr5=#D@b7u6wDjti5(MRFSJr@_<+i11c2PP=#MA?N=lIFDQ;Co~oO7pt3o}*p9BiVq z;?!vxeM##FgY04BL%(rMH=xa4qR~hZVbxE^iyBqZ;XMV`a&dp=rm`HL-; zf&F!zDi-Ch_kAp5e@v4aa7CwXjX4s}wY=`!sK(s7a-sCPgByFPQ<#ym&HEhx)_O?0 zB-y9fL18d|(Opn_gFI9EL|1vW=kd+=5k6+Z|NIzscbuPz9_LLiGioE(Pi~wS{-M>$ z&p*g(trpW3l4S00Qx;ticmhd8#vn;lPnABuqL##Y1T_9&e_1pzA6tQm5QG1PMWmI%p|5&~(MsqUiaPOD5D+qRMTH>8Do< z!;{aYrYuANmh?Sz8=#4v9|tp@t?cDg8rJRQL=^-YP6>M7UE^-KUVDFutUYx?SYPd@ zhEc+4oFCJjbC{9h4$=D+@2&^Z32}fcu$J$)zPY4WJa=ni@m7ONt5NaNS*~O6yBu6G zK-8R%d=t`qe03e$BfSz-D$rsiQfJ$pK0Gtz9cIzj$C8z4k4$-Ax%t4xGN<9dbNstk z4@QCIF50KLiE>6}irWNRe1Tv-v#u?HXjA+HhPp5+5T01y)lM=q; zd2B;&=6D8cUzAO?x6v47J+}KbY%}>e}-?WUj2IPLvj*^Up(SrT=E($}n+d25u$9?Am(%2?>e>4j z{t>}_atLCB&B3Q>FIt9NLT2_9TQ#RokQOMyOn9!vU7+2{##P$!|IK}mmcqudGMD>4 z9j}kZM6Yf=hU|>1FG2+jE5HWk*RdzPW#(7c0x%;=4PW}LMY);7wZG08Z)Or!1Ydt< zl9D{DdlVq+6w<_3VBPxU#i9K?h^tAL?cHszn*8w141I<{ZWLBjdO>{!sA)v*UkNuQ zAa#Er)u%Xw*|qvDVakS^bv(p|G(E94*uX89K0Q-#(s3pRPhGZTC^iZy^;e5B)w%kQ zsb0~T#eOtWsuNGn5Zwavf%LEd^`C;9dvmVkXmvKlxLvYjhMZ+rp&fSoAJFqWVg9S? 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+
+ +
+ +
+
+
+

All decays into two Kaons

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

In a recent paper review, the question came up how many resonances there are which decay into two kaons. It is not easy to look this up somewhere. My solution was to extract the information in the following way from the Pythia8 decay database. I have to first get the database, which is in XML format. I parse the XML database and then go through all entries and isolate the particles which exclusively decay into two kaons.

+
+
import xml.etree.ElementTree as ET
+from particle import literals as lp
+from matplotlib import pyplot as plt
+from pathlib import Path
+import chromo
+from chromo.models import Pythia8
+from chromo.util import _cached_data_dir
+from chromo.kinematics import CenterOfMass, TeV
+
+# HACK: ensure that Pythia8 data is downloaded without running Pythia8
+pythia8_data = Path(_cached_data_dir(Pythia8._data_url))
+
+
+
# TODO: we should make this information accessible via the chromo API
+fn = pythia8_data / "xmldoc" / "ParticleData.xml"
+lines = ["<root>"]
+with open(fn) as f:
+    for iline, line in enumerate(f.readlines()):
+        if iline < 166: continue
+        if "</chapter>" in line: break
+        lines.append(line)
+lines.append("</root>")
+
+tree = ET.fromstring("".join(lines))
+
+all_kk = []
+
+pid_k = int(lp.K_plus.pdgid)
+
+for p in tree.findall("particle"):
+    for ch in p.findall("channel"):
+        da = [int(x) for x in ch.attrib["products"].split()]
+        if len(da) != 2: continue
+        if all(abs(pi) == pid_k for pi in da):
+            pa = p.attrib
+            all_kk.append((pa["name"], float(pa["m0"])))
+
+

The plot shows where the resonances are located in the invariant mass distribution. The y-axis has no meaning, it just separates the points vertically.

+
+
from cycler import cycler
+from matplotlib.colors import TABLEAU_COLORS
+
+c = cycler(marker="osd") * cycler(color=TABLEAU_COLORS)
+
+plt.gca().set_prop_cycle(c)
+
+i = 0
+for kk in sorted(all_kk, key=lambda x: x[1]):
+    plt.plot(kk[1] * 1e3, i, label=kk[0])
+    i+=1
+plt.legend(ncol=3)
+plt.xlabel("mass / MeV$c^{-2}$");
+
+

+
+
+

And here is a sorted list of the masses.

+
+
for kk in sorted(all_kk, key=lambda x:x[1]):
+    print(kk[0], kk[1] * 1e3)
+
+
a_0(980)0 983.5
+f_0(980) 1000.0
+phi 1019.46
+f_2 1275.1
+a_20 1318.3
+f_0(1370) 1350.0
+a_0(1450)0 1474.0
+f'_2(1525) 1525.0
+f_0(1710) 1720.0
+rho(1700)0 1720.0
+D0 1864.86
+chi_0c 3414.75
+chi_2c 3556.2
+psi(2S) 3686.11
+B0 5279.58
+B_s0 5366.7699999999995
+
+
+ + + +
+ +
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z$GbOC3burKj%oY7neUMcgpane?ohnAQP}#=NHHK^D`v;G=8OYCXO+=`QoNMREp+Q; z(2fqI`%Gx_3Cv+&f_!$zhKt$-g0yk#u)?F&3%O6)4*nsbwvQY4Vl910#1+;b)(=h! zx4Oh7A`^@ScuVwH@NnU|Z^s%=jmQ80BMkoi+?n1<2AAyFd8?M2KEm;j49zXNYH2DK Gzx@yYxgD+m literal 0 HcmV?d00001 diff --git a/posts/benchmark of building an array with numba.html b/posts/benchmark of building an array with numba.html new file mode 100644 index 0000000..b540083 --- /dev/null +++ b/posts/benchmark of building an array with numba.html @@ -0,0 +1,624 @@ + + + + + + + + + +Hans Dembinski’s blog - Benchmark of building an array with numba + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Benchmark of building an array with numba

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

Here I try out different ways of making arrays of tuples with Numba where the number of tuples that need to be produced is not known in advance. The fastest way is to make a list of tuples, and then convert this into a Numpy array before returning. The baseline shown in the last plot is the fastest possible way of pre-allocating a Numpy array with the right size upfront, but this cannot be used if the number of tuples is not known in advance.

+
+
import numba as nb
+import numpy as np
+import awkward as ak
+print(f"{nb.__version__=}")
+print(f"{ak.__version__=}")
+
+
nb.__version__='0.53.1'
+ak.__version__='1.1.2'
+
+
+
+
@nb.njit
+def make0(n):
+    r = np.empty((n, 4))
+    for i in range(n):
+        # simulate some work
+        x = np.random.rand()
+        y = np.random.rand()
+        z = np.random.rand()
+        t = np.random.rand()
+        r[i,0] = x
+        r[i,1] = y
+        r[i,2] = z
+        r[i,3] = t
+    return r
+
+make0(3)
+
+
array([[0.25473852, 0.47593451, 0.04606011, 0.55805331],
+       [0.77287874, 0.52879803, 0.91944361, 0.98775869],
+       [0.01221509, 0.30608935, 0.90908168, 0.07629862]])
+
+
+
+
%timeit make0(1000)
+
+
26.1 µs ± 1.51 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
+
+
+
+
@nb.njit
+def make1(n):
+    r = []
+    for i in range(n):
+        # simulate some work
+        x = np.random.rand()
+        y = np.random.rand()
+        z = np.random.rand()
+        t = np.random.rand()
+        r.append((x, y, z, t))
+    return np.asarray(r)
+
+make1(3)
+
+
array([[0.40883519, 0.14577046, 0.20275019, 0.24327066],
+       [0.7325605 , 0.76214408, 0.47819782, 0.93920595],
+       [0.24872141, 0.92187207, 0.6691103 , 0.5643187 ]])
+
+
+
+
%timeit make1(1000)
+
+
36.9 µs ± 1.55 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
+
+
+
+
@nb.njit
+def make2(n):
+    r = []
+    for i in range(n):
+        ri = []
+        # simulate some work
+        x = np.random.rand()
+        y = np.random.rand()
+        z = np.random.rand()
+        t = np.random.rand()
+        ri.append(x)
+        ri.append(y)
+        ri.append(z)
+        ri.append(t)        
+        r.append(ri)
+    return np.asarray(r)
+
+make2(3)
+
+
array([[0.13088925, 0.25692562, 0.48062596, 0.39696723],
+       [0.3552985 , 0.99054454, 0.38778594, 0.94289909],
+       [0.76625383, 0.01588522, 0.43885749, 0.73488083]])
+
+
+
+
%timeit make2(1000)
+
+
419 µs ± 18.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
+
+
+
+
@nb.njit
+def make3(n):
+    r = []
+    for i in range(n):
+        # simulate some work
+        x = np.random.rand()
+        y = np.random.rand()
+        z = np.random.rand()
+        t = np.random.rand()
+        r.append(x)
+        r.append(y)
+        r.append(z)
+        r.append(t)
+    return np.array(r).reshape((n, 4))
+
+make3(3)
+
+
array([[0.60766162, 0.66840636, 0.2019663 , 0.92669818],
+       [0.69739471, 0.33933288, 0.9356699 , 0.61572307],
+       [0.43173953, 0.40320305, 0.80278753, 0.46843452]])
+
+
+
+
%timeit make3(1000)
+
+
49.8 µs ± 2.36 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
+
+
+
+
@nb.njit
+def _make4(b, n):
+    for i in range(n):
+        x = np.random.rand()
+        y = np.random.rand()
+        z = np.random.rand()
+        t = np.random.rand()
+        b.begin_tuple(4)
+        b.index(0).append(x)
+        b.index(1).append(y)
+        b.index(2).append(z)
+        b.index(3).append(t)
+        b.end_tuple()
+    return b
+
+def make4(n):
+    return _make4(ak.ArrayBuilder(), n).snapshot()
+
+make4(3)
+
+
<Array [(0.376, 0.269, 0.149, ... 0.409, 0.38)] type='3 * (float64, float64, flo...'>
+
+
+
+
%timeit make4(1000)
+
+
356 µs ± 11.1 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
+
+
+
+
@nb.njit
+def _make5(b, n):
+    for i in range(n):
+        x = np.random.rand()
+        y = np.random.rand()
+        z = np.random.rand()
+        t = np.random.rand()
+        b.begin_list()
+        b.append(x)
+        b.append(y)
+        b.append(z)
+        b.append(t)
+        b.end_list()
+    return b
+
+def make5(n):
+    return _make5(ak.ArrayBuilder(), n).snapshot()
+
+make5(3)
+
+
<Array [[0.216, 0.481, ... 0.247, 0.0629]] type='3 * var * float64'>
+
+
+
+
%timeit make5(1000)
+
+
263 µs ± 10.9 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
+
+
+
+
import timeit
+import matplotlib.pyplot as plt
+
+n = np.geomspace(1, 1000000, 20).astype(int)
+
+ys = []
+for fn in (make0, make1, make2, make3, make4, make5):
+    y = []
+    for ni in n:
+        m, t = timeit.Timer(lambda : fn(ni)).autorange()
+        y.append(t/m)
+    ys.append(y)
+
+
+
for y, label in zip(ys, ("baseline",
+                         "list+tuple+asarray",
+                         "list+list+asarray",
+                         "list+array+reshape",
+                         "ArrayBuilder+tuple",
+                         "ArrayBuilder+list")):
+    plt.plot(n, np.divide(y, 1e-6), label=label)
+plt.legend()
+plt.xlabel("array length")
+plt.ylabel("time / μs")
+plt.loglog();
+
+

+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/benchmark of building an array with numba_files/figure-html/cell-16-output-1.png b/posts/benchmark of building an array with numba_files/figure-html/cell-16-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..714689e96ce72a747a46c80e0768a7f6a8f3704a GIT binary patch literal 32665 zcmZr&bx@T-y9bd@rCSWTyGs-}lpK(5q`MoWr2Eh%-6bH>rKEs#cXxN)<(qr&A9wDI zjKhKRo_F`%eV$(}A)n+WG0{lS;NajerKKdGaB%Q};0p@{3A|#gjIIR!<8%C==BQ+2 z?C7FzZv-c&?`Ugj<7jDS@XFc9-oebqnuD2>nU(RCsiUK<13wFk)&KqivyHt8i*Rod z54Z@bt(3Y092`3C(-%BkN*WOyoDY(;gqX5x+Wx$YXT&6B+woBELI?(1tdzlF^LVj( zHJ;TlwCtf+V;)wlsPUmhS@CoA$Is*WV}(KnxKv_2IMW_;PToFqQn)YuBwh~Qp7-r8 z+|F$ZUkdgPoHmq@p*P#l5BwnZ10o`L^<5Ym z8u%p$0X`ny2SpeoJ?+zb)RjMd0^j~f!gqo{z!6V^(}l%>Aml=q&n+szR+PW$)FCUqmZLAEfpbb4y=65^!QUe=ds@p`X zy_SO1>!K`sO>R(&M#D}>zTy~Qc4>&CeD9r;%$#Hk#Od^1hSSl}xw^i_cdWwypqODU z9!%;cq>4vAu+E0`RY07ht_+zUoyz<9#pbyI-MToP(HroOr%nqY9ZbbZaHyPQHH;mf 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splits + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Chance of seeing a one-sigma deviation in random splits

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

We randomly split a sample into two parts and check whether their respective arithmetic means deviate by more than one standard deviation from each other. The chance for this two happen appears to be independent of the distribution from which the original samples are drawn.

+
+
import numpy as np
+
+
+
rng = np.random.default_rng(0)
+
+for distribution in (rng.normal, rng.uniform, rng.exponential):
+    x = rng.uniform(size=20)
+    
+    deltas = []
+    ntry = 10000
+    for itry in range(ntry):
+        rng.shuffle(x)
+        a = x[:int(len(x) / 2) ]
+        b = x[int(len(x) / 2):]
+        ma = np.mean(a)
+        mb = np.mean(b)
+        va = np.var(a) / len(a)
+        vb = np.var(b) / len(b)
+        v = va + vb
+        deltas.append((ma - mb) / v ** 0.5)
+    deltas = np.array(deltas)
+    
+    print(distribution.__name__, "fraction with < 1 sigma deviation", np.sum(deltas < 1) / len(deltas))
+
+
normal fraction with < 1 sigma deviation 0.8255
+uniform fraction with < 1 sigma deviation 0.8238
+exponential fraction with < 1 sigma deviation 0.8281
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/comparison_chisquare_test_statistics.html b/posts/comparison_chisquare_test_statistics.html new file mode 100644 index 0000000..fb139c5 --- /dev/null +++ b/posts/comparison_chisquare_test_statistics.html @@ -0,0 +1,752 @@ + + + + + + + + + +Hans Dembinski’s blog - Comparison of asymptotically chisquare-distributed test statistics + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Comparison of asymptotically chisquare-distributed test statistics

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

This notebook is released under the MIT license. See LICENSE.

+

For GoF tests, we often use a test statistic that is asymptotically \(\chi^2\) distributed.

+

We compute the test statistic on binned data with \(k\) bins with Poisson-distributed counts \(n_i\) for which we have estimates of the expected counts \(\nu_i\), typically obtained from a fitted model with \(m\) free parameters. If the data is truely sampled from the model (the \(H_0\) hypothesis), the test statistic \(X\) is asymptotically distributed as \(\chi^2(k - m)\).

+

Small p-values \(P = \int_{X}^\infty \chi^2(X'; k - m) \, \text{d}X'\) can be used as evidence against the hypothesis \(H_0\).

+

Note: the asymptotic limit is assumed to be taken while keeping the binning constant. If the binning is adaptive to the sample size so that \(\nu_i\) remains constant, \(X\) will not approach the \(\chi^2(k - m)\) distribution.

+

There are candidates for an asymptotically \(\chi^2\)-distributed test statistic.

+
    +
  • Pearson’s test statistic \[ +X_P = \sum_i \frac{(n_i - \nu_i)^2}{\nu_i} +\]

  • +
  • Pearson’s test statistic with \(\nu_i\) replaced by bin-wise estimate \(n_i\) \[ +X_N = \sum_i \frac{(n_i - \nu_i)^2}{n_i} +\]

  • +
  • Likelihood ratio to saturated model (see e.g. introduction of https://doi.org/10.1088/1748-0221/4/10/P10009) \[ +X_{L} = 2\sum_i \Big( n_i \ln\frac{n_i}{\nu_i} - n_i + \nu_i \Big) +\] The statistic is equal to \(-2\ln(L/L_\text{saturated})\), with \(L = \prod_i P_\text{poisson}(n_i; \nu_i)\) and \(L_\text{saturated} = \prod_i P_\text{poisson}(n_i; n_i)\), the likelihood for the saturated model with \(\nu_i = n_i\). The saturated model has the largest possible likelihood given the data.

  • +
  • Transform-based test statistic \[ +X_T = \sum_i z_i^2 \quad\text{with}\quad z_i = q_\text{norm}\bigg(\sum_{k=0}^{n_i} P_\text{poisson}(k; v_i)\bigg) +\] where \(q_\text{norm}\) is the quantile of the standard normal distribution. The double-transform converts a Poisson distributed count \(k\) into a variable \(z\) that has a standard normal distribution.

  • +
+
+
from sympy import *
+
+n = symbols("n", integer=True, positive=True)
+v = symbols("v", real=True, positive=True)
+
+
+def poisson(n, v):
+    return v ** n * exp(-v) / factorial(n)
+
+poisson(n, v)
+
+

\(\displaystyle \frac{v^{n} e^{- v}}{n!}\)

+
+
+
+
X_L = simplify(- 2 * (log(poisson(n, v)) - log(poisson(n, n)))); X_L
+
+

\(\displaystyle 2 n \log{\left(n \right)} - 2 n \log{\left(v \right)} - 2 n + 2 v\)

+
+
+

Given this choice, it is fair to ask which test statistic performs best. The test statistics may approach the \(\chi^2(k - m)\) distribution with different speeds.

+

We check this empirically with toy simulations. For each toy experiment, we use a fixed expected value \(\nu_i = \mu\) and draw \(k\) Poisson-distributed numbers \(n_i\) from \(\mu\), corresponding to \(k\) bins in the toy experiment. We then compute the test statistics \(X\) and its p-value, assuming it is \(\chi^2(k)\) distributed (\(m = 0\) here). This is repeated many times to get a histogram of p-values. The distribution is uniform if the test statistic \(X\) is indeed \(\chi^2(k)\) distributed. Deviations from uniformity indicate that the test statistic has not reached the asymptotic limit yet.

+

We then quantify the agreement of the p-value distribution with a flat distribution with the reduced \(\chi^2\) value.

+
+
import numpy as np
+import numba as nb
+from scipy.stats import chi2, poisson, norm
+import matplotlib.pyplot as plt
+from pyik.mplext import plot_hist
+
+
+
# numba currently does not support scipy, so we cannot access
+# scipy.stats.norm.ppf and scipy.stats.poisson.cdf in a JIT'ed
+# function. As a workaround, we wrap special functions from
+# scipy to implement the needed functions here.
+
+from numba.extending import get_cython_function_address
+import ctypes
+
+
+def get(name, narg):
+    addr = get_cython_function_address("scipy.special.cython_special", name)
+    functype = ctypes.CFUNCTYPE(ctypes.c_double, *([ctypes.c_double] * narg))
+    return functype(addr)
+
+
+erfinv = get("erfinv", 1)
+gammaincc = get("gammaincc", 2)
+
+
+@nb.vectorize('float64(float64)')
+def norm_ppf(p):
+    return np.sqrt(2) * erfinv(2 * p - 1)
+
+
+@nb.vectorize('float64(intp, float64)')
+def poisson_cdf(k, m):
+    return gammaincc(k + 1, m)
+
+
+# check implementations
+m = np.linspace(0.1, 3, 20)[:,np.newaxis]
+k = np.arange(10)
+got = poisson_cdf(k, m)
+expected = poisson.cdf(k, m)
+np.testing.assert_allclose(got, expected)
+
+p = np.linspace(0, 1, 10)
+got = norm_ppf(p)
+expected = norm.ppf(p)
+np.testing.assert_allclose(got, expected)
+
+
+
@nb.njit
+def xn(n, v):
+    r = 0.0
+    k = 0
+    for ni in n:
+        if ni > 0:
+            k += 1
+            r += (ni - v) ** 2 / ni
+    return r, k
+
+
+@nb.njit
+def xp(n, v):
+    return np.sum((n - v) ** 2 / v), len(n)
+
+
+@nb.njit
+def xl(n, v):
+    r = 0.0
+    for ni in n:
+        if ni > 0:
+            r += ni * np.log(ni / v) + v - ni
+        else:
+            r += v
+    return 2 * r, len(n)
+
+
+@nb.njit
+def xt(n, v):
+    p = poisson_cdf(n, v)
+    z = norm_ppf(p)
+    return np.sum(z ** 2), len(n)
+        
+
+@nb.njit(parallel=True)
+def run(n, mu, nmc):
+    args = []
+    for i in range(len(n)):
+        for j in range(len(mu)):
+            for imc in range(nmc):
+                args.append((i, j, imc))
+    results = np.empty((4, len(n), len(mu), nmc, 2))
+    for m in nb.prange(len(args)):
+        i, j, imc = args[m]
+        ni = n[i]
+        mui = mu[j]
+        np.random.seed(imc)
+        x = np.random.poisson(mui, size=ni)
+        rp = xp(x, mui)
+        rn = xn(x, mui)
+        rl = xl(x, mui)
+        rt = xt(x, mui)
+        results[0, i, j, imc, 0] = rp[0]
+        results[0, i, j, imc, 1] = rp[1]
+        results[1, i, j, imc, 0] = rn[0]
+        results[1, i, j, imc, 1] = rn[1]
+        results[2, i, j, imc, 0] = rl[0]
+        results[2, i, j, imc, 1] = rl[1]
+        results[3, i, j, imc, 0] = rt[0]
+        results[3, i, j, imc, 1] = rt[1]
+    return results
+
+
+mu = np.geomspace(1e-1, 1e3, 17)
+n = 1, 3, 10, 30, 100, 1000
+nmc = 10000
+
+result = run(n, mu, nmc)
+
+
+
def reduced_chi2(r):
+    p = 1 - chi2.cdf(*np.transpose(r))
+    bins = 20
+    xe = np.linspace(0, 1, bins + 1)
+    n = np.histogram(p, bins=xe)[0]
+    v = len(p) / bins
+    return np.sum((n - v) ** 2 / v) / bins
+
+matrix = np.empty((len(result), len(n), len(mu)))
+for k in range(len(result)):
+    for i, ni in enumerate(n):
+        for j, mui in enumerate(mu):
+            matrix[k, i, j] = reduced_chi2(result[k, i, j])
+
+
+
def plot(r, **kwargs):
+    p = 1 - chi2.cdf(*np.transpose(r))
+    bins = 20
+    xe = np.linspace(0, 1, bins + 1)
+    n = np.histogram(p, bins=xe)[0]
+    plot_hist(xe, n, **kwargs)
+
+
+for i, ni in enumerate(n):
+    for j, mui in enumerate(mu):
+        if j % 4 == 0:  # draw only every forth value of mu
+            fig, ax = plt.subplots(1, 4, figsize=(15, 5))
+            plt.suptitle(f"k = {ni} mu = {mui:.2f}")
+            for k, t in enumerate("PNLT"):
+                plt.sca(ax[k])
+                plot(result[k, i, j])
+                plt.title(f"$X_{t}$ ($\chi^2/n_\mathrm{{dof}} = ${matrix[k, i, j]:.2g})")
+
+
RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`).
+  fig, ax = plt.subplots(1, 4, figsize=(15, 5))
+
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We can by eye that \(X_P\) gives the best results. Its p-value distribution converges to a uniform distribution for lower values of \(\mu\) and \(n\) than the other two test statistics. We summarize this by plotting the reduced \(\chi^2\) for flatness as a function of \(\mu\) and \(n\).

+
+
fig, ax = plt.subplots(1, len(result), figsize=(15, 5), sharex=True, sharey=True)
+for i, ni in enumerate(n):
+    for axi, matrixi in zip(ax, matrix):
+        axi.plot(mu, matrixi[i],
+                 label=f"$k = {ni:.1f}$")
+        
+for axi, t in zip(ax, "PNLT"):
+    axi.set_title(f"$X_{t}$")
+    axi.axhline(1, ls=":", color="0.5", zorder=0)
+    axi.set_xlabel("$\mu$")
+ax[0].set_ylabel("$\chi^2 / n_\mathrm{dof}$")
+ax[1].legend()
+plt.loglog();
+
+

+
+
+
+
for i, ni in enumerate(n):
+    plt.figure()
+    plt.title(f"k = {ni}")
+    for j, t in enumerate("PNLT"):
+        plt.plot(mu, matrix[j, i] / matrix[0, i],
+                 label=f"$X_{t}$")
+    plt.loglog()
+    plt.xlabel("$\mu$")
+    plt.ylabel("red. $\chi^2$ ratio")
+    plt.legend()
+
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+

The classic test statistics \(X_P\) perform best overall, it converges to the asymptotic \(\chi^2\) distribution for smaller values of \(\mu\) and \(k\) and is also the simplest to compute.

+

The statistic \(X_L\) shows a curious behavior, it converges well around \(k=10\), on par with \(X_P\), but the convergence gets worse again for \(k > 10\).

+

The study confirms that none of the test statistics is doing well if the expected count per bin is smaller than about 10. If the original binned data contains bins with small counts, it is recommended to compute the distribution of the test statistic via parametric bootstrapping from the fitted model instead of using the asymptotic \(\chi^2\) distribution. Parametric bootstrapping in this case simplifies to drawing new samples for each data bin from a Poisson distribution whose expectation is equal to the expected count per bin predicted by the model. For many such samples one computes the test statistic to obtain a distribution. The p-value is then given by the fraction of sample values that are higher than the original value.

+ + + +
+ +
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+
+ +
+ +
+
+
+

Fitting mixture models with COWs

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

This is a demo of how to use custom orthogonal weight functions (COWs) to generate weights which extract one component of a mixture. Find out more in our paper.

+
+
import numpy as np
+from numba_stats import bernstein, norm, expon
+import matplotlib.pyplot as plt
+from scipy.integrate import quad
+from typing import Callable
+
+
+
rng = np.random.default_rng(1)
+
+s_x = rng.normal(1, 0.1, size=10000)
+b_x = rng.exponential(1, size=100000)
+s_y = rng.exponential(1, size=len(s_x))
+b_y = rng.normal(0, 1, size=len(b_x))
+
+x = np.append(s_x, b_x)
+y = np.append(s_y, b_y)
+id = np.append(np.ones_like(s_x, dtype=bool), np.zeros_like(b_x, dtype=bool))
+
+xrange = 0, 2
+yrange = 0, 4
+m = (x > xrange[0]) & (x < xrange[1]) & (y > yrange[0]) & (y < yrange[1])
+x = x[m]
+y = y[m]
+id = id[m]
+
+s_x = x[id]
+s_y = y[id]
+b_x = x[~id]
+b_y = y[~id]
+
+plt.figure()
+plt.hist(x, bins=100, range=xrange)
+plt.hist(s_x, bins=100, range=xrange)
+
+plt.figure()
+plt.hist(y, bins=100, range=yrange)
+plt.hist(s_y, bins=100, range=yrange);
+
+

+
+
+

+
+
+
+
def gs(m):
+    return norm.pdf(m, 1, 0.1)
+
+def gb(m):
+    return expon.pdf(m, 0, 1)
+
+def g(m):
+    return gs(m) + 10 * gb(m) / 11.
+
+def Gs(m):
+    return norm.pdf(m, 1.1, 0.2)
+
+def flat(m):
+    return np.ones_like(m)
+
+
+
def _A(I, S, B, xrange=(-np.inf, np.inf)):
+    nB = len(B)
+    W = np.empty((1 + nB, 1 + nB))
+    F = [S] + B
+    for i in range(1 + nB):
+        for j in range(i + 1):
+            W[i, j] = quad(lambda m: F[i](m) * F[j](m) / I(m), *xrange)[0]
+            if i != j:
+                W[j, i] = W[i, j]
+    return np.linalg.inv(W)
+
+def cow(m, I, S, B, xrange=(-np.inf, np.inf)):
+    if isinstance(B, Callable):
+        B = [B]
+    a = _A(I, S, B, xrange=xrange)
+    F = [S(m)] + [Bi(m) for Bi in B]
+    return (a[0] @ F) / I(m)
+
+
+
w0 = cow(x, g, gs, gb, xrange=xrange)
+plt.plot(x, w0, ".", ms=1, mew=0);
+
+

+
+
+
+
plt.hist(y, weights=w0, density=True, label="estimate")
+plt.hist(s_y, density=True, edgecolor="r", facecolor="None", label="truth")
+plt.legend(title=f"var(w0) = {np.var(w0):.2f}");
+
+

+
+
+
+
w0 = cow(x, g, Gs, gb, xrange=xrange)
+plt.plot(x, w0, ".", ms=1, mew=0);
+
+

+
+
+
+
plt.hist(y, weights=w0, density=True, label="estimate")
+plt.hist(s_y, density=True, edgecolor="r", facecolor="None", label="truth")
+plt.legend(title=f"var(w0) = {np.var(w0):.2f}");
+
+

+
+
+
+
w0 = cow(x, flat, Gs, gb, xrange=xrange)
+plt.plot(x, w0, ".", ms=1, mew=0);
+
+

+
+
+
+
plt.hist(y, weights=w0, density=True, label="estimate")
+plt.hist(s_y, density=True, edgecolor="r", facecolor="None", label="truth")
+plt.legend(title=f"var(w0) = {np.var(w0):.2f}");
+
+

+
+
+
+
import numba as nb
+
+class Base:
+    def __init__(self, n, i, xrange):
+        self.beta = np.zeros(n)
+        self.beta[i] = 1
+        self.xrange = xrange
+
+    def __call__(self, m):
+        return bernstein.density(m, self.beta, *self.xrange)
+
+def make_basis(n, xrange):
+    return [Base(n, i, xrange) for i in range(n)]
+
+N = 7
+B = make_basis(N, xrange)
+for Bi in B:
+    plt.plot(x, Bi(x), ".", ms=1, mew=0)
+
+w0 = cow(x, flat, gs, B, xrange)
+plt.figure()
+plt.plot(x, w0, ".", ms=1, mew=0);
+
+

+
+
+

+
+
+
+
plt.hist(y, weights=w0, density=True, label="estimate")
+plt.hist(s_y, density=True, edgecolor="r", facecolor="None", label="truth")
+plt.legend(title=f"var(w0) = {np.var(w0):.2f}");
+
+

+
+
+ + + +
+ +
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+
+
+

Statistical issues in naive calibration analyses

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

The statistical problem of calibrating a sample with observable pairs \(\hat a\) and \(\hat b\) is challenging if random stochastic fluctuations affect both numbers of the pair. Even if the unknown true values \((a, b)\) have a perfectly linear relationship and if the stochastic fluctuations around \((a, b)\) are symmetric, a binned profile of the \(\hat b\) values as a function of the \(\hat a\) values may indicate a departure from a linear dependence. This happens if the stochastic fluctuations of \(\hat a\) around \(a\) are not negligible compared to the bin size.

+

We demonstrate this in the following with toy simulation. We assume fluctuations that are relative to the true values and that the true pairs \((a,b)\) are not uniformly distributed, but follow an exponential distribution. This is commonly the case in particle physics.

+
+
import numpy as np
+from matplotlib import pyplot as plt
+import boost_histogram as bh
+from iminuit import Minuit
+from iminuit.cost import LeastSquares
+
+rng = np.random.default_rng(1)
+
+x_true = rng.exponential(1, size=1000)
+# x_true = rng.uniform(0, 10, size=1000)
+y_true = x_true.copy()
+
+def sigma_x(x_true):
+    return x_true * 0.1
+
+def sigma_y(y_true):
+    return y_true * 0.5
+
+x0 = rng.normal(x_true, sigma_x(x_true))
+y0 = rng.normal(y_true, sigma_y(y_true))
+
+x = x0
+y = y0
+
+h = bh.Histogram(bh.axis.Variable([0, 0.1, 0.3, 0.5, 0.7, 1, 1.5, 1.75, 2, 3, 4, 6, 10]), storage=bh.storage.Mean())
+h.fill(x, sample=y)
+
+def line(x, a, b):
+    return a + x * b
+
+m = Minuit(LeastSquares(h.axes[0].centers, h.values(), h.variances() ** 0.5, line), 0, 0)
+m.migrad()
+
+plt.scatter(x, y, marker=".", color="C1")
+m.visualize()
+print(m)
+
+
┌─────────────────────────────────────────────────────────────────────────┐
+│                                Migrad                                   │
+├──────────────────────────────────┬──────────────────────────────────────┤
+│ FCN = 5.194 (χ²/ndof = 0.5)      │              Nfcn = 31               │
+│ EDM = 1.42e-21 (Goal: 0.0002)    │                                      │
+├──────────────────────────────────┼──────────────────────────────────────┤
+│          Valid Minimum           │   Below EDM threshold (goal x 10)    │
+├──────────────────────────────────┼──────────────────────────────────────┤
+│      No parameters at limit      │           Below call limit           │
+├──────────────────────────────────┼──────────────────────────────────────┤
+│             Hesse ok             │         Covariance accurate          │
+└──────────────────────────────────┴──────────────────────────────────────┘
+┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐
+│   │ Name │   Value   │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit-  │ Limit+  │ Fixed │
+├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤
+│ 0 │ a    │   0.005   │   0.005   │            │            │         │         │       │
+│ 1 │ b    │   0.997   │   0.020   │            │            │         │         │       │
+└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘
+┌───┬─────────────────────┐
+│   │         a         b │
+├───┼─────────────────────┤
+│ a │  2.18e-05 -0.050e-3 │
+│ b │ -0.050e-3  0.000401 │
+└───┴─────────────────────┘
+
+
+

+
+
+

We find excellent linearity, the fitted line is compatible with zero offset and a slope of one.

+

Now we swap the roles of x and y and perform the exact same analysis again. An analysis based on a profile is not treating x and y symmetrically, therefore it is possible that we get another answer, although fundamentally it does not matter whether we study pairs (x, y) or (y, x).

+
+
x = y0
+y = x0
+
+h = bh.Histogram(bh.axis.Variable([0, 0.1, 0.3, 0.5, 0.7, 1, 1.5, 1.75, 2, 3, 4, 6, 10]), storage=bh.storage.Mean())
+h.fill(x, sample=y)
+
+m = Minuit(LeastSquares(h.axes[0].centers, h.values(), h.variances(), line), 0, 0)
+m.migrad()
+
+plt.scatter(x, y, marker=".", color="C1")
+m.visualize()
+print(m)
+
+
┌─────────────────────────────────────────────────────────────────────────┐
+│                                Migrad                                   │
+├──────────────────────────────────┬──────────────────────────────────────┤
+│ FCN = 8884 (χ²/ndof = 888.4)     │              Nfcn = 35               │
+│ EDM = 1.01e-20 (Goal: 0.0002)    │                                      │
+├──────────────────────────────────┼──────────────────────────────────────┤
+│          Valid Minimum           │   Below EDM threshold (goal x 10)    │
+├──────────────────────────────────┼──────────────────────────────────────┤
+│      No parameters at limit      │           Below call limit           │
+├──────────────────────────────────┼──────────────────────────────────────┤
+│             Hesse ok             │         Covariance accurate          │
+└──────────────────────────────────┴──────────────────────────────────────┘
+┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐
+│   │ Name │   Value   │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit-  │ Limit+  │ Fixed │
+├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤
+│ 0 │ a    │ 163.2e-3  │  0.6e-3   │            │            │         │         │       │
+│ 1 │ b    │  0.7905   │  0.0014   │            │            │         │         │       │
+└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘
+┌───┬───────────────────┐
+│   │        a        b │
+├───┼───────────────────┤
+│ a │ 3.05e-07 -0.50e-6 │
+│ b │ -0.50e-6 1.93e-06 │
+└───┴───────────────────┘
+
+
+

+
+
+

Both eye inspection and the fit indicate that the data is not linear, although it actually is. This surprising result is obtained even though the fluctuations in both x and y are symmetric around the true value. It originates from bin migration effects, which are not taken into account in this simple study.

+

It is possible to avoid the bias with a more sophisticated analysis that properly treats this problem with the maximum likelihood method. Such an analysis was developed to calibrate the Pierre Auger Observatory, which can be transferred to other calibration problems.

+

H. Dembinski, B. Kégl, I.C. Maris, M. Roth, D. Veberič, Astropart.Phys. 73 (2016) 44-51

+

If the proper likelihood analysis of the problem cannot be performed, one should always bin in the variable which depicts smaller fluctuations and try to make the bins large compared to these fluctuations to minimize bin migration. However, this can also be counter-productive as making bins larger increases another bias from mixing samples that have locally different expectation values and densities.

+

Even then, however, it is not guaranteed that biases are absent. This needs to confirmed with toy simulations.

+

If the extra effort needed to patch up a naive analysis is included, the amount of work is as great or larger to using the proper likelihood analysis described in the paper, which makes the likelihood analysis the economic choice.

+
+

Partial solution: modified least-squares method

+

A partial solution that may produce good results on some data is the modified least-squares method. It requires point estimates for the uncertainties of the x and y values. The standard least-squares method assumes that the x-values are known and only the y-values carry Gaussian uncertainties. Gaussian uncertainties on x-values can be approximately taken into account via error-propagation inside the cost function, using the current estimate of the fitting function.

+

This method works well, if the point estimates of the errors are fairly accurate and if the fitted function is not far from linear on the scale of the x-errors.

+

In our toy example, we get good results if we use the true uncertainties per point, but the results are poor if we try to estimate the uncertainties from the observed values.

+
+
# with small error on x
+sx_true = sigma_x(x_true)
+sy_true = sigma_y(y_true)
+
+x, y = x0, y0
+sx, sy = sx_true, sy_true
+
+def chi2(a, b):
+    ym = line(x, a, b)
+    dy_dx = b
+    var = (dy_dx * sx) ** 2 + sy ** 2
+    res = (y - ym) ** 2 / var
+    return np.sum(res)
+
+chi2.ndata = len(x)
+
+
+def visualize(args):
+    a, b = args
+    ym = line(x, a, b)
+    plt.plot(x, ym, zorder=3)
+    plt.errorbar(x, y, np.abs(sigma_y(y)), np.abs(sigma_x(x)), fmt=".", alpha=0.5)
+
+chi2.visualize = visualize
+
+m = Minuit(chi2, 0, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 953.6 (χ²/ndof = 1.0)Nfcn = 31
EDM = 1.57e-07 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0a-1.2e-30.8e-3
1b1.0130.017
+ + + + + + + + + + + + + + + + + + +
ab
a6.42e-07-3.5e-6 (-0.260)
b-3.5e-6 (-0.260)0.000277
+ + + + + + + + 2023-07-10T11:27:45.162470 + image/svg+xml + + + Matplotlib v3.6.3, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+
+
+
# with large error on x
+sx_true = sigma_x(x_true)
+sy_true = sigma_y(y_true)
+
+y, x = x0, y0
+sy, sx = sx_true, sy_true
+
+m = Minuit(chi2, 0, 1)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 953.6 (χ²/ndof = 1.0)Nfcn = 38
EDM = 1.13e-07 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0a1.2e-30.8e-3
1b0.9870.016
+ + + + + + + + + + + + + + + + + + +
ab
a6.17e-07-3.0e-6 (-0.236)
b-3.0e-6 (-0.236)0.000263
+ + + + + + + + 2023-07-10T11:27:45.502446 + image/svg+xml + + + Matplotlib v3.6.3, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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This fit is already slightly biased due to the points at negative x-values.

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The fit gets badly biased if we don’t know the true uncertainties per point and have to estimate them based on the observed x- and y-values. For the y-values, we can do a bit better by using the predicted y-value from the model to compute the uncertainty estimate, but this trick does not work for the x-values.

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We are not recovering the expected (0, 1) parameters.

+
+
+

General solution: maximum-likelihood method with hierarchical statistical model

+

The correct solution from first principles is to write down a statistical model of the calibration process. This model is hierarchical. The first step is to draw a sample of unknown true values \(x, y\) from the underlying frequency distribution (here an exponential). This needs to be folded with normal distributions to model the observed distribution.

+

Following section A of H. Dembinski, B. Kégl, I.C. Maris, M. Roth, D. Veberič, Astropart.Phys. 73 (2016) 44-51, we get for this toy example

+

\[ +f(x, y; \vec p) = \int \text{d}x_\text{true} \, g(x; x_\text{true}, \sigma_x) \, g(y; f(x_\text{true}, \vec{p}), \sigma_y) \, h(x_\text{true}) +\]

+

where \(g\) is a normal distribution and \(h\) is an exponential. This solution treats x- and y-values on equal footing and is derived from first principles.

+

As explained in the paper, solving this equation numerically is slow. However, a fast approximation is given with approximation B (Eq. 26) which replaces \(h(x_\text{true})\) with its empirical density (an idea borrowed from bootstrap theory). With this we get

+

\[ +f(x, y; \vec p) = \frac 1 N \sum_i \, g(x; x_i, \sigma_{x,i}) \, g(y; f(x_i), \sigma_{y,i}) +\]

+

where \((x_i, y_i)\) are the observed values and the sum \(\sum_i\) runs over the observed sample. The log-likelihood to maximize is then

+

\[ +\ln L(\vec p) = \sum_k \ln f(x_k, y_k; \vec p) +\]

+

The approximate solution for this problem contains a curious double sum over the event sample. This is not an error, but the consequence of using the empirical distribution to replace \(h(x_\text{true})\).

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+
import numba as nb
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+
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+        p += norm_pdf(xi, xk, sxk) * norm_pdf(yi, ym, syk)
+    return p
+
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+@nb.njit
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+    result = 0
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+        result += np.log(pdf(xi, yi, a, b))
+    return -2 * result
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+m = Minuit(chi2, 0, 1)
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Migrad
FCN = -1.14e+04Nfcn = 40
EDM = 7.52e-05 (Goal: 0.0002)time = 1.1 sec
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
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NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
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1b1.0180.008
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ab
a2.55e-07-0.52e-6 (-0.136)
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p-value computation and conversion: one-sided, two-sided?

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This is an attempt to clear up the confusion around p-value computation and based on the authorative source on that matter:

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G. Cowan, K. Cranmer, E. Gross, O. Vitells, Eur.Phys.J.C 71 (2011) 1554

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The point of that paper is that one should use two different test statistics based on the likelihood-ratio for deviations that are always positive and for deviations where the sign is not known a priori. One gets two different p-values accordingly. The conversion to significance is then always the same.

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I generate a flat distribution from \((-1,1)\) which represents background-only samples, then I fit a Gaussian peak centered at 0 with width 0.1 to this data test the two hypotheses

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  • \(H_0\): there is only background
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  • \(H_1\): there is background and a signal
  • +
+

The \(H_1\) in these kinds of tests is never arbitrary in terms of the signal, we do not seek for any deviation, but a rather specific one. In our example, we seek for a signal in particular location (here centered at 0). This is important for the computation of the p-value later. The less constrained \(H_1\) is, the more of the Look-Elsewhere Effect I get, because more random fluctuations of the background can be confused with a signal.

+

When fitting the toys, I allow the amplitude \(\mu\) of the signal to be positive or negative (following Cowan et al.). As a test statistic for a deviation from the background-only case I use \(t_0\) (again following Cowan et al. in the notation) \[ +t_0 = -2\ln \lambda(0) = -2 \ln \frac{L(0, \hat {\hat \theta})}{L(\hat \mu, \hat \theta)} +\] where * \(\mu\) is the amplitude of the hypothetical signal * \(\theta\) are nuisance parameters * \(\hat \mu\) is the fitted value of \(\mu\) * \(\hat\theta\) are the fitted values of \(\theta\) when \(\mu\) is also fitted * \(\hat{\hat\theta}\) are the fitted values of \(\theta\) under the condition \(\mu = 0\).

+

If you search for \(t_0\) in the paper, it is not explicitly written there, \(t_0\) is \(t_\mu\) from section 2.1 with \(\mu = 0\).

+
+
import numpy as np
+from iminuit import Minuit
+from iminuit.cost import ExtendedUnbinnedNLL
+from numba_stats import uniform_pdf, norm_pdf
+import matplotlib.pyplot as plt
+import boost_histogram as bh
+
+
+
rng = np.random.default_rng(1)
+
+def model(x, mu, theta):
+    return mu + theta, mu * norm_pdf(x, 0, 0.1) + theta * uniform_pdf(x, -1, 2)
+
+positive = [0]
+negative = [0]
+t0 = []
+mu = []
+for imc in range(1000):
+    b = rng.uniform(-1, 1, size=1000)
+    c = ExtendedUnbinnedNLL(b, model)
+    m = Minuit(c, mu=0, theta=len(b))
+    m.limits["theta"] = (0, None)
+    m.fixed["mu"] = True
+    m.migrad()
+    assert m.valid
+    lnL_b = -m.fval
+    m.fixed["mu"] = False
+    m.migrad()
+    mui = m.values["mu"]
+    lnL_sb = -m.fval
+    t0i = 2 * (lnL_sb - lnL_b)
+    t0.append(t0i)
+    mu.append(mui)
+    if mui > 0 and t0i > positive[0]:
+        h = bh.Histogram(bh.axis.Regular(100, -1, 1))
+        h.fill(b)
+        positive = (t0i, h, m.values[:], m.covariance[:])
+    elif mui < 0 and t0i > negative[0]:
+        h = bh.Histogram(bh.axis.Regular(100, -1, 1))
+        h.fill(b)
+        negative = (t0i, h, m.values[:], m.covariance[:])
+        
+t0 = np.array(t0)
+mu = np.array(mu)
+
+

Let’s look at the two most extreme deviations from \(H_0\), the ones with the largest and smallest signal.

+
+
fig, ax = plt.subplots(1, 2, figsize=(14, 5), sharex=True, sharey=True)
+for (t0i, h, par, cov), axi in zip((negative, positive), ax):
+    plt.sca(axi)
+    scale = 1/h.axes[0].widths
+    plt.errorbar(h.axes[0].centers, h.values() * scale, h.variances()**0.5 * scale, fmt="ok", label="data")
+    scale = 1 / h.axes[0].widths
+    xm = np.linspace(-1, 1)
+    plt.plot(xm, model(xm, *par)[1], label="fit")
+    plt.legend()
+    dev = par[0] / cov[0,0] ** 0.5
+    plt.title(f"$\mu = {par[0]:.1f}$        $t_0 = {t0i:.2f}$")
+    plt.ylabel("density")
+    plt.xlabel("$x$")
+
+

+
+
+

Both upward and downward fluctuations are unlikely events if \(H_0\) is true and therefore both get a large value of our test statistic \(t_0\), which does not know in which way the deviation went.

+
+

Searches with \(\mu > 0\) (for a new particle, new decay mode, etc.)

+

In most cases we look for a positive peak (new particle, new decay mode, etc). If we see a downward fluctuation, we know for sure that this cannot be what we are looking for. Therefore, when we compute the p-value for some observation from a simulation like this one, we must not regard large values of the test statistic when the fluctuation was negative, since this is not a deviation of \(H_0\) that mimics our signal. Instead we should set \(t_0\) to zero in such cases. Cowan et al. use \(q_0\) for this modified test statistic \[ +q_0 = \begin{cases} +t_0 &\text{if } \hat\mu \ge 0 \\ +0 &\text{otherwise} +\end{cases} +\]

+
+
plt.hist(t0, alpha=0.5, bins=20, range=(0, 20), label="$t_0$")
+q0 = t0.copy()
+q0 = np.where(mu < 0, 0, t0)
+plt.hist(q0, alpha=0.5, bins=20, range=(0, 20), label="$q_0$")
+plt.legend()
+plt.axhline()
+plt.xlabel("$t_0, q_0$")
+plt.semilogy();
+
+

+
+
+

If we observed \(t_0 = q_0 = 15\) in a real experiment, we need to compare it with the \(q_0\) distribution and not with the \(t_0\) distribution to compute the p-value.

+
+
print(f"Wrong: p-value based on t0-distribution {np.mean(t0 > 15)}")
+print(f"Right: p-value based on q0-distribution {np.mean(q0 > 15)}")
+
+
Wrong: p-value based on t0-distribution 0.006
+Right: p-value based on q0-distribution 0.004
+
+
+

As we can see, the p-value is enhanced in this case, because we do not need to consider the negative fluctuations at all. The conversion to significance \(Z\) is done with a normal distribution.

+
+
from scipy.stats import norm
+
+p = np.mean(q0 > 15)
+print(f"Z = {norm.ppf(1 - p):.2f}")
+
+
Z = 2.65
+
+
+
+
+

Searches for deviations where the sign of \(\mu\) is not known a priori

+

If cannot exclude a priori that our signal has \(\mu < 0\), we need to use the \(t_0\) distribution instead of \(q_0\).

+

If we observed \(t_0 = 15\) in a real experiment, we need to compare it with \(t_0\) distribution to compute the p-value.

+
+
print(f"Right: p-value based on t0-distribution {np.mean(t0 > 15)}")
+
+
Right: p-value based on t0-distribution 0.006
+
+
+

As we can see, the p-value is diluted in this case, because we need to consider fluctuations in both directions. In other words, the Look-Elsewhere Effect is larger in this case, because more kinds of fluctuations in the background can be confused with a signal.

+
+
from scipy.stats import norm
+
+p = np.mean(t0 > 15)
+print(f"Z = {norm.ppf(1 - p):.2f}")
+
+
Z = 2.51
+
+
+

Accordingly, the significance is a bit lower compared to the previous case.

+ + +
+ +
+ +
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}^LxtDpWovmMul^5NahT)hoXFftY?q~_<%3=4uxcJ2>#NxH zudeRM*j0Sq{YUZm+EA|M%XmiY3uWR_s-&S)hOasz)$9KrfBZLl%D{skSc19Jc$G7E zr6b#H)P93)R5@!m<`%hYfE^yfa@cxgv`B#Pa6+nI1rg-+_!{rUs~Z^Jt032GG%GW2XXO()`lGtgp2ug@Kt%O`a(uA}3^QiU)0De3hT_>Qg79V) zZ#M2VTjVng_=o$o#W!jevT|ut})2ZH2zX_G`E5-QgS;x%8XjGyh@Y@*^ iDEIHJ;E#4^Ur0(EdTy7W#H{cX4S=wAfHRJJUHLD~)1H(7 literal 0 HcmV?d00001 diff --git a/posts/parse_latex.html b/posts/parse_latex.html new file mode 100644 index 0000000..b5b47ff --- /dev/null +++ b/posts/parse_latex.html @@ -0,0 +1,750 @@ + + + + + + + + + +Hans Dembinski’s blog - Transform LaTeX to Unicode + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+ +
+ +
+
+
+

Transform LaTeX to Unicode

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

It is possible to transform a subset of LaTeX to Unicode, as demonstrated by unicodeit website. Unfortunately, unicodeit only works on short LaTeX strings.

+

Here, I write a simple parser with Lark to process more complex LaTeX strings. A more complete version of this prototype is available as unicodeitplus.

+
+
# Wikipedia: https://en.wikipedia.org/wiki/Unicode_subscripts_and_superscripts
+import string
+
+COMMANDS = {}
+
+superscript_numbers = "⁰¹²³⁴⁵⁶⁷⁸⁹"
+for i, ch in enumerate(superscript_numbers):
+    COMMANDS[f"^{{{i}}}"] = ch
+
+subscript_numbers = "₀₁₂₃₄₅₆₇₈₉"
+for i, ch in enumerate(subscript_numbers):
+    COMMANDS[f"_{{{i}}}"] = ch
+
+superscript_lowercase = "ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖ𐞥ʳˢᵗᵘᵛʷˣʸᶻ"
+superscript_uppercase = "ᴬᴮꟲᴰᴱꟳᴳᴴᴵᴶᴷᴸᴹᴺᴼᴾꟴᴿ ᵀᵁⱽᵂ   "
+for latex, ch in zip(
+    string.ascii_letters, superscript_lowercase + superscript_uppercase
+):
+    if ch != " ":
+        COMMANDS[f"^{{{latex}}}"] = ch
+
+subscript_lowercase = "ₐ   ₑ  ₕᵢⱼₖₗₘₙₒₚ ᵣₛₜᵤᵥ ₓ  "
+for latex, ch in zip(string.ascii_letters, subscript_lowercase * 2):
+    if ch != " ":
+        COMMANDS[f"_{{{latex}}}"] = ch
+
+greek_lowercase = (
+    r"\alpha",
+    r"\beta",
+    r"\gamma",
+    r"\delta",
+    r"\epsilon",
+    r"\zeta",
+    r"\eta",
+    r"\theta",
+    r"\iota",
+    r"\kappa",
+    r"\lambda",
+    r"\mu",
+    r"\nu",
+    r"\xi",
+    "o",
+    r"\pi",
+    r"\rho",
+    r"\sigma",
+    r"\tau",
+    r"\upsilon",
+    r"\phi",
+    r"\chi",
+    r"\psi",
+    r"\omega",
+)
+
+superscript_lowercase_greek = " ᵝᵞᵟᵋ  ᶿᶥ          ᶹᵠᵡ  "
+subscript_lowercase_greek = " ᵦᵧ             ᵨ   ᵩᵪ  "
+for latex, sup, sub in zip(
+    greek_lowercase, superscript_lowercase_greek, subscript_lowercase_greek
+):
+    if sup != " ":
+        COMMANDS[f"^{{{latex}}}"] = sup
+    if sub != " ":
+        COMMANDS[f"_{{{latex}}}"] = sub
+
+
+
# Symbols extracted from http://milde.users.sourceforge.net/LUCR/Math/data/unimathsymbols.txt, which is under Copyright 2011 by Günter Milde and licensed under the LaTeX Project Public License (LPPL)
+
+from pathlib import Path
+from urllib.request import urlopen
+
+
+def match(comments):
+    matches = [
+        ("PLUS", "+"),
+        ("MINUS", "-"),
+        ("EQUALS", "="),
+        ("LEFT PARENTHESIS", "("),
+        ("RIGHT PARENTHESIS", ")"),
+    ]
+    for match, latex in matches:
+        if match in comments:
+            return latex
+    assert False, f"unmatched: {comments}"  # never arrive here
+
+
+with urlopen("http://milde.users.sourceforge.net/LUCR/Math/data/unimathsymbols.txt") as response:
+    body = response.read().decode()
+    for line in body.split("\n"):
+        if not line or line.startswith("#"):
+            continue
+        items = line.split("^")
+        _, ch, latex, latex2, clas, category, requirements, comments = items
+        comments = comments[:-1]
+        if latex:
+            if len(ch) > 1:
+                COMMANDS[latex] = ch[1]
+            else:
+                COMMANDS[latex] = ch
+        elif latex2:
+            COMMANDS[latex2] = ch
+        elif comments.startswith("SUPERSCRIPT"):
+            latex = f"^{{{match(comments)}}}"
+            COMMANDS[latex] = ch
+        elif comments.startswith("SUBSCRIPT"):
+            latex = f"_{{{match(comments)}}}"
+            COMMANDS[latex] = ch
+        else:
+            pass
+
+
+
# enhancements
+COMMANDS[r"\to"] = COMMANDS[r"\rightarrow"]
+COMMANDS[r"^{\ast}"] = "*"
+COMMANDS[r"\hbar"] = COMMANDS[r"\hslash"]
+COMMANDS["h"] = "ℎ"
+
+

Lark is awesome, because it generates the parser from an EMBL string, is fast and lightweight.

+
+
from lark import Lark
+from lark import Tree
+from lark.visitors import Transformer, Visitor, Discard, v_args, Interpreter
+from copy import deepcopy
+
+parser = Lark(r"""
+start: (item | math)*
+
+?atom: CHARACTER
+    | COMMAND
+
+?item: atom
+    | WS+
+    | group
+
+CHARACTER: /[^%#&\{\}^_]/ | ESCAPED
+ESCAPED: "\\\\" | "\\#" | "\\%" | "\\&"  | "\\{" | "\\}" | "\\_"
+group: "{" item* "}"
+math: "$" item* "$"
+SUBSCRIPT: "_"
+SUPERSCRIPT: "^"
+COMMAND: (("\\" WORD WS*) | SUBSCRIPT | SUPERSCRIPT)
+
+%import common.WS
+%import common.WORD
+""", parser="lalr")
+
+
+
HAS_ARG = {
+    r"_",
+    r"^",
+    r"\grave",
+    r"\acute",
+    r"\hat",
+    r"\tilde",
+    r"\bar",
+    r"\overline",
+    r"\breve",
+    r"\dot",
+    r"\ddot",
+    r"\mathring",
+    r"\check",
+    r"\utilde",
+    r"\underbar",
+    r"\underline",
+    r"\not",
+    r"\lvec",
+    r"\vec",
+    r"\LVec",
+    r"\vec",
+    r"\dddot",
+    r"\ddddot",
+    r"\overleftrightarrow",
+    r"\underleftarrow",
+    r"\underrightarrow",
+    r"\mathbf",
+    r"\text",
+    r"\mathrm",
+    r"\left",
+    r"\right",
+    r"\big",
+    r"\Big",
+    r"\Bigg",
+    r"\sqrt",
+}
+
+IGNORE_AS_FALLBACK = {
+    r"\text",
+    r"\mathbf",
+    r"\mathrm",
+    r"\left",
+    r"\right",
+    r"\big",
+    r"\Big",
+    r"\Bigg",
+}
+
+ESCAPED = {
+    r"\}": "}",
+    r"\{": "{",
+    "\\\\": "\\",
+}
+
+
+def handle_cmd(state, x):
+    # - x can be a character or a command, like \alpha
+    # - state["command"] contains stack with commands, may be empty
+    # - to transform ^{\alpha} or \text{x} correctly, we first try to 
+    #   convert innermost command and x as a unit
+    # - they are treated independently only if previous step fails
+    cmd_stack = state["command"].copy()
+    if state["math"]:
+        cmd = cmd_stack[-1] if cmd_stack else ""
+        latex = f"{cmd}{{{x}}}"
+        if cmd and latex in COMMANDS:
+            x = COMMANDS[latex]
+            cmd_stack.pop()
+        elif x.startswith(r"\\"):
+            x = COMMANDS.get(x, x)
+        elif cmd in (r"\text", r"\mathrm"):
+            cmd_stack.pop()
+        else:
+            x = COMMANDS.get(x, x)
+        for cmd in reversed(cmd_stack):
+            if cmd in COMMANDS:
+                # must be some unicode modifier, e.g. \dot, \vec
+                assert cmd in HAS_ARG
+                x += COMMANDS[cmd]
+            else:
+                latex = f"{cmd}{{{x}}}"
+                if latex in COMMANDS:
+                    x = COMMANDS[latex]
+                elif cmd not in IGNORE_AS_FALLBACK:
+                    x = latex
+    else:
+        for cmd in reversed(state["command"]):
+            x = f"{cmd}{{{x}}}"
+    if state["command"] and not state["group"]:
+        state["command"].pop()
+    return x
+
+
+def transform(ch, state=None):
+    if state is None:
+        state = {
+            "math": False,
+            "command": [],
+            "group": False,
+        }
+
+    if isinstance(ch, Tree):
+        r = []
+        undo_math = False
+        if ch.data == "math":
+            state["math"] = True
+        if ch.data == "group":
+            state["group"] = True
+        for x in ch.children:
+            r.append(transform(x, state))
+        if ch.data == "math":
+            state["math"] = False
+        if ch.data == "group":
+            state["group"] = False
+            if state["command"]:
+                state["command"].clear()
+        return "".join(r)
+
+    if ch.type == "CHARACTER":
+        x = ESCAPED.get(ch.value, ch.value)
+        return handle_cmd(state, x)
+    if ch.type == "WS":
+        return "" if state["math"] else " "
+    if ch.type == "COMMAND":
+        x = ch.value.strip()
+        if x in HAS_ARG:
+            if x == r"\sqrt":
+                state["command"].append(r"\overline")
+                return COMMANDS[r"\sqrt"]
+            state["command"].append(x)
+            return ""
+        return handle_cmd(state, x)
+    # never arrive here
+    assert False, f"unknown token {ch}" 
+
+
+def parse(s):
+    tree = parser.parse(s)
+    return transform(tree)
+
+

Let’s see how well (or not) this works on a few examples.

+
+
s = r"foo?!-1+2. \}  \\ $\left(\mathbf{\alpha + 1}^2_x y\right)$ bar $\beta^{12}$ $\bar p {}^foo$ $\bar \mathrm{t}$ "
+print(s)
+parse(s)
+
+
foo?!-1+2. \}  \\ $\left(\mathbf{\alpha + 1}^2_x y\right)$ bar $\beta^{12}$ $\bar p {}^foo$ $\bar \mathrm{t}$ 
+
+
+
'foo?!-1+2. } \\ (𝛂+𝟏²ₓ𝑦) bar 𝛽¹² 𝑝̄ᶠ𝑜𝑜 t̄ '
+
+
+
+
parse(r"$D^{\ast\ast} \to hhee$")
+
+
'𝐷**→ℎℎ𝑒𝑒'
+
+
+
+
parse(r"$\mathbf{xyz + 1}$")
+
+
'𝐱𝐲𝐳+𝟏'
+
+
+
+
parse(r"$\sqrt {1Aas\alpha}$")
+
+
'√1̅𝐴̅𝑎̅𝑠̅𝛼̅'
+
+
+
+
parse(r"$\vec{x} b^2 \vec\alpha\overline\alpha K^0_S p_\text{T} \text T$")
+
+
'𝑥⃗𝑏²𝛼⃗𝛼̅𝐾⁰ₛ𝑝ₜT'
+
+
+
+
parse(r"$\sqrt{abcd}$")
+
+
'√𝑎̅𝑏̅𝑐̅𝑑̅'
+
+
+
+
parse(r"$p_T / \text{GeV}c^{-1}$")
+
+
'𝑝ₜ/GeV𝑐⁻¹'
+
+
+
+
parse(r"Search for resonant $ \mathrm{t}\overline{\mathrm{t}} $ production in proton-proton collisions at $ \sqrt{s}=13 $ TeV")
+
+
'Search for resonant tt̅ production in proton-proton collisions at √𝑠̅=13 TeV'
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/plots_with_hyperlinks.html b/posts/plots_with_hyperlinks.html new file mode 100644 index 0000000..39c8adc --- /dev/null +++ b/posts/plots_with_hyperlinks.html @@ -0,0 +1,439 @@ + + + + + + + + + +Hans Dembinski’s blog - How to make a plot with legend entries that are hyperlinks + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

How to make a plot with legend entries that are hyperlinks

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

This is a demo on what the title says, also see this matplotlib issue. The trick works with SVG and PDF. Try to click on “BBC” in the legend.

+
+
from matplotlib import pyplot as plt
+import numpy as np
+
+# generate SVG images instead of PNG
+%config InlineBackend.figure_formats = ['svg']
+
+# required for SVG to accept a click on the text area and not just on the text path
+plt.rcParams["svg.fonttype"] = "none"
+
+plt.figure()
+plt.scatter([1, 2], [4, 6], label="BBC")
+plt.scatter([1, 2, 3], [6, 5, 4], label="Google")
+
+urls = {"BBC": 'https://www.bbc.com/news'}
+
+leg = plt.legend()
+for ta in leg.texts:
+    t = ta.get_text()
+    try:
+        url = urls[t]
+        ta.set_url(url)
+    except KeyError:
+        pass
+
+# plt.savefig('scatter.svg')
+# plt.savefig('scatter.pdf')
+
+

+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/plots_with_hyperlinks_files/figure-html/cell-2-output-1.svg b/posts/plots_with_hyperlinks_files/figure-html/cell-2-output-1.svg new file mode 100644 index 0000000..0389f81 --- /dev/null +++ b/posts/plots_with_hyperlinks_files/figure-html/cell-2-output-1.svg @@ -0,0 +1,333 @@ + + + + + + + + 2024-05-12T19:16:53.354679 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1.00 + + + + + + + + + + 1.25 + + + + + + + + + + 1.50 + + + + + + + + + + 1.75 + + + + + + + + + + 2.00 + + + + + + + + + + 2.25 + + + + + + + + + + 2.50 + + + + + + + + + + 2.75 + + + + + + + + + + 3.00 + + + + + + + + + + + + + + + 4.00 + + + + + + + + + + 4.25 + + + + + + + + + + 4.50 + + + + + + + + + + 4.75 + + + + + + + + + + 5.00 + + + + + + + + + + 5.25 + + + + + + + + + + 5.50 + + + + + + + + + + 5.75 + + + + + + + + + + 6.00 + + + + + + + + + + + + + + + + + + + + + + + + + + + BBC + + + + + + + + + Google + + + + + + + + + + diff --git a/posts/ratio bias.html b/posts/ratio bias.html new file mode 100644 index 0000000..7377190 --- /dev/null +++ b/posts/ratio bias.html @@ -0,0 +1,471 @@ + + + + + + + + + +Hans Dembinski’s blog - Ratio bias + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Ratio bias

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

When you compute a ratio of two random numbers, the ratio is biased in general, unless the relative error on the denominator is very small. This is a general consequence of the fact that any non-linear transformation of a random variable introduces a bias in general.

+

The good news is that we can correct for this bias to first order with a simple formula, and thus get an approximately unbiased result.

+
+
import numpy as np
+import matplotlib.pyplot as plt
+
+
+
rng = np.random.default_rng(1)
+
+a_true = 100.
+b_factor = 20.
+
+a = rng.poisson(a_true, size=10000)
+b = rng.poisson(a_true / b_factor, size=len(a)) * b_factor
+b[b==0] = np.nan
+
+
+
plt.hist(a / b, bins=50, label="toys")
+plt.axvline(1, ls="--", color="k", label="true")
+plt.axvline(np.nanmean(a/b), ls="-", color="r", label="avg")
+plt.legend();
+
+

+
+
+
+
@np.vectorize
+def ratio_bias(a, a_var, b, b_var):
+    if np.isnan(a) or np.isnan(b) or a == 0 or b == 0:
+        return np.nan
+
+    scale_a = a_var / a
+    n_eff_a = a / scale_a
+    scale_b = b_var / b
+    n_eff_b = b / scale_b
+
+    rng = np.random.default_rng(1)
+    a_toy = rng.poisson(n_eff_a, size=10000) * scale_a
+    b_toy = rng.poisson(n_eff_b, size=len(a_toy)) * scale_b
+    b_toy[b_toy == 0] = np.nan
+    r_toy = np.nanmean(a_toy / b_toy)
+    return r_toy / (a / b)
+
+
+
a = 10.
+b = 10.
+
+for a_rel_err in (0.01, 0.5):
+    b_rel_err = np.geomspace(0.01, 0.5, 10)
+    bias = ratio_bias(a, (a * a_rel_err) ** 2, b, (b * b_rel_err) ** 2)
+    plt.plot(b_rel_err, bias, label=f"$\\sigma(a)/a = {a_rel_err:.2f}$")
+
+# analytical formula for bias
+bias = (1+b_rel_err**2)
+plt.plot(b_rel_err, bias, label=f"analytical")
+
+plt.ylabel(r"$\langle a/b \rangle$ / truth")
+plt.xlabel(r"$\sigma(b)/b$")
+plt.legend();
+
+

+
+
+

The analytical formula is a first order correction, therefore it fails to describe the bias for large relative uncertainties.

+ + + +
+ +
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+ + + + +

+
+ +
+ +
+
+
+

Regression example with a neural network

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +
+

Prepare data

+
+
import pickle
+import gzip
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import StandardScaler
+
+
+
with gzip.open("rec_hist_pt_25.pkl.gz") as f:
+    rec = pickle.load(f)
+
+with gzip.open("gen_hist_pt_25.pkl.gz") as f:
+    gen = pickle.load(f)
+
+# fix for divide by zero
+gen.values()[gen.values() == 0] = 1
+
+n_gen = gen.values()
+n_rec = rec.values()
+# fix for rec > n
+n_rec = np.where(n_rec > n_gen, n_gen, n_rec)
+
+X = []
+data_k = []
+data_n = []
+for ieta, eta in enumerate(rec.axes[0].centers):
+    for ipt, pt in enumerate(rec.axes[1].centers):
+        for iphi, phi in enumerate(rec.axes[2].centers):
+            for ich, _ in enumerate(rec.axes[3]):
+                for isp, _ in enumerate(rec.axes[4]):
+                    X.append((eta, np.log(pt), np.cos(phi), np.sin(phi), ich, isp))
+                    data_k.append(n_rec[ieta, ipt, iphi, ich, isp])
+                    data_n.append(n_gen[ieta, ipt, iphi, ich, isp])
+
+X0 = np.array(X).astype(np.float32)
+data_k = np.array(data_k).astype(np.float32)
+data_n = np.array(data_n).astype(np.float32)
+y = data_k / data_n
+
+scaler = StandardScaler()
+scaler.fit(X0)
+
+X = scaler.transform(X0)
+
+print("data points", X.shape[0])
+
+
data points 14400
+
+
+
+
def draw(model):
+    labels = ("π", "K", "p", "other")
+    for phii in rec.axes[2].centers[:]:
+        if phii < 0:
+            continue
+        fig, axes = plt.subplots(3, 2, sharex=True, sharey=True, layout="compressed")
+        plt.suptitle(rf"$\phi = {np.degrees(phii):.0f}$ deg")
+        for etai, axi in zip(rec.axes[0].centers, axes.flat):
+            plt.sca(axi)
+            for ich, charge in enumerate((-1, 1)):
+                for isp, label in enumerate(labels):
+                    ma = X0[:, 0] == etai
+                    ma &= X0[:, 2] == np.cos(phii)
+                    ma &= X0[:, 3] == np.sin(phii)
+                    ma &= X0[:, 4] == ich
+                    ma &= X0[:, 5] == isp
+                    plt.plot(np.exp(X0[ma, 1]), charge * y[ma], "o", ms=4, color=f"C{isp}", label=f"{label}" if ich else None)
+
+                    mpt = np.geomspace(10, 1e4, 2000)
+                    Xp = np.empty((len(mpt), 6), dtype=np.float32)
+                    Xp[:, 0] = etai
+                    Xp[:, 1] = np.log(mpt)
+                    Xp[:, 2] = np.cos(phii)
+                    Xp[:, 3] = np.sin(phii)
+                    Xp[:, 4] = ich
+                    Xp[:, 5] = isp
+                    yp = model(scaler.transform(Xp))
+                    plt.plot(np.exp(Xp[:, 1]), charge * yp, color=f"C{isp}")
+        plt.sca(axes[0,0])
+        plt.semilogx()
+        plt.ylim(-1.1, 1.1)
+        plt.legend(fontsize="xx-small", ncol=2, frameon=False);
+
+
+
+

Scikit-Learn

+

It turns out that the simple MLPRegressor in Scikit-Learn works very well on small datasets.

+
+
from sklearn.neural_network import MLPRegressor
+from sklearn.metrics import mean_squared_error
+
+clf = MLPRegressor(
+    hidden_layer_sizes=(2**6,) * 4,
+    batch_size=1000,
+    alpha=0,
+    tol=1e-6,
+    max_iter=1000,
+    verbose=1,
+)
+
+clf.fit(X, y)
+
+# coefficients only available after training
+n = 0
+for c in clf.coefs_:
+    n += np.prod(c.shape)
+for c in clf.intercepts_:
+    n += c.shape[0]
+print("number of parameters", n)
+
+print("MLPRegressor", mean_squared_error(clf.predict(X), y))
+
+
Iteration 1, loss = 0.03872877
+Iteration 2, loss = 0.01838284
+Iteration 3, loss = 0.01322917
+Iteration 4, loss = 0.00992924
+Iteration 5, loss = 0.00764789
+Iteration 6, loss = 0.00596758
+Iteration 7, loss = 0.00476454
+Iteration 8, loss = 0.00403427
+Iteration 9, loss = 0.00354749
+Iteration 10, loss = 0.00312269
+Iteration 11, loss = 0.00285058
+Iteration 12, loss = 0.00266859
+Iteration 13, loss = 0.00250857
+Iteration 14, loss = 0.00243164
+Iteration 15, loss = 0.00227876
+Iteration 16, loss = 0.00218516
+Iteration 17, loss = 0.00204908
+Iteration 18, loss = 0.00195609
+Iteration 19, loss = 0.00188667
+Iteration 20, loss = 0.00181428
+Iteration 21, loss = 0.00181260
+Iteration 22, loss = 0.00172098
+Iteration 23, loss = 0.00166783
+Iteration 24, loss = 0.00162889
+Iteration 25, loss = 0.00162617
+Iteration 26, loss = 0.00151559
+Iteration 27, loss = 0.00148309
+Iteration 28, loss = 0.00146949
+Iteration 29, loss = 0.00142209
+Iteration 30, loss = 0.00137413
+Iteration 31, loss = 0.00133488
+Iteration 32, loss = 0.00132600
+Iteration 33, loss = 0.00133857
+Iteration 34, loss = 0.00133807
+Iteration 35, loss = 0.00125656
+Iteration 36, loss = 0.00125801
+Iteration 37, loss = 0.00124734
+Iteration 38, loss = 0.00116203
+Iteration 39, loss = 0.00112787
+Iteration 40, loss = 0.00113262
+Iteration 41, loss = 0.00111350
+Iteration 42, loss = 0.00109534
+Iteration 43, loss = 0.00109662
+Iteration 44, loss = 0.00103106
+Iteration 45, loss = 0.00099346
+Iteration 46, loss = 0.00098687
+Iteration 47, loss = 0.00103844
+Iteration 48, loss = 0.00099100
+Iteration 49, loss = 0.00096235
+Iteration 50, loss = 0.00097157
+Iteration 51, loss = 0.00092747
+Iteration 52, loss = 0.00091701
+Iteration 53, loss = 0.00089235
+Iteration 54, loss = 0.00087139
+Iteration 55, loss = 0.00087560
+Iteration 56, loss = 0.00087910
+Iteration 57, loss = 0.00090950
+Iteration 58, loss = 0.00085914
+Iteration 59, loss = 0.00082885
+Iteration 60, loss = 0.00081474
+Iteration 61, loss = 0.00084764
+Iteration 62, loss = 0.00081955
+Iteration 63, loss = 0.00083712
+Iteration 64, loss = 0.00080643
+Iteration 65, loss = 0.00078607
+Iteration 66, loss = 0.00077217
+Iteration 67, loss = 0.00080849
+Iteration 68, loss = 0.00081811
+Iteration 69, loss = 0.00078761
+Iteration 70, loss = 0.00077427
+Iteration 71, loss = 0.00075709
+Iteration 72, loss = 0.00075573
+Iteration 73, loss = 0.00075636
+Iteration 74, loss = 0.00074296
+Iteration 75, loss = 0.00075769
+Iteration 76, loss = 0.00078198
+Iteration 77, loss = 0.00074714
+Iteration 78, loss = 0.00076107
+Iteration 79, loss = 0.00073871
+Iteration 80, loss = 0.00071247
+Iteration 81, loss = 0.00073748
+Iteration 82, loss = 0.00074470
+Iteration 83, loss = 0.00070695
+Iteration 84, loss = 0.00069866
+Iteration 85, loss = 0.00068365
+Iteration 86, loss = 0.00067912
+Iteration 87, loss = 0.00068739
+Iteration 88, loss = 0.00067988
+Iteration 89, loss = 0.00070646
+Iteration 90, loss = 0.00067668
+Iteration 91, loss = 0.00068584
+Iteration 92, loss = 0.00066119
+Iteration 93, loss = 0.00065713
+Iteration 94, loss = 0.00079503
+Iteration 95, loss = 0.00070443
+Iteration 96, loss = 0.00069255
+Iteration 97, loss = 0.00064721
+Iteration 98, loss = 0.00066456
+Iteration 99, loss = 0.00066503
+Iteration 100, loss = 0.00064768
+Iteration 101, loss = 0.00063161
+Iteration 102, loss = 0.00065152
+Iteration 103, loss = 0.00063627
+Iteration 104, loss = 0.00062377
+Iteration 105, loss = 0.00063346
+Iteration 106, loss = 0.00065331
+Iteration 107, loss = 0.00063847
+Iteration 108, loss = 0.00061856
+Iteration 109, loss = 0.00061806
+Iteration 110, loss = 0.00060691
+Iteration 111, loss = 0.00061332
+Iteration 112, loss = 0.00061673
+Iteration 113, loss = 0.00064613
+Iteration 114, loss = 0.00060685
+Iteration 115, loss = 0.00059055
+Iteration 116, loss = 0.00060077
+Iteration 117, loss = 0.00062042
+Iteration 118, loss = 0.00065500
+Iteration 119, loss = 0.00060160
+Iteration 120, loss = 0.00060204
+Iteration 121, loss = 0.00059747
+Iteration 122, loss = 0.00061649
+Iteration 123, loss = 0.00060509
+Iteration 124, loss = 0.00063298
+Iteration 125, loss = 0.00060878
+Iteration 126, loss = 0.00061471
+Training loss did not improve more than tol=0.000001 for 10 consecutive epochs. Stopping.
+number of parameters 12993
+MLPRegressor 0.0011639113
+
+
+
+
draw(clf.predict)
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+
+
+

PyTorch

+
+
import torch
+from torch import nn
+from torch.optim.lr_scheduler import ReduceLROnPlateau
+from torch.optim import Adam
+
+device = "cuda" if torch.cuda.is_available() else "cpu"
+torch.set_default_device(device)
+torch.set_default_dtype(torch.float32)
+
+nonlin = nn.ReLU()
+num = 2**6
+torch_model = nn.Sequential(
+    nn.Linear(6, num),
+    nonlin,
+    nn.Linear(num, num),
+    nonlin,
+    nn.Linear(num, num),
+    nonlin,
+    nn.Linear(num, num),
+    nonlin,
+    nn.Linear(num, 1),
+    # adding a ReLU here makes training unstable
+    nn.Flatten(0, 1),
+)
+
+n = 0
+for par in torch_model.parameters():
+    n += np.prod(par.shape)
+print("number of parameters", n)
+
+# must make copies here
+torch_X = torch.tensor(X.copy())
+torch_y = torch.tensor(y.copy())
+
+max_epoch = 5000
+shrink_patience = 10
+abort_patience = 100
+learning_rate = 1e-3
+learning_rate_shrink_factor = 0.8
+tol = 1e-6
+
+loss_fn = nn.MSELoss()
+opt = Adam(torch_model.parameters(), lr=learning_rate)
+scheduler = ReduceLROnPlateau(
+    opt, patience=shrink_patience, factor=learning_rate_shrink_factor, verbose=True
+)
+
+losses = []
+no_improvement_below_tolerance = 0
+prev_loss = np.inf
+for epoch in range(1, max_epoch + 1):
+    y_pred = torch_model(torch_X)
+    loss = loss_fn(y_pred, torch_y)
+    opt.zero_grad()
+    loss.backward()
+    opt.step()
+    scheduler.step(loss)
+
+    loss = loss.item()
+    losses.append(loss)
+    if epoch == 1 or epoch % 100 == 0:
+        print("epoch", epoch, "loss", loss)
+
+    if loss < prev_loss - tol:
+        no_improvement_below_tolerance = 0
+        prev_loss = loss
+    else:
+        no_improvement_below_tolerance += 1
+    if no_improvement_below_tolerance > abort_patience:
+        break
+
+plt.plot(losses)
+plt.semilogy();
+
+
number of parameters 12993
+epoch 1 loss 0.25884100794792175
+epoch 100 loss 0.010155524127185345
+epoch 200 loss 0.003915281500667334
+epoch 300 loss 0.002415228867903352
+epoch 400 loss 0.0017484889831393957
+epoch 500 loss 0.0014784386148676276
+Epoch 00508: reducing learning rate of group 0 to 8.0000e-04.
+epoch 600 loss 0.0013304103631526232
+Epoch 00695: reducing learning rate of group 0 to 6.4000e-04.
+epoch 700 loss 0.001243863138370216
+epoch 800 loss 0.0011801073560491204
+epoch 900 loss 0.0011410759761929512
+Epoch 00905: reducing learning rate of group 0 to 5.1200e-04.
+epoch 1000 loss 0.00109128060285002
+epoch 1100 loss 0.0010561983799561858
+Epoch 01152: reducing learning rate of group 0 to 4.0960e-04.
+epoch 1200 loss 0.0010263388976454735
+epoch 1300 loss 0.001002235570922494
+epoch 1400 loss 0.0009775016224011779
+Epoch 01450: reducing learning rate of group 0 to 3.2768e-04.
+epoch 1500 loss 0.0009563906933180988
+epoch 1600 loss 0.0009381748968735337
+epoch 1700 loss 0.0009208598639816046
+epoch 1800 loss 0.0009038595599122345
+Epoch 01887: reducing learning rate of group 0 to 2.6214e-04.
+epoch 1900 loss 0.0008835234912112355
+epoch 2000 loss 0.0008701192564330995
+epoch 2100 loss 0.0008563698502257466
+epoch 2200 loss 0.000845507369376719
+epoch 2300 loss 0.0008326928946189582
+Epoch 02305: reducing learning rate of group 0 to 2.0972e-04.
+epoch 2400 loss 0.0008199398289434612
+epoch 2500 loss 0.0008091804338619113
+epoch 2600 loss 0.000800078792963177
+epoch 2700 loss 0.0007896188762970269
+epoch 2800 loss 0.000779114430770278
+Epoch 02864: reducing learning rate of group 0 to 1.6777e-04.
+epoch 2900 loss 0.0007669864571653306
+epoch 3000 loss 0.0007585080456919968
+epoch 3100 loss 0.0007498746854253113
+epoch 3200 loss 0.0007412403356283903
+epoch 3300 loss 0.0007329632644541562
+epoch 3400 loss 0.0007246120949275792
+Epoch 03408: reducing learning rate of group 0 to 1.3422e-04.
+epoch 3500 loss 0.0007171895122155547
+epoch 3600 loss 0.0007101951632648706
+epoch 3700 loss 0.0007029927219264209
+epoch 3800 loss 0.0006956694996915758
+Epoch 03889: reducing learning rate of group 0 to 1.0737e-04.
+epoch 3900 loss 0.0006882176967337728
+epoch 4000 loss 0.0006816069362685084
+epoch 4100 loss 0.0006751783657819033
+epoch 4200 loss 0.0006688352441415191
+epoch 4300 loss 0.0006628871196880937
+epoch 4400 loss 0.0006569336983375251
+epoch 4500 loss 0.0006518965819850564
+epoch 4600 loss 0.0006464265170507133
+epoch 4700 loss 0.00063846365083009
+epoch 4800 loss 0.0006334380595944822
+epoch 4900 loss 0.0006282057147473097
+epoch 5000 loss 0.0006211302825249732
+
+
+

+
+
+
+
print("MLPRegressor", mean_squared_error(clf.predict(X), y))
+print("PyTorch     ", mean_squared_error(torch_model(torch_X).detach().numpy(), y))
+
+
MLPRegressor 0.0011639113
+PyTorch      0.00062100013
+
+
+
+
draw(lambda x: torch_model(torch.tensor(x)).detach().numpy())
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+
+
+

Flax / JAX

+

Flax with JAX is a modern functional programming library which is great for trying out new things. It comes with fewer high-level functionality than PyTorch. Where PyTorch offers ReduceLROnPlateau, we need to implement something analog from scratch.

+
+
from typing import Callable, Iterable
+from types import SimpleNamespace
+import jax
+from jax import numpy as jnp
+from flax import linen as nn
+import optax
+
+
+class Model(nn.Module):
+    sizes: Iterable[int]  # Number of hidden neurons
+    nonlin: Callable
+
+    @nn.compact  # Tells Flax to look for defined submodules
+    def __call__(self, x):
+        for size in self.sizes:
+            x = nn.Dense(size)(x)
+            x = self.nonlin(x)
+        return nn.Dense(1)(x).flatten()
+
+
+def train(config: SimpleNamespace, x, y):
+    x = jax.device_put(x)
+    y = jax.device_put(y)
+
+    model = Model(config.layers, config.nonlin)
+
+    @jax.jit
+    def loss_fn(theta, x, y):
+        def squared_error(x, y):
+            y_pred = model.apply(theta, x)
+            return jnp.square(y - y_pred)
+        return jnp.mean(jax.vmap(squared_error, in_axes=0)(x, y))
+
+    key1, key2 = jax.random.split(jax.random.PRNGKey(0))
+    theta = model.init(key2, jax.random.normal(key1, x.shape[1:]))
+
+    n = 0
+    for layer in theta["params"].values():
+        n += np.prod(layer["kernel"].shape) + layer["bias"].shape[0]
+    print("number of parameters", n)
+
+    tx = optax.inject_hyperparams(lambda learning_rate: optax.chain(
+        optax.clip(1),  # equivalent to using huber loss
+        optax.adabelief(learning_rate),
+    ))(learning_rate=config.learning_rate)
+    tx_state = tx.init(theta)
+    loss_grad_fn = jax.value_and_grad(loss_fn)
+
+    losses = []
+    prev_loss = np.inf
+    epochs_without_improvement = 0
+    for epoch in range(1, config.max_epoch + 1):
+
+        loss, grads = loss_grad_fn(theta, x, y)
+        updates, tx_state = tx.update(grads, tx_state)
+        theta = optax.apply_updates(theta, updates)
+
+        losses.append(loss)
+
+        if epoch == 1 or epoch % config.print_freq == 0:
+            print(f"epoch {epoch} loss={loss}")
+
+        if loss < prev_loss:
+            epochs_without_improvement = 0
+            prev_loss = loss
+        else:
+            epochs_without_improvement += 1
+        if epochs_without_improvement >= config.shrink_patience:
+            tx_state.hyperparams["learning_rate"] *= config.learning_rate_shrink_factor
+            print(f"epoch {epoch } learning rate={tx_state.hyperparams['learning_rate']}")
+            epochs_without_improvement = 0
+            continue
+        if tx_state.hyperparams["learning_rate"] < config.tolerance:
+            break
+
+    return model, theta, losses
+
+print(f"JAX process: {jax.process_index()} / {jax.process_count()}")
+print(f"JAX local devices: {jax.local_devices()}")
+
+config = SimpleNamespace()
+config.layers = (2 ** 6,) * 4
+config.nonlin = nn.relu
+config.max_epoch = 5000
+config.learning_rate = 1e-2
+config.shrink_patience = 10
+config.learning_rate_shrink_factor = 0.8
+config.print_freq = 100
+config.tolerance = 1e-6
+
+prev_losses = locals().get("prev_losses", None)
+
+flax_model, flax_theta, losses = train(config, X, y)
+
+best_losses = locals().get("best_losses", losses)
+if best_losses[-1] < losses[-1]:
+    best_losses = losses
+
+plt.plot(losses, label="now")
+if prev_losses is not None:
+    plt.plot(prev_losses, label="previous")
+plt.plot(best_losses, label="best")
+plt.legend()
+plt.semilogy();
+
+
JAX process: 0 / 1
+JAX local devices: [CpuDevice(id=0)]
+number of parameters 12993
+epoch 1 loss=0.1615249365568161
+epoch 100 loss=0.004056446719914675
+epoch 200 loss=0.0022707092575728893
+epoch 237 learning rate=0.00800000037997961
+epoch 247 learning rate=0.006400000303983688
+epoch 300 loss=0.0015574624994769692
+epoch 400 loss=0.0013330463552847505
+epoch 500 loss=0.0012052441015839577
+epoch 600 loss=0.0011462707770988345
+epoch 604 learning rate=0.005120000336319208
+epoch 700 loss=0.0010661932174116373
+epoch 773 learning rate=0.0040960004553198814
+epoch 800 loss=0.0010216122027486563
+epoch 900 loss=0.0009868466295301914
+epoch 982 learning rate=0.0032768005039542913
+epoch 1000 loss=0.0009563774801790714
+epoch 1100 loss=0.0009271150338463485
+epoch 1200 loss=0.0008946977904997766
+epoch 1256 learning rate=0.002621440449729562
+epoch 1300 loss=0.0008701515616849065
+epoch 1400 loss=0.0008422626415267587
+epoch 1500 loss=0.0008212537504732609
+epoch 1600 loss=0.0008094692602753639
+epoch 1602 learning rate=0.002097152406349778
+epoch 1700 loss=0.0007778366561979055
+epoch 1800 loss=0.0007623148267157376
+epoch 1867 learning rate=0.0016777219716459513
+epoch 1900 loss=0.0007488274131901562
+epoch 2000 loss=0.0007354258559644222
+epoch 2100 loss=0.0007237782701849937
+epoch 2200 loss=0.0007126012351363897
+epoch 2236 learning rate=0.0013421776238828897
+epoch 2300 loss=0.0007015920709818602
+epoch 2400 loss=0.0006939161103218794
+epoch 2500 loss=0.000684640312101692
+epoch 2544 learning rate=0.001073742168955505
+epoch 2600 loss=0.0006755936774425209
+epoch 2700 loss=0.0006690087029710412
+epoch 2768 learning rate=0.0008589937351644039
+epoch 2800 loss=0.0006616609753109515
+epoch 2853 learning rate=0.0006871949881315231
+epoch 2900 loss=0.0006564840441569686
+epoch 3000 loss=0.0006521755130961537
+epoch 3100 loss=0.0006475721020251513
+epoch 3200 loss=0.0006428791675716639
+epoch 3300 loss=0.0006390072521753609
+epoch 3344 learning rate=0.0005497559905052185
+epoch 3386 learning rate=0.000439804804045707
+epoch 3400 loss=0.000634271593298763
+epoch 3500 loss=0.0006310778553597629
+epoch 3600 loss=0.0006278276559896767
+epoch 3700 loss=0.0006245431723073125
+epoch 3800 loss=0.000620958860963583
+epoch 3900 loss=0.0006179117481224239
+epoch 4000 loss=0.0006144311628304422
+epoch 4091 learning rate=0.00035184386069886386
+epoch 4100 loss=0.0006107029039412737
+epoch 4200 loss=0.0006076746503822505
+epoch 4300 loss=0.0006047412171028554
+epoch 4400 loss=0.0006017453270033002
+epoch 4500 loss=0.0005980761488899589
+epoch 4510 learning rate=0.00028147510602138937
+epoch 4600 loss=0.0005954525549896061
+epoch 4700 loss=0.0005925073637627065
+epoch 4800 loss=0.0005897046066820621
+epoch 4900 loss=0.0005866923602297902
+epoch 5000 loss=0.0005837904172949493
+
+
+

+
+
+
+
print("MLPRegressor", mean_squared_error(clf.predict(X), y))
+print("PyTorch     ", mean_squared_error(torch_model(torch_X).detach().numpy(), y))
+print("Flax        ", mean_squared_error(flax_model.apply(flax_theta, X), y))
+
+
MLPRegressor 0.0011639113
+PyTorch      0.00062100013
+Flax         0.0005836923
+
+
+
+
draw(lambda x: flax_model.apply(flax_theta, x))
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

Another version of learning with FLAX using the meta-learning technique, where the learning rate of the model optimization is learned by an outer learning step. Meta-learning outperforms our previous approach: the loss after 5000 epochs is smaller, and the run-time is a bit shorter.

+
+
from typing import Callable, Iterable
+from types import SimpleNamespace
+import jax
+from jax import numpy as jnp
+from flax import linen as nn
+import optax
+
+
+class Model(nn.Module):
+    sizes: Iterable[int]  # Number of hidden neurons
+    nonlin: Callable
+
+    @nn.compact  # Tells Flax to look for defined submodules
+    def __call__(self, x):
+        for size in self.sizes:
+            x = nn.Dense(size)(x)
+            x = self.nonlin(x)
+        return nn.Dense(1)(x).flatten()
+
+
+def train(config, x, y):
+    x = jax.device_put(x)
+    y = jax.device_put(y)
+    model = Model(config.layers, config.nonlin)
+
+    root_key = jax.random.PRNGKey(0)
+    param_key, perm_key = jax.random.split(root_key)
+    theta = model.init(param_key, x)
+
+    opt = optax.inject_hyperparams(optax.rmsprop)(learning_rate=config.learning_rate)
+
+    meta_opt = optax.adabelief(learning_rate=config.meta_learning_rate)
+
+    state = opt.init(theta)
+    eta = -np.log(1.0 / config.learning_rate - 1)
+    meta_state = meta_opt.init(eta)
+
+    @jax.jit
+    def loss_fn(theta, x, y):
+        def squared_error(x, y):
+            y_pred = model.apply(theta, x)
+            return jnp.square(y - y_pred)
+        return jnp.mean(jax.vmap(squared_error, in_axes=0)(x, y))
+
+    @jax.jit
+    def step(theta, state, x, y):
+        loss, grad = jax.value_and_grad(loss_fn)(theta, x, y)
+        updates, state = opt.update(grad, state)
+        theta = optax.apply_updates(theta, updates)
+        return loss, theta, state
+
+    @jax.jit
+    def outer_loss(eta, theta, state, x, y):
+        state.hyperparams["learning_rate"] = jax.nn.sigmoid(eta)
+        loss, theta, state = step(theta, state, x[:-1], y[:-1])
+        return loss_fn(theta, x[-1:], y[-1:]), (loss, theta, state)
+
+    @jax.jit
+    def outer_step(eta, theta, meta_state, state, x, y):
+        grad, (loss, theta, state) = jax.grad(outer_loss, has_aux=True)(eta, theta, state, x, y)
+        meta_updates, meta_state = meta_opt.update(grad, meta_state)
+        eta = optax.apply_updates(eta, meta_updates)
+        return eta, theta, meta_state, state, loss
+
+    learning_rates = []
+    losses = []
+
+    for epoch in range(1, config.max_epoch + 1):
+        perm_train_key = jax.random.fold_in(perm_key, epoch)
+        perm = jax.random.permutation(perm_train_key, len(x))
+        eta, theta, meta_state, state, loss = outer_step(eta, theta, meta_state, state, x[perm], y[perm])
+        learning_rate = jax.nn.sigmoid(eta)
+
+        losses.append(loss)
+        learning_rates.append(learning_rate)
+
+        if epoch == 1 or epoch % config.print_freq == 0:
+            print(f"epoch {epoch} loss={loss} lr={learning_rate}")
+
+    return model, theta, losses, learning_rates
+
+config = SimpleNamespace()
+config.layers = (2 ** 6,) * 4
+config.nonlin = nn.relu
+config.max_epoch = 5000
+config.learning_rate = 1e-2
+config.meta_learning_rate = 0.03
+config.print_freq = 100
+
+flax_model, flax_ml_theta, losses, learning_rates = train(config, X, y)
+
+fig, ax = plt.subplots(2, 1, sharex=True)
+ax[0].plot(losses)
+ax[0].set(ylabel="loss", yscale="log")
+ax[1].plot(learning_rates)
+ax[1].set(xlabel="epoch", ylabel="lr", yscale="log");
+
+
epoch 1 loss=0.31326305866241455 lr=0.00970732606947422
+epoch 100 loss=0.022382477298378944 lr=0.008342958986759186
+epoch 200 loss=0.009440391324460506 lr=0.008291025646030903
+epoch 300 loss=0.005982980597764254 lr=0.008259016089141369
+epoch 400 loss=0.004722251556813717 lr=0.008235295303165913
+epoch 500 loss=0.0026633746456354856 lr=0.008185734041035175
+epoch 600 loss=0.0032396698370575905 lr=0.008146436884999275
+epoch 700 loss=0.002778511494398117 lr=0.008116255514323711
+epoch 800 loss=0.003227198962122202 lr=0.00809383299201727
+epoch 900 loss=0.0033369455486536026 lr=0.008060149848461151
+epoch 1000 loss=0.0021190992556512356 lr=0.008024145849049091
+epoch 1100 loss=0.002638544887304306 lr=0.007992972619831562
+epoch 1200 loss=0.002345192711800337 lr=0.007957393303513527
+epoch 1300 loss=0.001486414228565991 lr=0.007931879721581936
+epoch 1400 loss=0.0017564770532771945 lr=0.007895255461335182
+epoch 1500 loss=0.0017623923486098647 lr=0.007862220518290997
+epoch 1600 loss=0.0017055724747478962 lr=0.007831801660358906
+epoch 1700 loss=0.0017303635831922293 lr=0.007792978081852198
+epoch 1800 loss=0.0011820418294519186 lr=0.007761101704090834
+epoch 1900 loss=0.0019437490263953805 lr=0.007730470504611731
+epoch 2000 loss=0.0014381277142092586 lr=0.007678112480789423
+epoch 2100 loss=0.001700665452517569 lr=0.007647846359759569
+epoch 2200 loss=0.0012748426524922252 lr=0.007603880017995834
+epoch 2300 loss=0.0014644829789176583 lr=0.007567934691905975
+epoch 2400 loss=0.0017952200723811984 lr=0.007531826850026846
+epoch 2500 loss=0.0014581787399947643 lr=0.0074939788319170475
+epoch 2600 loss=0.0011716835433617234 lr=0.0074319071136415005
+epoch 2700 loss=0.0018534348346292973 lr=0.007394589018076658
+epoch 2800 loss=0.0011852614115923643 lr=0.0073562441393733025
+epoch 2900 loss=0.0014049195451661944 lr=0.007317622657865286
+epoch 3000 loss=0.0016386691713705659 lr=0.007295084185898304
+epoch 3100 loss=0.0010843385243788362 lr=0.007261158432811499
+epoch 3200 loss=0.0009666284313425422 lr=0.007225275970995426
+epoch 3300 loss=0.0010213557397946715 lr=0.0071740285493433475
+epoch 3400 loss=0.001257327850908041 lr=0.0071229999884963036
+epoch 3500 loss=0.0009237469639629126 lr=0.007060170639306307
+epoch 3600 loss=0.0010209324536845088 lr=0.007001352030783892
+epoch 3700 loss=0.0010023850481957197 lr=0.006939604412764311
+epoch 3800 loss=0.0011092709610238671 lr=0.006892509292811155
+epoch 3900 loss=0.0008320825872942805 lr=0.006856516934931278
+epoch 4000 loss=0.0015411889180541039 lr=0.006792154163122177
+epoch 4100 loss=0.000866543676238507 lr=0.00673663429915905
+epoch 4200 loss=0.0009100940660573542 lr=0.00666508823633194
+epoch 4300 loss=0.0008933907374739647 lr=0.006604321300983429
+epoch 4400 loss=0.0006370239425450563 lr=0.00655223848298192
+epoch 4500 loss=0.0007405806099995971 lr=0.006423715502023697
+epoch 4600 loss=0.0006541237817145884 lr=0.006394413765519857
+epoch 4700 loss=0.0005402541719377041 lr=0.006318734493106604
+epoch 4800 loss=0.0009636411559768021 lr=0.00621804129332304
+epoch 4900 loss=0.0007088639540597796 lr=0.006063143257051706
+epoch 5000 loss=0.0007585420971736312 lr=0.006008874159306288
+
+
+

+
+
+
+
print("MLPRegressor       ", mean_squared_error(clf.predict(X), y))
+print("PyTorch            ", mean_squared_error(torch_model(torch_X).detach().numpy(), y))
+print("Flax               ", mean_squared_error(flax_model.apply(flax_theta, X), y))
+print("Flax: meta-learning", mean_squared_error(flax_model.apply(flax_ml_theta, X), y))
+
+
MLPRegressor        0.0011639113
+PyTorch             0.00062100013
+Flax                0.0005836923
+Flax: meta-learning 0.0007262042
+
+
+
+
draw(lambda x: flax_model.apply(flax_ml_theta, x))
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+
+
from typing import Callable, Iterable, Any
+from types import SimpleNamespace
+import jax
+from jax import numpy as jnp
+from flax import struct
+from flax.core import FrozenDict
+from flax import linen as nn
+import optax
+
+
+class Model(nn.Module):
+    sizes: Iterable[int]  # Number of hidden neurons
+    nonlin: Callable
+    dropout_rate: float = 0.0
+
+    @nn.compact  # Tells Flax to look for defined submodules
+    def __call__(self, x, train):
+        for size in self.sizes:
+            x = nn.Dense(size)(x)
+            x = self.nonlin(x)
+        x = nn.Dropout(rate=self.dropout_rate, deterministic=not train)(x)
+        return nn.Dense(1, use_bias=False)(x).flatten()
+
+
+class TrainState(struct.PyTreeNode):
+    dropout_key: jax.Array
+    params: FrozenDict[str, Any] = struct.field(pytree_node=True)
+    opt_state: optax.OptState = struct.field(pytree_node=True)
+    meta_params: float
+    meta_opt_state: optax.OptState = struct.field(pytree_node=True)
+
+
+def predict(model, theta, x):
+    y = model.apply({"params": theta}, x, train=False)
+    with np.errstate(over="ignore"):
+        return 1 / (1 + np.exp(-y))
+
+
+def train(config, x, k, n):
+    x = jax.device_put(x)
+    k = jax.device_put(k)
+    n = jax.device_put(n)
+
+    model = Model(config.layers, config.nonlin, config.dropout_rate)
+
+    root_key = jax.random.PRNGKey(0)
+    params_key, dropout_key, perm_key = jax.random.split(root_key, num=3)
+    variables = model.init(params_key, x, train=False)
+
+    opt = optax.inject_hyperparams(optax.rmsprop)(learning_rate=config.learning_rate)
+    meta_opt = optax.adabelief(learning_rate=config.meta_learning_rate)
+    eta = -np.log(1.0 / config.learning_rate - 1)
+
+    state = TrainState(
+        dropout_key=dropout_key,
+        params=variables["params"],
+        opt_state=opt.init(variables["params"]),
+        meta_params=eta,
+        meta_opt_state=meta_opt.init(eta),
+    )
+
+    @jax.jit
+    def loss_fn(params, state, x, k, n):
+        def fn(x, k, n):
+            y = model.apply(
+                {"params": params},
+                x,
+                train=True,
+                rngs={"dropout": state.dropout_key},
+            )
+            p_sat = jnp.clip(k / (n + 1e-6), 1e-6, 1 - 1e-6)
+            y_sat = jnp.log(p_sat / (1 - p_sat))
+            l = k * (y - y_sat) - n * (jnp.logaddexp(0, y) - jnp.logaddexp(0, y_sat))
+            return l
+
+        l = jax.vmap(fn, in_axes=0)(x, k, n)
+        return -2 * jnp.mean(l)
+
+    @jax.jit
+    def inner_step(state, x, k, n):
+        loss, grad = jax.value_and_grad(loss_fn)(state.params, state, x, k, n)
+        updates, opt_state = opt.update(grad, state.opt_state)
+        params = optax.apply_updates(state.params, updates)
+        state = state.replace(params=params, opt_state=opt_state)
+        return loss, state
+
+    @jax.jit
+    def outer_loss(eta, state, x, k, n):
+        state.opt_state.hyperparams["learning_rate"] = jax.nn.sigmoid(eta)
+        loss, state = inner_step(state, x[:-1], k[:-1], n[:-1])
+        return loss_fn(state.params, state, x[-1:], k[-1:], n[-1:]), (loss, state)
+
+    @jax.jit
+    def outer_step(state, x, k, n):
+        grad, (loss, state) = jax.grad(outer_loss, has_aux=True)(
+            state.meta_params, state, x, k, n
+        )
+        meta_updates, meta_opt_state = meta_opt.update(grad, state.meta_opt_state)
+        meta_params = optax.apply_updates(state.meta_params, meta_updates)
+        state = state.replace(meta_params=meta_params, meta_opt_state=meta_opt_state)
+        return loss, state
+
+    learning_rates = []
+    losses = []
+
+    for epoch in range(1, config.max_epoch + 1):
+        perm_train_key = jax.random.fold_in(key=perm_key, data=epoch)
+        state = state.replace(
+            dropout_key=jax.random.fold_in(key=dropout_key, data=epoch),
+        )
+        perm = jax.random.permutation(perm_train_key, len(x))
+        loss, state = outer_step(state, x[perm], k[perm], n[perm])
+        learning_rate = jax.nn.sigmoid(state.meta_params)
+        losses.append(loss)
+        learning_rates.append(learning_rate)
+
+        if epoch == 1 or epoch % config.print_freq == 0:
+            print(f"epoch {epoch} loss={loss} lr={learning_rate}")
+
+    return model, state.params, losses, learning_rates
+
+
+config = SimpleNamespace()
+config.layers = (2**6,) * 4
+config.nonlin = nn.relu
+config.dropout_rate = 0.01
+config.max_epoch = 5000
+config.learning_rate = 1e-2
+config.meta_learning_rate = 0.03
+config.print_freq = 100
+
+flax_model_2, flax_theta_2, losses, learning_rates = train(config, X, data_k, data_n)
+
+fig, ax = plt.subplots(2, 1, sharex=True)
+ax[0].plot(losses)
+ax[0].set(ylabel="loss", yscale="log")
+ax[1].plot(learning_rates)
+ax[1].set(xlabel="epoch", ylabel="lr", yscale="log");
+
+
epoch 1 loss=8698.1416015625 lr=0.00970732606947422
+epoch 100 loss=879.1326904296875 lr=0.006159865763038397
+epoch 200 loss=270.08587646484375 lr=0.004554468207061291
+epoch 300 loss=317.17718505859375 lr=0.0036662158090621233
+epoch 400 loss=91.02045440673828 lr=0.0032094777561724186
+epoch 500 loss=47.988033294677734 lr=0.0026605320163071156
+epoch 600 loss=49.83475112915039 lr=0.0023816365282982588
+epoch 700 loss=116.31311798095703 lr=0.0021545598283410072
+epoch 800 loss=35.58114242553711 lr=0.0017672122921794653
+epoch 900 loss=64.84205627441406 lr=0.0015509785152971745
+epoch 1000 loss=10.535590171813965 lr=0.001485868007875979
+epoch 1100 loss=26.29697036743164 lr=0.0013874215073883533
+epoch 1200 loss=16.937255859375 lr=0.0013577654026448727
+epoch 1300 loss=15.62310791015625 lr=0.0013214262435212731
+epoch 1400 loss=13.242799758911133 lr=0.001257105148397386
+epoch 1500 loss=11.61507797241211 lr=0.001235545496456325
+epoch 1600 loss=14.939531326293945 lr=0.0012049111537635326
+epoch 1700 loss=27.3996639251709 lr=0.001163122127763927
+epoch 1800 loss=21.23556900024414 lr=0.0010698930127546191
+epoch 1900 loss=32.21183776855469 lr=0.0010345007758587599
+epoch 2000 loss=9.131669044494629 lr=0.001013571978546679
+epoch 2100 loss=13.851628303527832 lr=0.0009823944419622421
+epoch 2200 loss=10.54258918762207 lr=0.0009545115754008293
+epoch 2300 loss=9.792181968688965 lr=0.000919334648642689
+epoch 2400 loss=4.94473123550415 lr=0.000911602983251214
+epoch 2500 loss=46.33889389038086 lr=0.0008922962588258088
+epoch 2600 loss=7.025729179382324 lr=0.0008748381515033543
+epoch 2700 loss=15.959792137145996 lr=0.0008621393935754895
+epoch 2800 loss=48.963592529296875 lr=0.0008530952618457377
+epoch 2900 loss=5.237392902374268 lr=0.0008423625258728862
+epoch 3000 loss=15.789849281311035 lr=0.0008185050683096051
+epoch 3100 loss=12.49224853515625 lr=0.0007375497953034937
+epoch 3200 loss=5.135715961456299 lr=0.0007307584746740758
+epoch 3300 loss=3.9630870819091797 lr=0.0006970498943701386
+epoch 3400 loss=9.80554485321045 lr=0.0006649661809206009
+epoch 3500 loss=4.468692302703857 lr=0.000663100800011307
+epoch 3600 loss=48.7807502746582 lr=0.0006592243444174528
+epoch 3700 loss=3.8573150634765625 lr=0.000651187205221504
+epoch 3800 loss=5.408437728881836 lr=0.0006451716180890799
+epoch 3900 loss=4.103397846221924 lr=0.0006489546503871679
+epoch 4000 loss=7.220167636871338 lr=0.0006393020157702267
+epoch 4100 loss=8.81872272491455 lr=0.0006356375524774194
+epoch 4200 loss=3.7671053409576416 lr=0.0006279195658862591
+epoch 4300 loss=3.080355167388916 lr=0.0006261372473090887
+epoch 4400 loss=26.2066650390625 lr=0.0006219103815965354
+epoch 4500 loss=2.989922046661377 lr=0.0006011227960698307
+epoch 4600 loss=5.964871406555176 lr=0.00058519042795524
+epoch 4700 loss=2.8790411949157715 lr=0.0005803073872812092
+epoch 4800 loss=3.9400908946990967 lr=0.0005809014546684921
+epoch 4900 loss=10.3778715133667 lr=0.0005587802152149379
+epoch 5000 loss=5.031178951263428 lr=0.0005468810559250414
+
+
+

+
+
+
+
print("MLPRegressor              ", mean_squared_error(clf.predict(X), y))
+print("PyTorch                   ", mean_squared_error(torch_model(torch_X).detach().numpy(), y))
+print("Flax                      ", mean_squared_error(flax_model.apply(flax_theta, X), y))
+print(" + meta-learning          ", mean_squared_error(flax_model.apply(flax_ml_theta, X), y))
+print(" + binomial loss, dropout ", mean_squared_error(predict(flax_model_2, flax_theta_2, X), y))
+
+
MLPRegressor               0.0011639113
+PyTorch                    0.00062100013
+Flax                       0.0005836923
+ + meta-learning           0.0007262042
+ + binomial loss, dropout  0.001051374
+
+
+
+
draw(lambda x: predict(flax_model_2, flax_theta_2, x))
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+

In a well-trained model, the parameters should not deviate too much from a normal distribution. We plot the parameter distributions for the last model.

+
+
for i, layer in enumerate(flax_theta_2.values()):
+    fig, ax = plt.subplots(1, len(layer), figsize=(10, 4))
+    ax = np.atleast_1d(ax)
+    for axi, (key, val) in zip(ax, layer.items()):
+        axi.hist(np.ravel(val), bins=20)
+        axi.set_title(f"layer {i+1}: {key}")
+
+

+
+
+

+
+
+

+
+
+

+
+
+

+
+
+ + +
+ +
+ +
+ + + + \ No newline at end of file diff --git a/posts/regression_files/figure-html/cell-10-output-2.png b/posts/regression_files/figure-html/cell-10-output-2.png new file mode 100644 index 0000000000000000000000000000000000000000..8bfbaaf87dfb2d75548fe3c5d8b10989c559f6d5 GIT binary patch literal 15262 zcmb7r2RPO5|M!tf5x#{a8JQV5B|>%~CwuRsLN;X{yMeNE#L3={Ejt_(*()nKR`%X| z9G?5*`fQxaJpv}bw_K!9lf!8)+z?rm<2_1 z?&Sj78wL#t?sT_FgRfSkKK-jfPU1sR z(1tnZFC+{mrM)g}^i96l)_{bPQBzGVoVU%P@&*ic=WOFOXIIyp%1V8{fy3RrDMOS9 zYAwmi$|{bJ7zVpk|Av@Ez-?n{cZ74cP66~wn|crD<8%3X9rUF(GBV1}%0hUB)h5P~ z!HF?}fk8nR!JMYi$(Qz@e&hR}E14Sp4AvzVdsl89Wf&k+H?LmgKeQaxG)Wge^ch;F z{3C(u{X_P(c{Mc>ScR4c$*S^HoXVfNW<}E{q4(EVhRXW!c<%C{yoCj;eNPFy4-dc& zWK-1`Bs_KV^73NtnlzG(epn%#wob7S;2tW|($Gj%y*1?|qQ`#iQtRTj)mV*Jpj;~H z*)#PLSBYJSVDi0LTi2(G!>xq$nf^Bt`rjHJNTv0DRkc%FSw9ctuG!iba}$3YSTq$)yI`Jycl)|d@k-6kybQIh;KZIV$s@|Xc zU;|T|yV)gaj4^aTaDUA;C|3r=S5i`{+-T(~by?Pr{PZa^KVR8;Aiw`;f3wtaR#`79 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+
+ +
+ +
+
+
+

Render LaTeX to SVG with matplotlib backend

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

We use matplotlib’s mathtext system and SVG backend to generate an SVG rendering of LaTeX code.

+
+
from matplotlib.backends.backend_svg import RendererSVG
+from matplotlib.font_manager import FontProperties
+from io import StringIO
+from IPython.display import display_svg, display_html
+
+
+
fp = FontProperties()
+
+s = r"$D^{\ast\ast}\ell\nu$"
+
+# get bounding box of rendered text
+r = RendererSVG(0, 0, StringIO())
+w, h, _ = r.get_text_width_height_descent(s, fp, ismath=True)
+
+with StringIO() as f:
+    # render LaTeX as path
+    r = RendererSVG(w, h, f)
+    r.draw_text(r.new_gc(), 0, h, s, fp, angle=0, ismath=True)
+    f.seek(0)
+    svg_code = f.read()
+
+display_svg(svg_code, raw=True)
+
+

+
+
+

We can insert the SVG image into an HTML table.

+
+
display_html(f"""
+<html>
+<body>
+<table>
+<tr>
+    <td style="fill: #f0f0f0"> {svg_code} </td> <td> foo </td>
+</tr>
+<tr>
+    <td> bar </td> <td> baz </td>
+</tr>
+</table>
+</body>
+</html>
+""", raw=True)
+
+ + + + + + + + + + + +
+ + + + + + + 2024-05-12T19:17:48.899755 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + foo
bar baz
+ + +
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/posts/render_latex_to_svg_files/figure-html/cell-3-output-1.svg b/posts/render_latex_to_svg_files/figure-html/cell-3-output-1.svg new file mode 100644 index 0000000..c85456a --- /dev/null +++ b/posts/render_latex_to_svg_files/figure-html/cell-3-output-1.svg @@ -0,0 +1,116 @@ + + + + + + + + 2024-05-12T19:17:48.899755 + image/svg+xml + + + Matplotlib v3.8.2, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + diff --git a/posts/resample and numba.html b/posts/resample and numba.html new file mode 100644 index 0000000..78dec98 --- /dev/null +++ b/posts/resample and numba.html @@ -0,0 +1,497 @@ + + + + + + + + + +Hans Dembinski’s blog - Numba-accelerated Jackknife + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Numba-accelerated Jackknife

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

My resample library implements the jackknife and bootstrap resampling techniques. The implementations use pure-Python. Since resampling techniques are computationally expensive, it is important to ask whether we can speed up these methods with Numba. In this notebook I compare a JIT-compiled implementation of the Jackknife with the pure Python implementation from resample.

+

Since Numba currently does not support JIT-compiled functions which accept other functions as arguments, the JIT-compiled version has to work a little differently from the pure-Python version. It creates a two-dimensional array with all leave-one-out samples at once inside the compiled code. This array is then passed to Python and the function is evaluated over this array. This has a downside: For large samples, creating an N x (N-1) array in memory becomes dangerous since the memory allocation may bring the computer down and very slow.

+

Fortunately, it turns out that there is nothing to gain from JIT-compiling and thus there is no need to accept this caveat.

+
+
from resample.jackknife import jackknife
+import numpy as np
+import numba as nb
+from typing import Callable
+from timeit import timeit
+from matplotlib import pyplot as plt
+
+
+
@nb.njit(nogil=True)
+def _jackknife_jit(a: np.ndarray):
+    n = len(a)
+    x = np.empty((n, n - 1), dtype=a.dtype)
+    for i in range(n):
+        for j in range(n-1):
+            shift = 0 if j < i else 1
+            x[i, j] = a[j + shift]
+    return x
+
+
+@nb.njit
+def _jackknife_gen(a):
+    n = len(a)
+    x = np.empty(n - 1, a.dtype)
+    for i in range(n - 1):
+        x[i] = a[i + 1]
+    yield x.view(x.dtype)
+
+    # update needs to change values only up to i
+    # for a = [0, 1, 2, 3]
+    # x0 = [1, 2, 3]
+    # x1 = [0, 2, 3]
+    # x2 = [0, 1, 3]
+    # x3 = [0, 1, 2]
+    for i in range(1, n):
+        for j in range(i):
+            x[j] = a[j]
+        yield x.view(x.dtype)
+
+
+def jackknife_jit(fn: Callable, a: np.ndarray) -> np.ndarray:
+    return np.asarray([fn(x) for x in _jackknife_gen(a)])
+
+# force compilation
+jackknife_jit(np.mean, np.random.randn(10));
+
+
+
x = np.random.rand(10000)
+
+
+
%%timeit
+jackknife(np.mean, x)
+
+
105 ms ± 10.6 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
+
+
+
+
%%timeit
+jackknife_jit( np.mean, x)
+
+
119 ms ± 6.34 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
+
+
+

The results are comparable. This also remains so when we benchmark samples of varying size.

+
+
n = np.asarray(np.geomspace(3, 20000, 50), dtype=int)
+times = {
+    ("jackknife", "np.mean"): np.empty(len(n)),
+    ("jackknife_jit", "np.mean"): np.empty(len(n)),
+}
+for i, ni in enumerate(n):
+    x = np.random.randn(ni)
+    for s_fcn, s_arg in times:
+        t = timeit(f"{s_fcn}({s_arg}, x)", f"from __main__ import {s_fcn}, x; import numpy as np", number=1)
+        times[(s_fcn, s_arg)][i] = t
+
+
+
plt.figure()
+for (s_fcn, s_arg), ts in times.items():
+    plt.plot(n, ts, label=f"{s_fcn} {s_arg}")
+plt.loglog()
+plt.ylabel("time / sec")
+plt.legend();
+
+

+
+
+ + + +
+ +
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zS7osK;t`Ml22U4QeP*jlGQp70r!?!!!nPTLJ{tx+)ar$%PjdJmZH)!3Q&CZYKe-wl zKC5UU7h;-ZB^T)y;w;@(cB%g_#-a?1H9*LEry`cmTE_)O@v1U{@eXawPMyxk8G zON8p)htNgW9P#6=i-52G3=ZSe%jjc`hi@?)=B{hOOk&9$WB=Eu+|K!#5a1n`bI z7jGyO_GM}GXqMSon8A0+f+Kk!^umaIF<_TObCuQR-nKBc0en_HN~8L z*!mNBIt9RN3zHh}T~t(4jDVTxNJ+lVR74GH8wzmBLfzfnAQ|>99&-xB@)W_{;q0T0 zgjX3LD=iH?>WddHyfbu%W>_$m26LUa7wYNlM=(E9WivFsrF}@@%i-yK4apr2aLPlk zEHcOsz_sMOfZbby1!h+uU@AG1qL$n@N_9WY2N + + + + + + + + +Hans Dembinski’s blog - Which data structure is faster: array of structs or struct of arrays? + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Which data structure is faster: array of structs or struct of arrays?

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

In the high-performance computing community, it is well known that certain algorithms run faster if the data are structured in such a way that the CPU can read them efficiently. Ideally, the CPU wants to process consecutive memory, not memory which is scattered. In other words, adding two numbers will be much faster on average if the two numbers are next to each other in memory than if they are distant in memory.

+

This means that, depending on the kind of computation, an array of structs (AoS) or a struct of arrays (SoA) performs better. Which one is better depends on the kind of calculation.

+

I demonstrate this for making steps in a Monte-Carlo simulation of particles. Particles are stored on the stack of this simulation, which have a location and a momentum in 3D space, six numbers per particle. In the AoS case, the six numbers for each particle are consecutive in memory, while in the SoA case all x-components of the momentum are consecutive, then all y-components and so on. When the simulation makes a time step, the particles need to be moved along the direction of their momentum. To do the calculation we have to compute the unit vector from the momentum and then move the particle while taking its velocity into account. Whether AoS or SoA works better is not obvious, since arguments can be made for both access patterns.

+

Below, I define the data structures and implement the move algorithms. For move_aos I try out two alternatives, the second is an attempt to help the compiler with autovectorization.

+
+
import numpy as np
+import numba as nb
+from numpy.testing import assert_allclose
+from matplotlib import pyplot as plt
+from timeit import Timer
+
+
+n = int(1e5)
+aos = np.empty((n, 6))
+soa = np.empty((6, n))
+for i in range(6):
+    aos[:, i] = np.arange(n) + i
+    soa[i, :] = np.arange(n) + i
+
+assert_allclose(aos, soa.T)
+
+aos_32 = aos.astype(np.float32)
+soa_32 = soa.astype(np.float32)
+
+
+@nb.njit(error_model="numpy")
+def move_aos(aos, step):
+    for i in range(len(aos)):
+        r = aos[i, :3]
+        p = aos[i, 3:]
+        pn = np.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])
+        ps = step / pn
+        r += p * ps
+    return aos
+
+
+@nb.njit(error_model="numpy")
+def move_aos_alt(aos, step):
+    r = aos[:, :3]
+    p = aos[:, 3:]
+    pn = np.sqrt(p[:,0] * p[:,0] + p[:,1] * p[:,1] + p[:,2] * p[:,2])
+    ps = step / pn
+    for i in range(3):
+        r[:, i] += p[:, i] * ps
+    return aos
+
+
+@nb.njit(error_model="numpy")
+def move_soa(soa, step):
+    r = soa[:3]
+    p = soa[3:]
+    pn = np.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])
+    ps = step / pn
+    for i in range(3):
+        r[i] += p[i] * ps
+    return soa
+
+
+assert_allclose(move_aos(aos, 3), move_soa(soa, 3).T)
+assert_allclose(move_aos_alt(aos, 3), move_soa(soa, 3).T)
+assert_allclose(move_aos(aos_32, 3), move_soa(soa_32, 3).T, rtol=1e-6)
+assert_allclose(move_aos_alt(aos_32, 3), move_soa(soa_32, 3).T, rtol=1e-6)
+
+fns = [x for x in locals() if x.startswith("move")]
+
+

I now run benchmarks for the calculations, using 32 and 64 bit floats and stacks of varying size. It turns out that move_soa in 32 bit performs best. Despite the fact that we need to compute the norm of the momentum for each particle, which in this case means adding momentum components squared from distant memory locations, computing the squares for each component in one go leads to better performance, probably because of pipelining in the CPU. The update of the particle locations can also be done with an efficient memory access pattern, since the respective inputs are consecutive in memory.

+
+
data = {}
+for bits in (32, 64):
+    data[bits] = {}
+    for ifn, fn in enumerate(fns):
+        times = []
+        n = np.geomspace(1, 1e6, 10).astype(int)
+        for ni in n:
+            arg = "aos" if "aos" in fn else "soa"
+            dtype = np.float32 if bits == 32 else np.float64
+            aos = np.empty((ni, 6), dtype=dtype)
+            soa = np.empty((6, ni), dtype=dtype)
+            for i in range(6):
+                aos[:, i] = np.arange(ni) + i
+                soa[i, :] = np.arange(ni) + i
+            t = Timer(f"{fn}({arg}, 3)", f"from __main__ import {fn}, {arg}")
+            m, dt = t.autorange()
+            tmin = min(dt, *t.repeat(3, m))
+            times.append(tmin / m)
+        data[bits][fn] = (n, times)
+
+
+
for bits in data:
+    for ifn, fn in enumerate(fns):
+        n, times = data[bits][fn]
+        plt.plot(n, times, color=f"C{ifn}",
+                 ls="-" if bits == 64 else "--",
+                 label=f"{bits} bit: {fn}")
+plt.legend()
+plt.loglog()
+plt.xlabel("number of particles")
+plt.ylabel("run-time (smaller is better)");
+
+

+
+
+

Another way to present this is to look at the speed-up factor compared to the worst performing AoS implementation. We gain a factor of 40 in the best case.

+
+
for bits in data:
+    for ifn, fn in enumerate(fns):
+        if fn == "move_aos":
+            continue
+        n, times = data[bits][fn]
+        ratio = np.divide(data[bits]["move_aos"][1], times) 
+        plt.plot(n, ratio, color=f"C{ifn}",
+                 ls="-" if bits == 64 else "--",
+                 label=f"{bits} bit: {fn}")
+        i = np.argmax(ratio)
+        plt.text(n[i], ratio[i] - 1, f"{ratio[i]:.0f}", ha="center", va="top")
+plt.plot(n, np.ones_like(n), color="C0")
+plt.legend()
+plt.semilogx()
+plt.xlabel("number of particles")
+plt.ylabel("speed-up");
+
+

+
+
+

To understand where these improvements come from, we investigate the assembler code for each implementation and look for SIMD instructions (single input multiple data), which are often responsible for speed gains. We find that move_soa does not really stand out here. The speed does not come from the instructions used, it comes from optimized access patterns and pipelining.

+
+
import boost_histogram as bh
+
+h = bh.Histogram(bh.axis.StrCategory([], growth=True), bh.axis.Integer(0, 3))
+
+for ifn, fn in enumerate(fns):
+    cmds = []
+    for line in list(globals()[fn].inspect_asm().values())[0].split("\n"):
+        line = line.strip("\t ")
+        if line.startswith(".") or line.startswith("__"):
+            continue
+        if "\t" not in line:
+            continue
+        cmd = line.split("\t")[0]
+        if not cmd.startswith("v"):
+            continue
+        if cmd.endswith("sd"):
+            continue
+        cmds.append(cmd)
+
+    h.fill(cmds, ifn)
+
+
+
import matplotlib.pyplot as plt
+
+xlabels = []
+values = []
+
+for x, v in sorted(zip(h.axes[0], h.values())):
+    xlabels.append(x)
+    values.append(v)
+
+values = np.array(values)
+
+xe = np.arange(len(xlabels) + 1)
+v = 0
+for ifn, fn in enumerate((move_aos, move_aos_alt, move_soa)):
+    before = v
+    v = v + values[:, ifn]
+    plt.stairs(v, 1-xe, baseline=before, fill=True, label=fn.__name__, orientation="horizontal")
+plt.legend()
+plt.gca().yaxis.set_ticks(-np.arange(len(v)) + 0.5)
+plt.gca().yaxis.set_ticklabels(xlabels);
+
+

+
+
+ + + +
+ +
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z8HVBB@bl+oEQSL48X*jvC;%7%Wb)=(TU(QFRDzfm8ycRF{0$FO5VkFXi0G;f&CQ>2 zPRD_*3=9ne;R$%>9eIaJv32WKQw`t))>R8MgxdfmxJWp zuBtd!xXbg;Mr)o%G`qWXA3M7pT>hI}vklrwe{b$yOdHq6+nF>qkQa*sQibY-p$nbp13EuSzatFJ%53tbf0VjVhp#|GRi`p4 zv$eVTDwB}udmUivSUh2$(9qCs)OFrY%y!qdwzOzW!g$pN`VQK2@Zb+w+**~2?exX_ z+qiAW#teo1q61_ci%GdEl483I{Gkm=fT!~sS9lMM;#gq%^pOlYu?k8R=2G=0f$+3p z3pgTlT%8o7Tk~V&>ruK&6iICYCFiP;2u66sho7BX_#l+3jSv;Nm&(TgGLyaq_s;X-3?7V?wpIYuxIuTW*o>?|I^K0W^#*$^e6svt*N?Mvbu$VzbP*5-y{GtN$?AjnWv5=iB z@ihV_moY?gPSVKS1Rs;!IV%e5xlA*fUbhw;SsPU#3ExIk(2C$KlpvpOEP!~Q&Ph7Q zf{{4MdN6zY9GznCKYtV1e}jDfJ#U?urCF7*#<)%S6!~sC$_eo^$HI + + + + + + + + +Hans Dembinski’s blog - Template fits with distortion correction + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+
+ +
+ +
+
+
+

Template fits with distortion correction

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

I recently gave a review talk on template fits in CERN’s PHYSTAT seminar, where I presented results from our recent paper on this subject. In the summary, I discussed on of the downsides of template fits: if the simulation does not match the experiment perfectly, one cannot easily adjust for these discrepancies, as one would do in a parametric fit. For example, if I fit a peak over a smooth background, and the peak location is slightly off in the simulation, a parametric fit would simply fit a different peak location and give a good fit, while a template fit with templates taken from simulation would give poor results.

+

We explore this on a toy example. We want to fit a peak over smooth background using templates. The sample which provides a template for the peak is distorted. Its mean is shifted and it also too narrow. We want to fit the yield of this peak under these conditions.

+
+
import numpy as np
+from iminuit.cost import template_chi2_da
+from iminuit import Minuit
+from numba_stats import truncexpon, truncnorm
+import matplotlib.pyplot as plt
+from resample.bootstrap import resample
+from scipy.optimize import differential_evolution
+import warnings
+from iminuit.warnings import IMinuitWarning
+
+s_sample = truncnorm.rvs(0, 1, 0.5, 0.1, size=1000, random_state=1)
+b_sample = truncexpon.rvs(0, 1, 0, 1, size=1000, random_state=1)
+x_sample = np.append(s_sample, b_sample)
+
+st_sample = truncnorm.rvs(0, 1, 0.6, 0.05, size=1000, random_state=2)
+bt_sample = truncexpon.rvs(0, 1, 0, 1, size=1000, random_state=2)
+
+n_data, xe, _ = plt.hist(
+    x_sample, bins=20, range=(0, 1), alpha=0.5, color="k", label="data"
+)
+template_s = plt.hist(st_sample, bins=xe, alpha=0.5, label="signal template")[0]
+template_b = plt.hist(bt_sample, bins=xe, alpha=0.5, label="background template")[0]
+plt.legend();
+
+

+
+
+

In practice, this situation is common. The simulation is a good, but not perfect approximation of reality. Template fits could work very well in these situations if we find a way to fit a slightly distorted template instead of the raw one. For this to work, the distortion has to be a simple function, for example, an affine transformation like in this case. Then, the transformation can be fitted together with the yields. The parameters of the transformation are nuisance parameters. Note that the uncertainty of the fitted yield will always increase when such a transformation has to be estimated from the data as well.

+
+

The catch

+

The idea to fit a transformed template is straight-forward, but there is a catch. Standard minimizers employed in HEP use a method based on gradient descent, and the gradient is computed numerically using a finite-difference formula. These minimizers only work correctly if the function to minimize is smooth: its first and second derivatives must be smoothly varying. But this is not the case the in the outlined approach, because the transformation will cause points from the samples that form the template to migrate from bin to bin. This causes jumps in the objective function and thus discontinuities in the derivatives. MINUIT does not like that at all. It can fail to converge, get stuck in a local minimum, or may report convergence to a point which is not a minimum. All of these outcomes are bad. The following cells demonstrate how MINUIT fails on such a fit. As objective function, we use the transformed approximate likelihood for template fits that we derived in our paper (implemented in template_chi2_da).

+
+
def model(s, b, offset, scale):
+    mu_st_sample = np.mean(st_sample)
+    st_sample_modified = scale * (st_sample - mu_st_sample) + mu_st_sample + offset
+    template_s_modified = np.histogram(st_sample_modified, bins=xe)[0]
+    # a small constant is added to avoid dividing by zero
+    ts = template_s_modified / (np.sum(template_s_modified) + 1e-6)
+    tb = template_b / np.sum(template_b)
+    mu = s * ts + b * tb
+    mu_var = s * ts**2 + b * tb**2
+    return mu, mu_var
+
+
+def objective(s, b, offset, scale):
+    mu, mu_var = model(s, b, offset, scale)
+    return template_chi2_da(n_data, mu, mu_var)
+
+
+# if errordef and ndata on the objective are set,
+# iminuit computes the chi2 gof test statistic for us
+objective.errordef = Minuit.LEAST_SQUARES
+objective.ndata = len(xe) - 1
+
+m = Minuit(objective, s=1000, b=1000, offset=0, scale=1)
+m.limits["s", "b"] = (0, None)
+m.limits["scale"] = (0.01, 2.5)
+m.limits["offset"] = (-0.2, 0.2)
+m.migrad()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
Migrad
FCN = 499 (χ²/ndof = 31.2)Nfcn = 296
EDM = 2.72e-12 (Goal: 0.0002)
Valid MinimumBelow EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0s494330
1b1.51e30.05e30
2offset-9.0380e-30.0033e-3-0.20.2
3scale999.17e-30.31e-30.012.5
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
sboffsetscale
s1.08e+03-0.4e3 (-0.299)-2.648168e-6 (-0.025)-124.42e-6 (-0.012)
b-0.4e3 (-0.299)2.02e+033.171095e-6 (0.022)149.02e-6 (0.011)
offset-2.648168e-6 (-0.025)3.171095e-6 (0.022)1.06e-110.497e-9 (0.498)
scale-124.42e-6 (-0.012)149.02e-6 (0.011)0.497e-9 (0.498)9.39e-08
+
+
+

MINUIT reports successful convergence, but signal and background yields are far from the truth. Also, the chi2/ndof ratio is far from 1. We can see that this a poor fit by drawing the fitted model and by looking at the profile-likelihood for each parameter.

+
+
# let's draw the fitted model
+plt.stairs(model(*m.values)[0], xe, label="fitted model")
+plt.stairs(n_data, xe, label="data")
+plt.legend();
+
+

+
+
+
+
plt.figure()
+m.draw_profile("s")
+plt.figure()
+m.draw_profile("b")
+# ignore some iminuit warnings about failing to converge during profile computation
+with warnings.catch_warnings():
+    warnings.simplefilter("ignore", IMinuitWarning)
+    plt.figure()
+    m.draw_mnprofile("scale", bound=(0.5, 3.5))
+    plt.figure()
+    m.draw_mnprofile("offset", bound=(-0.5, 0.2))
+
+

+
+
+

+
+
+

+
+
+

+
+
+

We can see in the profile that we are not in the global minimum, which is near offset = -0.1 and scale = 2.2.

+
+
+

Solution

+

Two solutions come to mind, both have caveats. I explore one below, which I consider to be the simpler one. The second solution is to replace the transformed template by a kernel density estimate. This would lead to additional hyperparameters introduced by the kernel density estimation, which would need to be carefully chosen for each problem, and require the derivation of a new complicated likelihood function for this case.

+
+

Use a global derivative-free minimizer and bootstrapping

+

Since we cannot use a local minimizer which uses gradients for our objective function, we switch to a global derivative-free minimizer. Global derivative-free minimizers require many more function calls than standard local minimizers like MINUIT’s MIGRAD, and this number moreover scales quadratically with the number of fitted parameters. Fortunately, our objective function is relative fast to compute and does not have many parameters, so global minimization is feasible. In the following, I apply the differential_evolution algorithm from scipy to this problem.

+
+
r = differential_evolution(
+    lambda x: objective(*x),
+    bounds=((0, 2000), (0, 2000), (-0.2, 0.2), (0.5, 3.5)),
+    polish=False,
+    seed=1,
+)
+print(r.message)
+
+chi2 = objective(*r.x)
+
+print(f"chi2/ndof = {chi2/(objective.ndata - m.nfit):.1f}")
+print(f"s         = {r.x[0]:.0f}")
+print(f"b         = {r.x[1]:.0f}")
+print(f"offset    = {r.x[2]:.2f}")
+print(f"scale     = {r.x[3]:.1f}")
+
+# let's draw the fitted model
+plt.stairs(model(*r.x)[0], xe, label="fitted model")
+plt.stairs(n_data, xe, label="data")
+plt.legend();
+
+
Optimization terminated successfully.
+chi2/ndof = 1.0
+s         = 1017
+b         = 987
+offset    = -0.09
+scale     = 2.0
+
+
+

+
+
+

This looks great! We got an excellent chi2/ndof ratio and the fitted model agrees well with the data. Our idea works, and the global minimizer did not take too much time either.

+

Now we need to address the next challenge: we also want to know the parameter uncertainties. We cannot use the usual technique of inverting the appropriately scaled Hesse matrix of the objective function at the minimum to get the covariance matrix, because this would again require a numerical computation of second derivatives.

+

I will nevertheless attempt to compute the covariance matrix with Minuit’s HESSE algorithm to see what happens.

+
+
m.values = r.x
+m.hesse()
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
External
FCN = 15.5 (χ²/ndof = 1.0)Nfcn = 9332
EDM = 0.0125 (Goal: 0.0002)
INVALID MinimumABOVE EDM threshold (goal x 10)
No parameters at limitBelow call limit
Hesse okCovariance accurate
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
0s1.02e30.04e30
1b990400
2offset-87.39e-30.14e-3-0.20.2
3scale1.98480.00150.012.5
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
sboffsetscale
s1.85e+03-0.7e3 (-0.380)4.261e-6-185.2e-6 (-0.003)
b-0.7e3 (-0.380)1.7e+03-28.018e-6 (-0.005)-217.3e-6 (-0.003)
offset4.261e-6-28.018e-6 (-0.005)2.08e-080.050e-6 (0.230)
scale-185.2e-6 (-0.003)-217.3e-6 (-0.003)0.050e-6 (0.230)2.29e-06
+
+
+

HESSE reports succeess, but the uncertainties on offset and scale are suspiciously small. We should not trust this result. This is one of the possible outcomes if the second derivatives are not well-defined bad. In other cases, HESSE may fail or report that the covariance matrix had to be forced to be positive definite.

+

Since this does not work, we employ the bootstrap technique, using the resample library. This library provides a function which readily computes the covariance for a function that depends on a single dataset, but we have a more complicated case here. We need to resample three samples simultaneously, which have different sizes, the data sample and the two samples that generate the templates. We use the “extended” resampling method to also vary the sizes of these samples, which is usually the correct thing in high-energy physics, but not how the standard bootstrap in the literature works.

+
+
# function which does the whole estimation; taking samples as input and
+# returning parameters of interest as output
+def fit(x_sample, st_sample, bt_sample):
+    n_data = np.histogram(x_sample, bins=xe)[0]
+    template_b = np.histogram(bt_sample, bins=xe)[0]
+
+    def model(s, b, offset, scale):
+        mu_st_sample = np.mean(st_sample)
+        st_sample_modified = scale * (st_sample - mu_st_sample) + mu_st_sample + offset
+        template_s_modified = np.histogram(st_sample_modified, bins=xe)[0]
+        # a small constant is added to avoid dividing by zero
+        ts = template_s_modified / (np.sum(template_s_modified) + 1e-6)
+        tb = template_b / np.sum(template_b)
+        mu = s * ts + b * tb
+        mu_var = s * ts**2 + b * tb**2
+        return mu, mu_var
+
+    def objective(s, b, offset, scale):
+        mu, mu_var = model(s, b, offset, scale)
+        return template_chi2_da(n_data, mu, mu_var)
+
+    r = differential_evolution(
+        lambda x: objective(*x),
+        bounds=((0, 2000), (0, 2000), (-0.2, 0.2), (0.5, 3.5)),
+        polish=False,
+        seed=1,
+    )
+    assert r.success
+    return r.x
+
+
+number_of_bootstrap_samples = 20
+kwargs = {"size": number_of_bootstrap_samples, "method": "extended", "random_state": 1}
+
+results = []
+for xs, sts, bts in zip(
+    resample(x_sample, **kwargs),
+    resample(st_sample, **kwargs),
+    resample(bt_sample, **kwargs),
+):
+    xb = fit(xs, sts, bts)
+    results.append(xb)
+
+cov = np.cov(results, rowvar=False)
+
+for name, val, err in zip(m.parameters, r.x, np.diag(cov) ** 0.5):
+    print(f"{name} = {val:.2f} +/- {err:.2f}")
+
+
s = 1016.88 +/- 71.60
+b = 987.33 +/- 51.06
+offset = -0.09 +/- 0.00
+scale = 1.98 +/- 0.16
+
+
+

The calculation takes a while but eventually completes. We now find larger uncertainties compared to the HESSE estimate above, which essentially treated the transform like it was fixed. We can see that the increase in the uncertainties indeed originates from the fitted transform, if we repeat the bootstrap with fixed the transformation parameters.

+
+
def fit(x_sample, st_sample, bt_sample):
+    n_data = np.histogram(x_sample, bins=xe)[0]
+    template_b = np.histogram(bt_sample, bins=xe)[0]
+
+    def model(s, b, offset, scale):
+        mu_st_sample = np.mean(st_sample)
+        st_sample_modified = scale * (st_sample - mu_st_sample) + mu_st_sample + offset
+        template_s_modified = np.histogram(st_sample_modified, bins=xe)[0]
+        # a small constant is added to avoid dividing by zero
+        ts = template_s_modified / (np.sum(template_s_modified) + 1e-6)
+        tb = template_b / np.sum(template_b)
+        mu = s * ts + b * tb
+        mu_var = s * ts**2 + b * tb**2
+        return mu, mu_var
+
+    def objective(s, b, offset, scale):
+        mu, mu_var = model(s, b, offset, scale)
+        return template_chi2_da(n_data, mu, mu_var)
+
+    r = differential_evolution(
+        lambda x: objective(*x, -0.09, 1.99), bounds=((0, 2000), (0, 2000)), 
+        polish=False,
+        seed=1
+    )
+    assert r.success
+    return r.x
+
+
+kwargs = {"size": number_of_bootstrap_samples, "method": "extended", "random_state": 1}
+
+results = []
+for xs, sts, bts in zip(
+    resample(x_sample, **kwargs),
+    resample(st_sample, **kwargs),
+    resample(bt_sample, **kwargs),
+):
+    xb = fit(xs, sts, bts)
+    results.append(xb)
+
+cov = np.cov(results, rowvar=False)
+
+for name, val, err in zip(m.parameters, r.x, np.diag(cov) ** 0.5):
+    print(f"{name} = {val:.2f} +/- {err:.2f}")
+
+
s = 1016.88 +/- 50.19
+b = 987.33 +/- 39.07
+
+
+
+
+
+

Conclusion

+

At least in some cases, it is possible to fit distorted templates and estimate the distortion parameters as nuisance parameters, and get good results with modest effort. The solution described here will work best if the components in the mixture are well separated and if the same distortion can be applied to all templates (for example, because it models a calibration mismatch).

+

In the following cases, this method will introduce large uncertainties on the fitted yields: - the components are strongly overlapping - different distortions are needed for each template - the distortion function is very flexible and has many parameters

+

In some of these difficult cases, reasonable results may be attainable if the distortion parameters are further constrained with penalty terms, that prevent the distortion function from diverging too much from the identity.

+ + +
+ +
+ +
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zJ3aTI*f{z;!6l`SKe&)TRxS*m0{D$?*GRCf6@h>1E_7rSVN)zd)NX&g&6;GANfK|3 zyi`v~ODhU}BAIcxswnij@0K_Y54{6WUcGX3DgXlL)Ue-P_3zw<53b#rb!>ocrQ<+D;%5{gBL~;+t`s!D4w-(}v!brrP!4_&PjJbFfH9Z}s!?F` z#AS4Yod#c3!Pbi&w&|V%xye08_0U-uMKIa^1F?{7W+SAP#OxwpUe+GF;|{Crmt_=izYyPkj5?9u-Kv+hP} literal 0 HcmV?d00001 diff --git a/posts/tracking_efficiency.html b/posts/tracking_efficiency.html new file mode 100644 index 0000000..abd7937 --- /dev/null +++ b/posts/tracking_efficiency.html @@ -0,0 +1,537 @@ + + + + + + + + + +Hans Dembinski’s blog - Tracking efficiency for two-body decay + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Tracking efficiency for two-body decay

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  • Consider decay \(A \to B C\)
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  • Want to know corrected efficiency \(\epsilon'_A\) +
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    • Given: \(\epsilon_A(x)\), \(\epsilon_B(x)\), \(\epsilon_C(x)\), correction factor \(\gamma(x)\) for B, C
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    • Efficiency and correction factors depend on variable \(x\) (can be multidimensional)
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    • \(f(x_B, x_C)\) is probability density to find decay products \(B\) and \(C\) at locations \(x_B\) and \(x_C\), respectively
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\[ +\epsilon'_A = \int \text{d}x_B\! \int \text{d}x_C\, \epsilon'_B(x_B) \, \epsilon'_C(x_C) \; f(x_B, x_C) +\]

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\[ +\epsilon'_A = \int \text{d}x_B\! \int \text{d}x_C \, \gamma(x_B) \, \gamma(x_C) \, \epsilon_B(x_B) \, \epsilon_C(x_C) \; f(x_B, x_C) +\]

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\[ +\epsilon'_A = \text{E}[\gamma(x_B) \, \gamma(x_C) \, \epsilon_B(x_B) \, \epsilon_C(x_C)] +\]

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Compute sample estimator

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  • Apply plugin estimate \(\text{E}[f(x)] \to \frac 1N \sum_k f(x_k)\), \(N\) is generated number of A decays
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\[ +\epsilon'_A = \frac 1 N \sum_k \gamma(x_{B,k}) \, \gamma(x_{C,k}) \, \epsilon_B(x_{B,k}) \, \epsilon_C(x_{C,k}) +\]

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  • Apply second plugin estimate: \(\sum_k \epsilon(x_k) \to \sum_k \beta_k\), with \(\beta_k \in \{0, 1\}\), \(\beta_k\) is outcome of the Bernoulli process which determines whether child is reconstructed or not
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\[ +\epsilon'_A = \frac 1 N \sum_k \gamma(x_{B,k}) \, \gamma(x_{C,k}) \, \beta_{B,k} \, \beta_{C,k} +\]

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  • For \(\gamma(x)\) that is piece-wise constant over \(x\)-bins \(i\), this is equal to \[ +\epsilon'_A = \frac 1 N \sum_{i,j} n_{ij} \, \gamma_{i} \, \gamma_{j} \quad +\text{with} \quad n_{ij} = \sum_k \delta_{B,ik} \delta_{C,jk} \beta_{B,k} \beta_{C,k} +\]

    +
      +
    • \(\delta_{X,ik}\) is one if particle X from decay \(k\) is inside \(x\)-bin \(i\) and zero otherwise
    • +
  • +
  • If we only sum over decays \(\ell\) with \(\beta_{\ell,B} \, \beta_{\ell,C} = 1\), this further simplifies to \[ +n_{ij} = \sum_{\ell} \delta_{B,i\ell} \delta_{C,j\ell} +\]

  • +
  • Correction factor for \(A\) then is with \(n = \sum_{i,j} n_{i,j}\)

  • +
+

\[ +\gamma_A = \frac{\epsilon'_A}{\epsilon_A} = \frac{\sum_{i,j} n_{ij} \, \gamma_{i} \, \gamma_{`j}}{\sum_{i,j} n_{ij}} = \frac 1 n \sum_{i,j} n_{ij} \, \gamma_{i} \, \gamma_{j} +\]

+
+
+

Error propagation

+
    +
  • Neglect error on \(n_{ij}\) and \(n\) to simplify

  • +
  • Variance of \(\gamma_i\) is \(V_{\gamma,i}\)

  • +
  • Variance of \(\gamma_A\) is \[ +V_{\gamma_A} = \sum_k J_k^2 \, V_{\gamma,k} \quad \text{with} \quad J_k = \frac{\partial \epsilon'_A}{\partial \gamma_k} = \frac{1}{n} \sum_i \gamma_i (n_{ki} + n_{ik}) +\]

  • +
  • Check of analytical formula against numerical calculation

  • +
+
+
from jacobi import propagate
+import numpy as np
+
+
+
rng = np.random.default_rng(1)
+
+# in general, n is not symmetric, n_ij != n_ji
+n = rng.random(size=(3, 3)) * 100
+g = 0.5 + rng.random(size=3)
+vg = 1 + rng.random(size=3)
+
+m = np.sum(n)
+
+
+
def compute(x):
+    nx = np.reshape(x[:n.size], n.shape)
+    gx = x[n.size:]
+    return np.einsum("i,j,ij", gx, gx, nx) / m
+
+x = np.append(n.reshape(-1), g)
+vx = np.append(np.zeros(n.size), vg)
+
+y, vy = propagate(compute, x, vx)
+y, vy
+
+
(array(0.80687392), array(1.65110298))
+
+
+
+
def analytical(n, g, vg):
+    j = g @ (n + n.T) / m
+    vy = np.sum(j ** 2 * vg)
+    return vy
+
+vy2 = analytical(n, g, vg)
+vy2
+
+
1.6511029832764388
+
+
+

Numerical and analytical calculations agree.

+

Now I compare my formula with the formula on the TWiki page of the Tracking Group.

+
+
# Formula on TWiki seems to assume that g ~ 1
+g = np.ones(3)
+
+def analytical_twiki(n, vg):
+    s = 0
+    sg = vg ** 0.5
+    for i in range(n.shape[0]):
+        for j in range(n.shape[1]):
+            s += n[i,j] ** 2 * (sg[i] + sg[j]) ** 2
+    return s / m ** 2
+
+vy3 = analytical_twiki(n, vg)
+vy3
+
+
0.802978765901557
+
+
+
+
analytical(n, g, vg)
+
+
1.9900608390572314
+
+
+

The formula from the TWiki seems to give a variance that is much too small by more than a factor of two in this case.

+ + +
+ +
+ +
+ + + + \ No newline at end of file diff --git a/posts/visual_cross_section.html b/posts/visual_cross_section.html new file mode 100644 index 0000000..07b484b --- /dev/null +++ b/posts/visual_cross_section.html @@ -0,0 +1,616 @@ + + + + + + + + + +Hans Dembinski’s blog - Visual cross-section for particle production + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+
+
+

Visual cross-section for particle production

+
+
+ + +
+ + + + +
+ + +
+ + + + +
+ + + + +

When you look for a somewhat rare decay, you want to know how much luminosity is required to see N decays in your detector. For this one needs to compute the visual cross-section, this is the part of the production cross-section for the particle for which the decay products all fall into the acceptance of the detector.

+

We compute the visual cross-section here for the decay of Omega and Xi with chromo. We then go on and compute the expected rate of interactions for running LHCb in fixed-target mode.

+
+

Visual cross-section

+

We compute the inelastic cross-section for p-He collisions in fixed-target configuration where the proton has 13 TeV with chromo.

+
+
from chromo.models import Pythia8, EposLHC
+from chromo.kinematics import FixedTarget, TeV, GeV
+import numpy as np
+from particle import literals as lp
+from chromo.util import pdg2name
+from rich.progress import track
+
+
+
kin = FixedTarget(13*TeV, "p", "p")
+
+# epos = EposLHC(kin)
+gen = Pythia8(kin, seed=1)
+cs = gen.cross_section()
+
+

+ *------------------------------------------------------------------------------------* 
+ |                                                                                    | 
+ |  *------------------------------------------------------------------------------*  | 
+ |  |                                                                              |  | 
+ |  |                                                                              |  | 
+ |  |   PPP   Y   Y  TTTTT  H   H  III    A      Welcome to the Lund Monte Carlo!  |  | 
+ |  |   P  P   Y Y     T    H   H   I    A A     This is PYTHIA version 8.308      |  | 
+ |  |   PPP     Y      T    HHHHH   I   AAAAA    Last date of change: 16 Nov 2022  |  | 
+ |  |   P       Y      T    H   H   I   A   A                                      |  | 
+ |  |   P       Y      T    H   H  III  A   A    Now is 31 Mar 2023 at 15:19:42    |  | 
+ |  |                                                                              |  | 
+ |  |   Program documentation and an archive of historic versions is found on:     |  | 
+ |  |                                                                              |  | 
+ |  |                               https://pythia.org/                            |  | 
+ |  |                                                                              |  | 
+ |  |   PYTHIA is authored by a collaboration consisting of:                       |  | 
+ |  |                                                                              |  | 
+ |  |   Christian Bierlich, Nishita Desai, Leif Gellersen, Ilkka Helenius, Philip  |  | 
+ |  |   Ilten, Leif Lonnblad, Stephen Mrenna, Stefan Prestel, Christian Preuss,    |  | 
+ |  |   Torbjorn Sjostrand, Peter Skands, Marius Utheim and Rob Verheyen.          |  | 
+ |  |                                                                              |  | 
+ |  |   The complete list of authors, including contact information and            |  | 
+ |  |   affiliations, can be found on https://pythia.org/.                         |  | 
+ |  |   Problems or bugs should be reported on email at authors@pythia.org.        |  | 
+ |  |                                                                              |  | 
+ |  |   The main program reference is C. Bierlich et al,                           |  | 
+ |  |   'A comprehensive guide to the physics and usage of Pythia 8.3',            |  | 
+ |  |   SciPost Phys. Codebases 8-r8.3 (2022) [arXiv:2203.11601 [hep-ph]]          |  | 
+ |  |                                                                              |  | 
+ |  |   PYTHIA is released under the GNU General Public Licence version 2 or later.|  | 
+ |  |   Please respect the MCnet Guidelines for Event Generator Authors and Users. |  | 
+ |  |                                                                              |  | 
+ |  |   Disclaimer: this program comes without any guarantees.                     |  | 
+ |  |   Beware of errors and use common sense when interpreting results.           |  | 
+ |  |                                                                              |  | 
+ |  |   Copyright (C) 2022 Torbjorn Sjostrand                                      |  | 
+ |  |                                                                              |  | 
+ |  |                                                                              |  | 
+ |  *------------------------------------------------------------------------------*  | 
+ |                                                                                    | 
+ *------------------------------------------------------------------------------------* 
+
+
+
+
+
cs.inelastic
+
+
40.543016683445494
+
+
+

Now we compute which fraction of inelastic events produce the following decays so that all charged final state decay products are in the LHCb acceptance, given by \(p > 2\) GeV/c and \(2 < \eta < 5\). \[ +\Xi^- \to \Lambda \pi^- \; (99.887\,\%) +\] \[ +\Omega^- \to \Lambda K^- \; (67.8\,\%) +\] We don’t need to take special care of the branching fractions. We get these reduction factors automatically if we select the specific decay we are interested in from the generator output.

+
+
apid_xi = abs(lp.Xi_minus.pdgid)
+apid_omega = abs(lp.Omega_minus.pdgid)
+apid_lambda = abs(lp.Lambda.pdgid)
+apid_pi = abs(lp.pi_plus.pdgid)
+apid_proton = abs(lp.proton.pdgid)
+apid_K = abs(lp.K_plus.pdgid)
+
+gen.set_stable(apid_xi, False)
+gen.set_stable(-apid_xi, False)
+gen.set_stable(apid_omega, False)
+gen.set_stable(-apid_omega, False)
+gen.set_stable(apid_lambda, False)
+gen.set_stable(-apid_lambda, False)
+
+
+
def is_visible_decay(ptot, eta, apid, children, par, apid1, apid2):
+    ch = children[par]
+    if len(ch) != 2:
+        return False
+    for i in ch:
+        if apid[i] != apid1 and apid[i] != apid2:
+            return False
+        if apid[i] == apid_lambda:
+            if not is_visible_decay(ptot, eta, apid, children, i, apid_proton, apid_pi):
+                return False
+        if ptot[i] < 2*GeV or eta[i] < 2 or eta[i] > 5:
+            return False
+    return True
+
+n_inel = 100000
+n_decays = {"Xi": 0, "Omega": 0}
+for event in track(gen(n_inel), total=n_inel):
+    apid = np.abs(event.pid)
+    mask_xi = apid == apid_xi
+    mask_omega = apid == apid_omega
+
+    if np.any(mask_xi | mask_omega):
+        # collect children of each parent
+        children = {}
+        for i, par in enumerate(event.parents):
+            if par[0] == 0:
+                continue
+            children.setdefault(par[0] - 1, []).append(i)
+
+        ptot = event.p_tot
+        eta = event.eta
+
+        # investigate Xi decays
+        for par in np.arange(len(mask_xi))[mask_xi]:
+            if is_visible_decay(ptot, eta, apid, children, par, apid_lambda, apid_pi):
+                n_decays["Xi"] += 1
+
+        # investigate Omega decays
+        for par in np.arange(len(mask_omega))[mask_omega]:
+            if is_visible_decay(ptot, eta, apid, children, par, apid_lambda, apid_K):
+                n_decays["Omega"] += 1
+
+print(n_decays, "out of", n_inel)
+
+ +
+
+

+
+
+

+
+
+
{'Xi': 122, 'Omega': 3} out of 100000
+
+
+
+
sigma_vis = {k: v / n_inel * cs.inelastic for k, v in n_decays.items()}
+
+print("visible cross-section")
+for k, s in sigma_vis.items():
+    print(f"{k:10} {s:.4f} mb")
+
+
visible cross-section
+Xi         0.0495 mb
+Omega      0.0012 mb
+
+
+
+
+

LHCb in fixed-target mode

+

LHCb in fixed-target mode uses a gas target. We compute the event rate for the SMOG 1 configuration where the gas is injected directly into the LHCb Vertex Locator (VELO).

+

The average number of interactions per bunch crossing is \[ +\mu = \sigma L N p \frac{N_A}{22400} +\] where \(L = 60\) cm is the length of the detection region in the VELO, \(N=10^{11}\) is the proton bunch population in Run 2, \(p\) is gas pressure in bar, and \(N_A\) is Avogadro’s constant.

+
+
m = 1
+b = 1e-28 * m ** 2
+mb = 1e-3 * b
+cm = 1e-2 * m
+Pa = 1
+bar = 1e5 * Pa
+K = 1
+kb = 1.380649e-23  # Boltzmann constant in J/K
+
+
+
# cylinder volume in which a single proton would interact on average
+cross_section = 200 * mb
+detection_region_length = 60 * cm
+volume = cross_section * detection_region_length
+
+# number of collisions given by number of gas molecules per volume: 
+# pV = N k T -> N = p V / k T
+p = 1e-10 * bar
+T = 273 * K
+n_collisions_per_proton = volume * p / (kb * T)
+
+n_protons_per_bunch = 1e11
+n_collisions = n_protons_per_bunch * n_collisions_per_proton
+n_collisions
+
+
0.0031837233037538114
+
+
+
+
#  Helium injection, 24/11/2022, target pressure on the VELO 1.2E-8mbar
+p = 1.2e-8 * 1e-3
+
+ + +
+ +
+ +
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b/py311/lib/python3.11/site-packages/Markdown-3.5.2.dist-info/LICENSE.html new file mode 100644 index 0000000..4e2aa19 --- /dev/null +++ b/py311/lib/python3.11/site-packages/Markdown-3.5.2.dist-info/LICENSE.html @@ -0,0 +1,365 @@ + + + + + + + + + +Hans Dembinski's blog – license + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ + + +

Copyright 2007, 2008 The Python Markdown Project (v. 1.7 and later) Copyright 2004, 2005, 2006 Yuri Takhteyev (v. 0.2-1.6b) Copyright 2004 Manfred Stienstra (the original version)

+

All rights reserved.

+

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

+
    +
  • Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
  • +
  • Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
  • +
  • Neither the name of the Python Markdown Project nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
  • +
+

THIS SOFTWARE IS PROVIDED BY THE PYTHON MARKDOWN PROJECT ‘’AS IS’’ AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL ANY CONTRIBUTORS TO THE PYTHON MARKDOWN PROJECT BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/flax/core/flax_functional_engine.html b/py311/lib/python3.11/site-packages/flax/core/flax_functional_engine.html new file mode 100644 index 0000000..4667744 --- /dev/null +++ b/py311/lib/python3.11/site-packages/flax/core/flax_functional_engine.html @@ -0,0 +1,564 @@ + + + + + + + + + +Hans Dembinski's blog – flax_functional_engine + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ + + +
+
import functools
+import jax
+from jax import numpy as jnp, random, lax
+import numpy as np
+
+
+
from flax import linen as nn, struct
+
+
+
from flax.core import Scope, init, apply, Array, lift, unfreeze
+
+
+
def dense(scope: Scope, inputs: Array, features: int, bias: bool = True,
+          kernel_init=nn.linear.default_kernel_init,
+          bias_init=nn.initializers.zeros_init()):
+  kernel = scope.param('kernel', kernel_init, (inputs.shape[-1], features))
+  y = jnp.dot(inputs, kernel)
+  if bias:
+    y += scope.param('bias', bias_init, (features,))
+  return y
+
+model_fn = functools.partial(dense, features=3)
+
+x = jnp.ones((1, 2))
+y, params = init(model_fn)(random.key(0), x)
+print(params)
+
+def mlp(scope: Scope, inputs: Array, features: int):
+  hidden = scope.child(dense, 'hidden')(inputs, features)
+  hidden = nn.relu(hidden)
+  return dense(scope.push('out'), hidden, 1)
+
+init(mlp)(random.key(0), x, features=3)
+
+
FrozenDict({'params': {'kernel': DeviceArray([[ 0.15374057, -0.6807397 , -1.3350962 ],
+             [ 0.59940743, -0.69430196, -0.7663768 ]], dtype=float32), 'bias': DeviceArray([0., 0., 0.], dtype=float32)}})
+
+
+
(DeviceArray([[0.17045607]], dtype=float32),
+ FrozenDict({'params': {'hidden': {'bias': DeviceArray([0., 0., 0.], dtype=float32), 'kernel': DeviceArray([[-0.22119394,  0.22075175, -0.0925657 ],
+              [ 0.40571952,  0.27750877,  1.0542233 ]], dtype=float32)}, 'out': {'kernel': DeviceArray([[ 0.21448377],
+              [-0.01530595],
+              [ 0.14402702]], dtype=float32), 'bias': DeviceArray([0.], dtype=float32)}}}))
+
+
+
+
@struct.dataclass
+class Embedding:
+  table: np.ndarray
+
+  def lookup(self, indices):
+    return self.table[indices]
+
+  def attend(self, query):
+    return jnp.dot(query, self.table.T)
+
+# all the embedding module does is provide a convenient initializers
+
+def embedding(scope: Scope, num_embeddings: int, features: int, init_fn=nn.linear.default_embed_init) -> Embedding:
+  table = scope.param('table', init_fn, (num_embeddings, features))
+  return Embedding(table)
+
+embedding, _ = init(embedding)(random.key(0), num_embeddings=2, features=3)
+print(embedding.table)
+print(embedding.lookup(1))
+print(embedding.attend(jnp.ones((1, 3,))))
+
+
[[ 0.11575121 -0.51936364 -1.113899  ]
+ [ 0.45569834 -0.5300623  -0.5873911 ]]
+[ 0.45569834 -0.5300623  -0.5873911 ]
+[[-1.5175114 -0.6617551]]
+
+
+
+
def lstm(scope, carry, inputs,
+         gate_fn=nn.activation.sigmoid, activation_fn=nn.activation.tanh,
+         kernel_init=nn.linear.default_kernel_init,
+         recurrent_kernel_init=nn.initializers.orthogonal(),
+         bias_init=nn.initializers.zeros_init()):
+  r"""A long short-term memory (LSTM) cell.
+
+  the mathematical definition of the cell is as follows
+  .. math::
+      \begin{array}{ll}
+      i = \sigma(W_{ii} x + W_{hi} h + b_{hi}) \\
+      f = \sigma(W_{if} x + W_{hf} h + b_{hf}) \\
+      g = \tanh(W_{ig} x + W_{hg} h + b_{hg}) \\
+      o = \sigma(W_{io} x + W_{ho} h + b_{ho}) \\
+      c' = f * c + i * g \\
+      h' = o * \tanh(c') \\
+      \end{array}
+  where x is the input, h is the output of the previous time step, and c is
+  the memory.
+
+  Args:
+    carry: the hidden state of the LSTM cell,
+      initialized using `LSTMCell.initialize_carry`.
+    inputs: an ndarray with the input for the current time step.
+      All dimensions except the final are considered batch dimensions.
+    gate_fn: activation function used for gates (default: sigmoid)
+    activation_fn: activation function used for output and memory update
+      (default: tanh).
+    kernel_init: initializer function for the kernels that transform
+      the input (default: lecun_normal).
+    recurrent_kernel_init: initializer function for the kernels that transform
+      the hidden state (default: orthogonal).
+    bias_init: initializer for the bias parameters (default: zeros_init())
+  Returns:
+    A tuple with the new carry and the output.
+  """
+  c, h = carry
+  hidden_features = h.shape[-1]
+  # input and recurrent layers are summed so only one needs a bias.
+  dense_h = lambda name: scope.child(dense, name)(
+      h, features=hidden_features, bias=True,
+      kernel_init=recurrent_kernel_init, bias_init=bias_init)
+  dense_i = lambda name: scope.child(dense, name)(
+      inputs, features=hidden_features, bias=False,
+      kernel_init=kernel_init)
+  i = gate_fn(dense_i(name='ii') + dense_h(name='hi'))
+  f = gate_fn(dense_i(name='if') + dense_h(name='hf'))
+  g = activation_fn(dense_i(name='ig') + dense_h(name='hg'))
+  o = gate_fn(dense_i(name='io') + dense_h(name='ho'))
+  new_c = f * c + i * g
+  new_h = o * activation_fn(new_c)
+  return (new_c, new_h), new_h
+
+def lstm_init_carry(batch_dims, size, init_fn=jnp.zeros):
+  shape = batch_dims + (size,)
+  return init_fn(shape), init_fn(shape)
+
+x = jnp.ones((1, 2))
+carry = lstm_init_carry((1,), 3)
+y, variables = init(lstm)(random.key(0), carry, x)
+jax.tree_util.tree_map(np.shape, (y, variables))
+
+
((((1, 3), (1, 3)), (1, 3)),
+ FrozenDict({'params': {'hf': {'bias': (3,), 'kernel': (3, 3)}, 'hg': {'bias': (3,), 'kernel': (3, 3)}, 'hi': {'bias': (3,), 'kernel': (3, 3)}, 'ho': {'bias': (3,), 'kernel': (3, 3)}, 'if': {'kernel': (2, 3)}, 'ig': {'kernel': (2, 3)}, 'ii': {'kernel': (2, 3)}, 'io': {'kernel': (2, 3)}}}))
+
+
+
+
def simple_scan(scope: Scope, xs):
+  init_carry = lstm_init_carry(xs.shape[:1], xs.shape[-1])
+#   cell = scope.child(lstm, 'cell')
+#   ys = []
+#   for i in range(xs.shape[1]):
+#       x = xs[:, i]
+#       init_carry, y = cell(init_carry, x)
+#       ys.append(y)
+#   return init_carry, ys
+  lstm_scan = lift.scan(lstm, in_axes=1, out_axes=1, variable_broadcast='params', split_rngs={'params': False})
+  return lstm_scan(scope, init_carry, xs)
+
+key1, key2 = random.split(random.key(0), 2)
+xs = random.uniform(key1, (1, 5, 2))
+
+
+y, init_variables = init(simple_scan)(key2, xs)
+
+print('initialized parameter shapes:\n', jax.tree_util.tree_map(jnp.shape, unfreeze(init_variables)))
+
+
initialized parameter shapes:
+ {'params': {'hf': {'bias': (2,), 'kernel': (2, 2)}, 'hg': {'bias': (2,), 'kernel': (2, 2)}, 'hi': {'bias': (2,), 'kernel': (2, 2)}, 'ho': {'bias': (2,), 'kernel': (2, 2)}, 'if': {'kernel': (2, 2)}, 'ig': {'kernel': (2, 2)}, 'ii': {'kernel': (2, 2)}, 'io': {'kernel': (2, 2)}}}
+
+
+
+
y = apply(simple_scan)(init_variables, xs)[0]
+print('output:\n', y)
+
+
output:
+ (DeviceArray([[-0.35626447,  0.25178757]], dtype=float32), DeviceArray([[-0.17885922,  0.13063088]], dtype=float32))
+
+
+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/idna-3.6.dist-info/LICENSE.html b/py311/lib/python3.11/site-packages/idna-3.6.dist-info/LICENSE.html new file mode 100644 index 0000000..d387e8b --- /dev/null +++ b/py311/lib/python3.11/site-packages/idna-3.6.dist-info/LICENSE.html @@ -0,0 +1,365 @@ + + + + + + + + + +Hans Dembinski's blog – license + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ + + +

BSD 3-Clause License

+

Copyright (c) 2013-2023, Kim Davies and contributors. All rights reserved.

+

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

+
    +
  1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

  2. +
  3. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

  4. +
  5. Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

  6. +
+

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html b/py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html new file mode 100644 index 0000000..a6a38d9 --- /dev/null +++ b/py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html @@ -0,0 +1,709 @@ + + + + + + + + + +Hans Dembinski’s blog - UAT for NbAgg backend. + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ +
+
+

UAT for NbAgg backend.

+
+ + + +
+ + + + +
+ + +
+ +
+
from imp import reload
+
+

The first line simply reloads matplotlib, uses the nbagg backend and then reloads the backend, just to ensure we have the latest modification to the backend code. Note: The underlying JavaScript will not be updated by this process, so a refresh of the browser after clearing the output and saving is necessary to clear everything fully.

+
+
import matplotlib
+reload(matplotlib)
+
+matplotlib.use('nbagg')
+
+import matplotlib.backends.backend_nbagg
+reload(matplotlib.backends.backend_nbagg)
+
+
+

UAT 1 - Simple figure creation using pyplot

+

Should produce a figure window which is interactive with the pan and zoom buttons. (Do not press the close button, but any others may be used).

+
+
import matplotlib.backends.backend_webagg_core
+reload(matplotlib.backends.backend_webagg_core)
+
+import matplotlib.pyplot as plt
+plt.interactive(False)
+
+fig1 = plt.figure()
+plt.plot(range(10))
+
+plt.show()
+
+
+
+

UAT 2 - Creation of another figure, without the need to do plt.figure.

+

As above, a new figure should be created.

+
+
plt.plot([3, 2, 1])
+plt.show()
+
+
+
+

UAT 3 - Connection info

+

The printout should show that there are two figures which have active CommSockets, and no figures pending show.

+
+
print(matplotlib.backends.backend_nbagg.connection_info())
+
+
+
+

UAT 4 - Closing figures

+

Closing a specific figure instance should turn the figure into a plain image - the UI should have been removed. In this case, scroll back to the first figure and assert this is the case.

+
+
plt.close(fig1)
+plt.close('all')
+
+
+
+

UAT 5 - No show without plt.show in non-interactive mode

+

Simply doing a plt.plot should not show a new figure, nor indeed update an existing one (easily verified in UAT 6). The output should simply be a list of Line2D instances.

+
+
plt.plot(range(10))
+
+
+
+

UAT 6 - Connection information

+

We just created a new figure, but didn’t show it. Connection info should no longer have “Figure 1” (as we closed it in UAT 4) and should have figure 2 and 3, with Figure 3 without any connections. There should be 1 figure pending.

+
+
print(matplotlib.backends.backend_nbagg.connection_info())
+
+
+
+

UAT 7 - Show of previously created figure

+

We should be able to show a figure we’ve previously created. The following should produce two figure windows.

+
+
plt.show()
+plt.figure()
+plt.plot(range(5))
+plt.show()
+
+
+
+

UAT 8 - Interactive mode

+

In interactive mode, creating a line should result in a figure being shown.

+
+
plt.interactive(True)
+plt.figure()
+plt.plot([3, 2, 1])
+
+

Subsequent lines should be added to the existing figure, rather than creating a new one.

+
+
plt.plot(range(3))
+
+

Calling connection_info in interactive mode should not show any pending figures.

+
+
print(matplotlib.backends.backend_nbagg.connection_info())
+
+

Disable interactive mode again.

+
+
plt.interactive(False)
+
+
+
+

UAT 9 - Multiple shows

+

Unlike most of the other matplotlib backends, we may want to see a figure multiple times (with or without synchronisation between the views, though the former is not yet implemented). Assert that plt.gcf().canvas.manager.reshow() results in another figure window which is synchronised upon pan & zoom.

+
+
plt.gcf().canvas.manager.reshow()
+
+
+
+

UAT 10 - Saving notebook

+

Saving the notebook (with CTRL+S or File->Save) should result in the saved notebook having static versions of the figures embedded within. The image should be the last update from user interaction and interactive plotting. (check by converting with ipython nbconvert <notebook>)

+
+
+

UAT 11 - Creation of a new figure on second show

+

Create a figure, show it, then create a new axes and show it. The result should be a new figure.

+

BUG: Sometimes this doesn’t work - not sure why (@pelson).

+
+
fig = plt.figure()
+plt.axes()
+plt.show()
+
+plt.plot([1, 2, 3])
+plt.show()
+
+
+
+

UAT 12 - OO interface

+

Should produce a new figure and plot it.

+
+
from matplotlib.backends.backend_nbagg import new_figure_manager,show
+
+manager = new_figure_manager(1000)
+fig = manager.canvas.figure
+ax = fig.add_subplot(1,1,1)
+ax.plot([1,2,3])
+fig.show()
+
+
+
+

UAT 13 - Animation

+

The following should generate an animated line:

+
+
import matplotlib.animation as animation
+import numpy as np
+
+fig, ax = plt.subplots()
+
+x = np.arange(0, 2*np.pi, 0.01)        # x-array
+line, = ax.plot(x, np.sin(x))
+
+def animate(i):
+    line.set_ydata(np.sin(x+i/10.0))  # update the data
+    return line,
+
+#Init only required for blitting to give a clean slate.
+def init():
+    line.set_ydata(np.ma.array(x, mask=True))
+    return line,
+
+ani = animation.FuncAnimation(fig, animate, np.arange(1, 200), init_func=init,
+                              interval=100., blit=True)
+plt.show()
+
+
+

UAT 14 - Keyboard shortcuts in IPython after close of figure

+

After closing the previous figure (with the close button above the figure) the IPython keyboard shortcuts should still function.

+
+
+

UAT 15 - Figure face colours

+

The nbagg honours all colours apart from that of the figure.patch. The two plots below should produce a figure with a red background. There should be no yellow figure.

+
+
import matplotlib
+matplotlib.rcParams.update({'figure.facecolor': 'red',
+                            'savefig.facecolor': 'yellow'})
+plt.figure()
+plt.plot([3, 2, 1])
+
+plt.show()
+
+
+
+

UAT 16 - Events

+

Pressing any keyboard key or mouse button (or scrolling) should cycle the line while the figure has focus. The figure should have focus by default when it is created and re-gain it by clicking on the canvas. Clicking anywhere outside of the figure should release focus, but moving the mouse out of the figure should not release focus.

+
+
import itertools
+fig, ax = plt.subplots()
+x = np.linspace(0,10,10000)
+y = np.sin(x)
+ln, = ax.plot(x,y)
+evt = []
+colors = iter(itertools.cycle(['r', 'g', 'b', 'k', 'c']))
+def on_event(event):
+    if event.name.startswith('key'):
+        fig.suptitle('%s: %s' % (event.name, event.key))
+    elif event.name == 'scroll_event':
+        fig.suptitle('%s: %s' % (event.name, event.step))
+    else:
+        fig.suptitle('%s: %s' % (event.name, event.button))
+    evt.append(event)
+    ln.set_color(next(colors))
+    fig.canvas.draw()
+    fig.canvas.draw_idle()
+
+fig.canvas.mpl_connect('button_press_event', on_event)
+fig.canvas.mpl_connect('button_release_event', on_event)
+fig.canvas.mpl_connect('scroll_event', on_event)
+fig.canvas.mpl_connect('key_press_event', on_event)
+fig.canvas.mpl_connect('key_release_event', on_event)
+
+plt.show()
+
+
+
+

UAT 17 - Timers

+

Single-shot timers follow a completely different code path in the nbagg backend than regular timers (such as those used in the animation example above.) The next set of tests ensures that both “regular” and “single-shot” timers work properly.

+

The following should show a simple clock that updates twice a second:

+
+
import time
+
+fig, ax = plt.subplots()
+text = ax.text(0.5, 0.5, '', ha='center')
+
+def update(text):
+    text.set(text=time.ctime())
+    text.axes.figure.canvas.draw()
+    
+timer = fig.canvas.new_timer(500, [(update, [text], {})])
+timer.start()
+plt.show()
+
+

However, the following should only update once and then stop:

+
+
fig, ax = plt.subplots()
+text = ax.text(0.5, 0.5, '', ha='center') 
+timer = fig.canvas.new_timer(500, [(update, [text], {})])
+
+timer.single_shot = True
+timer.start()
+
+plt.show()
+
+

And the next two examples should never show any visible text at all:

+
+
fig, ax = plt.subplots()
+text = ax.text(0.5, 0.5, '', ha='center')
+timer = fig.canvas.new_timer(500, [(update, [text], {})])
+
+timer.start()
+timer.stop()
+
+plt.show()
+
+
+
fig, ax = plt.subplots()
+text = ax.text(0.5, 0.5, '', ha='center')
+timer = fig.canvas.new_timer(500, [(update, [text], {})])
+
+timer.single_shot = True
+timer.start()
+timer.stop()
+
+plt.show()
+
+
+
+

UAT 18 - stopping figure when removed from DOM

+

When the div that contains from the figure is removed from the DOM the figure should shut down it’s comm, and if the python-side figure has no more active comms, it should destroy the figure. Repeatedly running the cell below should always have the same figure number

+
+
fig, ax = plt.subplots()
+ax.plot(range(5))
+plt.show()
+
+

Running the cell below will re-show the figure. After this, re-running the cell above should result in a new figure number.

+
+
fig.canvas.manager.reshow()
+
+
+
+

UAT 19 - Blitting

+

Clicking on the figure should plot a green horizontal line moving up the axes.

+
+
import itertools
+
+cnt = itertools.count()
+bg = None
+
+def onclick_handle(event):
+    """Should draw elevating green line on each mouse click"""
+    global bg
+    if bg is None:
+        bg = ax.figure.canvas.copy_from_bbox(ax.bbox) 
+    ax.figure.canvas.restore_region(bg)
+
+    cur_y = (next(cnt) % 10) * 0.1
+    ln.set_ydata([cur_y, cur_y])
+    ax.draw_artist(ln)
+    ax.figure.canvas.blit(ax.bbox)
+
+fig, ax = plt.subplots()
+ax.plot([0, 1], [0, 1], 'r')
+ln, = ax.plot([0, 1], [0, 0], 'g', animated=True)
+plt.show()
+ax.figure.canvas.draw()
+
+ax.figure.canvas.mpl_connect('button_press_event', onclick_handle)
+
+ + +
+
+ +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/orbax/checkpoint/orbax_checkpoint.html b/py311/lib/python3.11/site-packages/orbax/checkpoint/orbax_checkpoint.html new file mode 100644 index 0000000..1c35be5 --- /dev/null +++ b/py311/lib/python3.11/site-packages/orbax/checkpoint/orbax_checkpoint.html @@ -0,0 +1,1072 @@ + + + + + + + + + +Hans Dembinski’s blog - Checkpointing with Orbax + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ +
+
+

Checkpointing with Orbax

+
+ + + +
+ + + + +
+ + +
+ +

Open In Colab

+

Take a look at https://orbax.readthedocs.io/en/latest/ for additional documentation on Orbax checkpointing APIs.

+
+
!pip install orbax-checkpoint
+
+
+
import time
+from collections import namedtuple
+import jax
+from jax.sharding import Mesh, PartitionSpec
+from jax.experimental.pjit import pjit
+import numpy as np
+import os
+import portpicker
+from etils import epath
+import orbax.checkpoint
+
+
+
port = portpicker.pick_unused_port()
+jax.distributed.initialize(f'localhost:{port}', num_processes=1, process_id=0)
+
+
+
devices = np.asarray(jax.devices())
+mesh = Mesh(devices, ('data',))
+axes = PartitionSpec('data',)
+sharding = jax.sharding.NamedSharding(mesh, axes)
+
+
+
directory = epath.Path('checkpoint_data')
+if directory.exists():
+  directory.rmtree()
+directory.mkdir(parents=True)
+
+
+
print(directory)
+
+
+

A Basic Example

+
+
basic_dir = directory / 'basic'
+if not basic_dir.exists():
+  basic_dir.mkdir()
+
+
+
options = orbax.checkpoint.CheckpointManagerOptions(save_interval_steps=1, max_to_keep=3)
+mngr = orbax.checkpoint.CheckpointManager(
+    basic_dir, orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler()), options)
+
+

The CheckpointManager is constructed with Checkpointer and CheckpointHandler objects. We will discuss these further below, but at a high level, the Checkpointer controls the manner in which the object is saved while the CheckpointHandler deals with type-specific logic and provides extra options for customization.

+

First, we’ll need to perform some setup to create a train state that mimics in a very basic form how a real model might look. We use jax.Array for this is example, but it is also possible to use scalars or numpy arrays, assuming they are replicated or not sharded.

+
+
def create_initial_state():
+  state = {
+      'layer_0': {
+          'bias': np.zeros(16),
+          'kernel': np.arange(16),
+      },
+      'layer_1': {
+          'bias': np.zeros(8),
+          'kernel': np.arange(8),
+      },
+  }
+
+  create_sharded_array = pjit(lambda x: x, in_shardings=None, out_shardings=sharding)
+  with Mesh(mesh.devices, mesh.axis_names):
+    state = jax.tree_map(create_sharded_array, state)
+
+  state['step'] = 0
+  return state
+
+def state_shape(state):
+  return jax.eval_shape(lambda: state)
+
+

Here’s our mock training step. At every step, we save a checkpoint. Since we specified max_to_keep=3 in our options, we expect to only have the latest 3 checkpoints at the end of training.

+
+
state = create_initial_state()
+
+def train(step, state):
+  # do some training, modify state
+  state['step'] = step
+
+  mngr.save(step, state)
+
+  return state
+
+for step in range(5):
+  state = train(step, state)
+
+print(f'Steps: {mngr.all_steps()}')
+
+

After saving, we can restore the latest step. Since this CheckpointManager is only managing a single item, the arguments provided to restore should just match the state (see the below cell). See ‘Multi-Object Checkpointing’ for further details on how these arguments are used when there are multiple items.

+
+
dummy_state = state_shape(state)
+restore_args = jax.tree_util.tree_map(
+    lambda _:
+        orbax.checkpoint.ArrayRestoreArgs(
+            restore_type=jax.Array,
+            mesh=mesh, 
+            mesh_axes=axes), 
+    state)
+restore_args['step'] = orbax.checkpoint.RestoreArgs(restore_type=int)
+mngr.restore(mngr.latest_step(), items=dummy_state, 
+             restore_kwargs={'restore_args': restore_args})
+
+

We can achieve the same result by just using Checkpointer.

+
+
ckptr = orbax.checkpoint.PyTreeCheckpointer()
+
+def _maybe_get_sharding(arr):
+  if isinstance(arr, jax.Array):
+    return arr.sharding
+  else:
+    return None
+
+shardings = jax.tree_util.tree_map(_maybe_get_sharding, state)
+# restore_args can also be constructed "automatically" from a target PyTree.
+restore_args = orbax.checkpoint.checkpoint_utils.construct_restore_args(
+                    state, shardings)
+print(restore_args)
+print()
+# CheckpointManager saved the checkpoint under /<directory>/<step>/default.
+# 'default' is used as the subdirectory name when the CheckpointManager has a
+# single item. See below for information on how to use multiple items or how
+# to customize this name.
+ckpt_path = mngr.directory / str(mngr.latest_step()) / 'default'
+ckptr.restore(ckpt_path, item=dummy_state, restore_args=restore_args)
+
+
+
+

Tracking Metrics

+

When saving checkpoints across many steps, we are often interested in keeping only the best n checkpoints based on some metric.

+
+
ckpt_metrics_dir = directory / 'ckpt_with_metrics'
+if not ckpt_metrics_dir.exists():
+  ckpt_metrics_dir.mkdir()
+
+
+
options = orbax.checkpoint.CheckpointManagerOptions(
+    max_to_keep=3, best_fn=lambda metrics: metrics['loss'], best_mode='min')
+mngr = orbax.checkpoint.CheckpointManager(
+    ckpt_metrics_dir, orbax.checkpoint.AsyncCheckpointer(orbax.checkpoint.PyTreeCheckpointHandler()),
+    options)
+
+
+
def get_metrics(step):
+  return {'accuracy': 1.0, 'loss': step * 1.5}
+
+
+def train(step, state):
+  # do some training, modify state
+  metrics = get_metrics(step)
+  state['step'] = step
+  state_save_args = jax.tree_map(lambda _: orbax.checkpoint.SaveArgs(), state)
+  mngr.save(step, state, metrics=metrics)
+  return state
+
+
+state = create_initial_state()
+for step in range(5):
+  state = train(step, state)
+
+mngr.wait_until_finished()
+print(f'Steps: {mngr.all_steps()}')
+
+

Now that we track metrics, we will only keep the best checkpoints saved, while deleting the rest. Since our loss is getting progressively worse, and best_mode=‘min’, we will keep the first checkpoints, rather than the most recent ones. The metrics may be an arbitrary PyTree; it is up to you to define how it is interpreted.

+
+
+

Multi-Object Checkpointing

+

In the following example, we will show how to checkpoint multiple different objects at once using CheckpointManager.

+
+
multi_dir = directory / 'multi'
+if not multi_dir.exists():
+  multi_dir.mkdir()
+
+
+
# Save every 3 steps.
+options = orbax.checkpoint.CheckpointManagerOptions(save_interval_steps=3)
+mngr = orbax.checkpoint.CheckpointManager(
+    multi_dir, {
+        'state': orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler()),
+        'metadata': orbax.checkpoint.Checkpointer(orbax.checkpoint.JsonCheckpointHandler())
+    }, options)
+
+

We can save multiple objects simply by specifying a Checkpointer/CheckpointHandler combination for each of them. While Checkpointers can typically be used with any CheckpointHandler, you’ll need to ensure that your object can be saved and restored by the given CheckpointHandler.

+
+
NUM_STEPS = 5
+
+def train(step, state):
+  # do some training, modify state
+  state['step'] = step
+  metadata['timestamp'] = time.time()
+
+  # save with default arguments for all params except 'step', which is
+  # aggregated and saved using msgpack.
+  state_save_args = jax.tree_map(lambda _: orbax.checkpoint.SaveArgs(), state)
+  state_save_args['step'] = orbax.checkpoint.SaveArgs(aggregate=True)
+
+  # with `force` a save will be performed even if it would not ordinarily do so,
+  # based on the step number.
+  force = False
+  if step == NUM_STEPS - 1:
+    force = True
+  save_performed = mngr.save(
+      step,
+      items={
+          'state': state,
+          'metadata': metadata
+      },
+      # save_kwargs must be a dict with the same keys as items.
+      # not all keys in items have to be provided, in which case default kwargs
+      # are used each value must be a dict with keyword args passed to the
+      # underlying CheckpointHandler for that item (see CheckpointManager
+      # object construction)
+      save_kwargs={'state': {
+          'save_args': state_save_args
+      }}, 
+      force=force)
+  print(f'Save performed: {save_performed}')
+
+  return state
+
+
+state = create_initial_state()
+metadata = {
+    'version': 1.1,
+    'exp_name': 'my_test_exp',
+    'timestamp': 0,
+}
+for step in range(NUM_STEPS):
+  state = train(step, state)
+
+print(f'Steps: {mngr.all_steps()}')
+
+

We also have extra arguments to customize saving for the step parameter within the train state. Because this is only an integer, using the default storage mechanism, Tensorstore might be somewhat overkill. It would be more efficient to store it, along with any other similarly small parameters, into a single file using msgpack). The parameters passed here should match the optional arguments for the provided CheckpointHandler.

+

Let’s try restoring the latest step. In the items argument, make sure to provide a key for each of the items you want to restore. The values of this items dictionary are what gets provided to CheckpointHandler.restore as the argument of item. As a result, it may be None if the CheckpointHandler does not depend on the value of item.

+

The restore_kwargs argument should be a dictionary with matching top-level keys, but keys can be omitted if no arguments are needed. The values of the restore_kwargs dictionary are provided as keyword args to the matching CheckpointHandler.restore. For example, if a CheckpointHandler subclass called FooBarCheckpointHandler takes kwargs foo and bar (in addition to standard args like path and item, the restore_kwargs for CheckpointManager would be:

+
restore_kwargs = {'foobar_item': {'foo': ???, 'bar': ???}}
+
+
mngr.restore(mngr.latest_step(), 
+             # Safe to provide None for `items` values because they are not
+             # really needed in this case. `restore_args` is needed though.
+             items={'state': None, 'metadata': None}, 
+             restore_kwargs={'state': {'restore_args': restore_args}})
+
+

If we skip providing ‘metadata’ in items, it will not be returned in the result.

+
+
mngr.restore(mngr.latest_step(), 
+             items={'state': None}, 
+             restore_kwargs={'state': {'restore_args': restore_args}})
+
+
+
+

Asynchronous Checkpointing

+

You may be wondering what the point of Checkpointer is, and why it is separate from CheckpointHandler. The reason for this is that Checkpointer may have different subclasses, each of which handles certain common logic that we would not want to reimplement for every different CheckpointHandler.

+

This logic may include ensuring save operation atomicity and managing a background thread for asynchronous saving.

+
+
async_dir = directory / 'async'
+if not async_dir.exists():
+  async_dir.mkdir()
+
+
+
mngr = orbax.checkpoint.CheckpointManager(
+    async_dir, {
+        'state': orbax.checkpoint.AsyncCheckpointer(orbax.checkpoint.PyTreeCheckpointHandler()),
+        'metadata': orbax.checkpoint.Checkpointer(orbax.checkpoint.JsonCheckpointHandler())
+    })
+
+

With the above configuration, our metadata will be saved synchronously, but our state will be saved in a background thread. After calling save, all files for the state may not have been written yet. In the meantime, we may continue training.

+

However, we need to call wait_until_finished before ending our training program to block for any outstanding save operations. Calling save again will do this automatically - you cannot have multiple saves for multiple steps running concurrently.

+
+
def train(step, state):
+  # do some training, modify state
+  state['step'] = step
+  metadata['timestamp'] = time.time()
+
+  mngr.save(
+      step,
+      items={
+          'state': state,
+          'metadata': metadata
+      })
+
+  return state
+
+
+state = create_initial_state()
+metadata = {
+    'version': 1.1,
+    'exp_name': 'my_test_exp',
+    'timestamp': 0,
+}
+for step in range(NUM_STEPS):
+  state = train(step, state)
+
+mngr.wait_until_finished()
+print(f'Steps: {mngr.all_steps()}')
+
+
+
+

Checkpointer

+

Checkpointer allows you to save an object to a specified directory without providing any of the structure or extra features that CheckpointManager does.

+
+
state = create_initial_state()
+ckptr = orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler())
+
+existing_checkpoint_dir = multi_dir / '0' / 'state'
+restore_args = jax.tree_map(lambda _: orbax.checkpoint.ArrayRestoreArgs(mesh=mesh, mesh_axes=axes), state)
+restore_args['step'] = orbax.checkpoint.RestoreArgs()
+restored = ckptr.restore(existing_checkpoint_dir, restore_args=restore_args)
+
+for k, v in orbax.checkpoint.utils.to_flat_dict(restored, sep='/').items():
+  print(f'{k}: {v}')
+
+

As shown in this example, this object may be useful for restoring a pre-existing checkpoint without requiring a CheckpointManager.

+
+
+

CheckpointHandler

+

Important: CheckpointHandler should not be used independently of Checkpointer or CheckpointManager.

+

As shown above, PyTreeCheckpointHandler provides support for most standard use-cases, where a PyTree consisting of jax.Array, scalars, or numpy arrays can be saved using a combination of Tensorstore and msgpack.

+

However, we may also wish to provide custom support for a novel type or storage medium. Below is an example of how CheckpointHandler can be overridden to support a sharded file concept.

+
+
from typing import Any, List, Mapping, Optional
+import asyncio
+from concurrent import futures
+import datetime
+
+
+
executor = futures.ThreadPoolExecutor(max_workers=2)
+def ctime():
+  return datetime.datetime.now(
+      tz=datetime.timezone.utc
+  ).isoformat(sep=' ', timespec='milliseconds')
+
+
+
class ShardedFile:
+
+  def __init__(self):
+    self.devices = [d for d in jax.devices() if d.host_id == jax.process_index()]
+
+  def get_device_shard(self, device):
+    return f'Data associated with device: {device}... '
+
+  def get(self):
+    data = f'Process {jax.process_index()}: '
+    for d in self.devices:
+      data += self.get_device_shard(d)
+    return data
+
+class ShardedFileWriter:
+
+  async def copy(self):
+    await asyncio.sleep(1)
+    self.local_file = self.sharded_file.get()
+    print(f'{ctime()}: done copy')
+
+  def commit(self):
+    path = epath.Path(self.path) / str(jax.process_index())
+
+    def _write():
+      time.sleep(5)
+      path.write_text(self.local_file)
+      print(f'{ctime()}: done commit')
+
+    future = executor.submit(_write)
+    print(f'{ctime()}: started commit')
+    return future
+    
+  def __init__(self, path: str, sharded_file: ShardedFile):
+    self.path = path
+    self.sharded_file = sharded_file
+
+
+
class ShardedFileCheckpointHandler(orbax.checkpoint.async_checkpoint_handler.AsyncCheckpointHandler):
+
+  async def async_save(self, directory: epath.Path, item: ShardedFile) -> List[futures.Future]:
+    """Saves the given item.
+
+    Args:
+      directory: save location directory.
+      item: nested dictionary.
+    """
+    fw = ShardedFileWriter(os.fspath(directory), item)
+    await fw.copy()
+    return [fw.commit()]
+
+  def save(self, directory: epath.Path, item: Any, *args, **kwargs):
+    async def async_save(*args, **kwargs):
+      commit_futures = await self.async_save(*args, **kwargs)
+      # Futures are already running, so sequential waiting is equivalent to
+      # concurrent waiting.
+      for future in commit_futures:
+        future.result()  # Block on result.
+    asyncio.run(async_save(directory, item, *args, **kwargs))
+    orbax.checkpoint.utils.sync_global_devices('ShardedFileCheckpointHandler:save')
+
+  def restore(self,
+              directory: epath.Path,
+              item: Optional[bytes] = None) -> str:
+    del item
+    path = directory / str(jax.process_index())
+    return path.read_text()
+
+  def structure(self, directory: epath.Path) -> int:
+    return len(list(directory.iterdir()))
+
+
+
handler_dir = directory / 'sync_handler'
+async_handler_dir = directory / 'async_handler'
+
+file = ShardedFile()
+
+
+
checkpointer = orbax.checkpoint.Checkpointer(ShardedFileCheckpointHandler())
+checkpointer.save(handler_dir, file)
+print(f'{ctime()}: done save')
+
+
+
checkpointer = orbax.checkpoint.AsyncCheckpointer(ShardedFileCheckpointHandler())
+checkpointer.save(async_handler_dir, file)
+print(f'{ctime()}: processing save')
+# do something else
+checkpointer.wait_until_finished()
+print(f'{ctime()}: done save')
+
+
+
executor.shutdown()
+
+
+
+

TypeHandler

+

TypeHandler as a concept exists in conjunction with PyTreeCheckpointHandler to provide additional customization options for advanced users with custom types they wish to save as part of a PyTree.

+

By default, Orbax includes handler implementations for jax.Array, np.ndarray, scalars, strings, and others. These are simply the leaf types supported by PyTreeCheckpointHandler.

+

By implementing a subclass of TypeHandler and registering a type using register_type_handler, we can add support for a novel type.

+
+
class Foo():
+  # Realistically we would use a dataclass for this, but this is just for
+  # illustration purposes.
+  def __init__(self, a, b, c):
+    self.a = a
+    self.b = b
+    self.c = c
+
+  def __str__(self):
+    return f'{self.a};{self.b};{self.c}'
+
+
+
from orbax.checkpoint.value_metadata import Metadata
+from orbax.checkpoint.type_handlers import ParamInfo
+from typing import Any, Tuple
+
+class FooHandler(orbax.checkpoint.type_handlers.TypeHandler):
+
+  def typestr(self) -> str:
+    return 'foo'
+
+  def metadata(self, value: Foo) -> Metadata:
+    return Metadata()
+
+  async def open(self, info: ParamInfo) -> Tuple[Metadata, Any]:
+    return Metadata(), None
+
+  async def serialize(
+      self,
+      value: Foo,
+      info: ParamInfo,
+      args: Optional[orbax.checkpoint.SaveArgs] = None) -> List[orbax.checkpoint.future.Future]:
+    # A more sophisticated implementation would make this write asynchronous.
+    (info.path / 'data.txt').write_text(str(value))
+    return []
+
+  async def deserialize(
+      self,
+      file: Any,
+      info: ParamInfo,
+      args: Optional[orbax.checkpoint.RestoreArgs] = None) -> Foo:
+    del file
+    entries = (info.path / 'data.txt').read_text().split(';')
+    assert len(entries) == 3
+    return Foo(*entries)
+
+
+
type_handler_dir = directory / 'type_handler'
+orbax.checkpoint.type_handlers.register_type_handler(Foo, FooHandler(), override=True)
+
+
+
foo_tree = {
+    'one_foo': Foo(2, 4, 6),
+    'two_foo': Foo(1, 2, 3),
+}
+ckptr = orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler())
+ckptr.save(type_handler_dir, foo_tree)
+
+
+
restore_args = jax.tree_util.tree_map(lambda _: orbax.checkpoint.RestoreArgs(restore_type=Foo), foo_tree)
+restored = ckptr.restore(type_handler_dir, restore_args=restore_args)
+jax.tree_util.tree_map(str, restored)
+
+
+
+

Transformations

+

A key component of the Orbax checkpointing library is PyTree transformations. While this functionality is designed to be as flexible as possible, it can be used to support:

+
    +
  • Partial restoration of checkpoints where some keys can be dropped and replaced with randomly initialized values.
  • +
  • Checkpoint version compatibility where newer checkpoints may have different structures than old ones.
  • +
  • Mappings over keys, including one-to-one, many-to-one, one-to-many, and many-to-many transformations.
  • +
+

The transformations library is discussed in detail here, so we will avoid discussing all possible features and will instead focus on concrete examples.

+

Let’s start with a simple example first.

+
+
from orbax.checkpoint.transform_utils import Transform
+
+
+
original_tree = {
+  'a': 1,
+  'b': {
+    'c': 5,
+    'd': [0, 1, 2, 3]
+  },
+  'f': 2,
+}
+transformations = {
+  'a1': Transform(original_key='a'),  # rename
+  'b': {
+    # doubled original
+    'c': Transform(value_fn=lambda v: v * 2)
+    # drop b/d
+  },
+  # one to many mapping
+  'x': Transform(multi_value_fn=lambda _, kv: kv['b']['d'][0]),
+  'y': Transform(multi_value_fn=lambda _, kv: kv['b']['d'][1:]),
+  # many to one mapping
+  'z': Transform(multi_value_fn=lambda _, kv: kv['a'] * 2 + sum(kv['b']['d'])),
+}
+new_tree = {  # defines the structure of the result
+  'a1': ...,
+  'b': {
+    'c': ...,
+  },
+  'x': ...,
+  'y': ...,
+  'z': ...,
+  # 'f' defined in original_tree and new_tree, but not in transforms. Value
+  # carried over from original_tree.
+  'f': ...,
+  # This value matters since it is not present in original_tree or
+  # transformations, so the value here will simply be preserved in the result.
+  'g': 5,
+}
+
+orbax.checkpoint.apply_transformations(original_tree, transformations, new_tree)
+
+

An important rule of thumb to remember is that the output of apply_transformations will always match the structure of new_tree. This provides an easy way to know exactly what your result will look like after applying transformations.

+

Often, users have to deal with very large PyTrees and it would become very burdensome to specify transformations for large numbers of keys. Our library provides two solutions to this: regexes and implicit transformations.

+

Implicit transformations have been alluded to in our first example, but let’s focus on them specifically.

+

In the following example, transformations is an empty dictionary, so we rely exclusively on implicit transformations. Key/value pairs present in new_tree but not in original_tree simply remain in place in the result, while a key present in original_tree but not in new_tree will be dropped from the result.

+
+
original_tree = {
+  'a': 1,
+  'b': {
+    'c': 5,
+    'd': 6,
+    'e': 7,
+  },
+  'f': 2,
+}
+transformations = {}
+new_tree = {
+  'a': ...,
+  'b': {
+    'c': ...,
+  },
+  'f': ...,
+  'g': {
+      'h': 3,
+      'i': 4,
+  },
+}
+
+orbax.checkpoint.apply_transformations(original_tree, transformations, new_tree)
+
+

We can also change the default_to_original argument to customize the behavior when keys are unspecified in the transformations tree. Setting default_to_original=False means that unspecified keys will be taken from new_tree, not original_tree.

+

This can be useful if we just want to take a few values from our original checkpoint, while using the rest from our new state.

+
+
original_tree = {
+  'a': 1,
+  'b': {
+    'c': 5,
+    'd': 6,
+    'e': 7,
+  },
+  'f': 2,
+}
+transformations = {'a': Transform(value_fn=lambda x: x*10)}
+new_tree = {
+  'a': 11,
+  'b': {
+    'c': 12,
+    'd': 13,
+    'e': 14,
+  },
+  'f': 15,
+}
+orbax.checkpoint.apply_transformations(original_tree, transformations, new_tree, default_to_original=False)
+
+

Returning to our other feature: support for regexes. Real model states often represent a conceptual parameter (a single layer, perhaps) with multiple actual key/value pairs. In this case, it can be useful to use a regex to refer to the parameter in question.

+

In the following example, we have one model (conceputally a pretrained checkpoint) with two layers, and another model (conceputally our in-memory model state) with four layers. We would like to insert the two layers of the checkpoint as the middle two layers of the new state, while leaving the bottom and top layers randomly initialized.

+
+
import flax.linen as nn
+from orbax.checkpoint import test_utils
+
+
+
from flax.training.train_state import TrainState
+import optax
+from jax import numpy as jnp
+
+def init_flax_model(model):
+  params = model.init(jax.random.PRNGKey(0), jnp.ones([8, 8]))
+  tx = optax.adamw(learning_rate=0.001)
+  state = TrainState.create(apply_fn=model.apply, params=params, tx=tx)
+  return jax.tree_util.tree_map(np.asarray, state)
+
+class SmallModel(nn.Module):
+  @nn.compact
+  def __call__(self, x):
+    x = x.reshape((x.shape[0], -1))
+    x = nn.Dense(features=8)(x)
+    x = nn.sigmoid(x)
+    x = nn.Dense(features=8)(x)
+    return x
+
+old_state = init_flax_model(SmallModel())
+# multiply by 100 to represent "training"
+old_state = jax.tree_util.tree_map(lambda x: x * 100, old_state)
+
+class LargeModel(nn.Module):
+  @nn.compact
+  def __call__(self, x):
+    x = x.reshape((x.shape[0], -1))  # flatten
+    x = nn.Dense(features=8)(x)
+    x = nn.sigmoid(x)
+    x = nn.Dense(features=4)(x)
+    x = nn.sigmoid(x)
+    x = nn.Dense(features=4)(x)
+    x = nn.sigmoid(x)
+    x = nn.Dense(features=2)(x)
+    return x
+
+new_state = init_flax_model(LargeModel())
+
+
+
transformations = {
+    # NewModel layer_0 is a newly inserted layer, thus use_fallback=True.
+    # The "fallback" tree in this case is new_tree.
+    # Since the layer_0 has the same name in old and new, we need to provide
+    # an indication that the value of layer_0 should come from new_tree rather
+    # than original_tree.
+    r'(.*)Dense_0(.*)': Transform(use_fallback=True),
+    # OriginalModel layer 0 maps to NewModel layer 1
+    r'(.*)Dense_1(.*)': Transform(original_key=r'\1Dense_0\2'),
+    # OriginalModel layer 1 maps to NewModel layer 2
+    r'(.*)Dense_2(.*)': Transform(original_key=r'\1Dense_1\2')
+}  # Note: NewModel layer 3 is newly added.
+restored_state = orbax.checkpoint.apply_transformations(old_state, transformations, new_state)
+print(restored_state.params['params'])
+
+

We can see in the result that layers 0 and 3 are still small, while layers 1 and 2 have large values, coming from the original checkpoint.

+ + +
+ +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html b/py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html new file mode 100644 index 0000000..c55b0d1 --- /dev/null +++ b/py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html @@ -0,0 +1,465 @@ + + + + + + + + + +Hans Dembinski's blog – authors + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ + + +
+

Authors

+

This project was started and continues to be led by Brian E. Granger (ellisonbg AT gmail DOT com). Min Ragan-Kelley (benjaminrk AT gmail DOT com) is the primary developer of pyzmq at this time.

+

The following people have contributed to the project:

+
    +
  • Alexander Else (alexander DOT else AT team DOT telstra DOT com)
  • +
  • Alexander Pyhalov (apyhalov AT gmail DOT com)
  • +
  • Alexandr Emelin (frvzmb AT gmail DOT com)
  • +
  • Amr Ali (amr AT ledgerx DOT com)
  • +
  • Andre Caron (andre DOT l DOT caron AT gmail DOT com)
  • +
  • Andrea Crotti (andrea DOT crotti DOT 0 AT gmail DOT com)
  • +
  • Andrew Gwozdziewycz (git AT apgwoz DOT com)
  • +
  • Baptiste Lepilleur (baptiste DOT lepilleur AT gmail DOT com)
  • +
  • Brandyn A. White (bwhite AT dappervision DOT com)
  • +
  • Brian E. Granger (ellisonbg AT gmail DOT com)
  • +
  • Brian Hoffman (hoffman_brian AT bah DOT com)
  • +
  • Carlos A. Rocha (carlos DOT rocha AT gmail DOT com)
  • +
  • Chris Laws (clawsicus AT gmail DOT com)
  • +
  • Christian Wyglendowski (christian AT bu DOT mp)
  • +
  • Christoph Gohlke (cgohlke AT uci DOT edu)
  • +
  • Curtis (curtis AT tinbrain DOT net)
  • +
  • Cyril Holweck (cyril DOT holweck AT free DOT fr)
  • +
  • Dan Colish (dcolish AT gmail DOT com)
  • +
  • Daniel Lundin (dln AT eintr DOT org)
  • +
  • Daniel Truemper (truemped AT googlemail DOT com)
  • +
  • Douglas Creager (douglas DOT creager AT redjack DOT com)
  • +
  • Eduardo Stalinho (eduardooc DOT 86 AT gmail DOT com)
  • +
  • Eren Güven (erenguven0 AT gmail DOT com)
  • +
  • Erick Tryzelaar (erick DOT tryzelaar AT gmail DOT com)
  • +
  • Erik Tollerud (erik DOT tollerud AT gmail DOT com)
  • +
  • FELD Boris (lothiraldan AT gmail DOT com)
  • +
  • Fantix King (fantix DOT king AT gmail DOT com)
  • +
  • Felipe Cruz (felipecruz AT loogica DOT net)
  • +
  • Fernando Perez (Fernando DOT Perez AT berkeley DOT edu)
  • +
  • Frank Wiles (frank AT revsys DOT com)
  • +
  • Félix-Antoine Fortin (felix DOT antoine DOT fortin AT gmail DOT com)
  • +
  • Gavrie Philipson (gavriep AT il DOT ibm DOT com)
  • +
  • Godefroid Chapelle (gotcha AT bubblenet DOT be)
  • +
  • Greg Banks (gbanks AT mybasis DOT com)
  • +
  • Greg Ward (greg AT gerg DOT ca)
  • +
  • Guido Goldstein (github AT a-nugget DOT de)
  • +
  • Ian Lee (IanLee1521 AT gmail DOT com)
  • +
  • Ionuț Arțăriși (ionut AT artarisi DOT eu)
  • +
  • Ivo Danihelka (ivo AT danihelka DOT net)
  • +
  • Iyed (iyed DOT bennour AT gmail DOT com)
  • +
  • Jim Garrison (jim AT garrison DOT cc)
  • +
  • John Gallagher (johnkgallagher AT gmail DOT com)
  • +
  • Julian Taylor (jtaylor DOT debian AT googlemail DOT com)
  • +
  • Justin Bronder (jsbronder AT gmail DOT com)
  • +
  • Justin Riley (justin DOT t DOT riley AT gmail DOT com)
  • +
  • Marc Abramowitz (marc AT marc-abramowitz DOT com)
  • +
  • Matthew Aburn (mattja6 AT gmail DOT com)
  • +
  • Michel Pelletier (pelletier DOT michel AT gmail DOT com)
  • +
  • Michel Zou (xantares09 AT hotmail DOT com)
  • +
  • Min Ragan-Kelley (benjaminrk AT gmail DOT com)
  • +
  • Nell Hardcastle (nell AT dev-nell DOT com)
  • +
  • Nicholas Pilkington (nicholas DOT pilkington AT gmail DOT com)
  • +
  • Nicholas Piël (nicholas AT nichol DOT as)
  • +
  • Nick Pellegrino (npellegrino AT mozilla DOT com)
  • +
  • Nicolas Delaby (nicolas DOT delaby AT ezeep DOT com)
  • +
  • Ondrej Certik (ondrej AT certik DOT cz)
  • +
  • Paul Colomiets (paul AT colomiets DOT name)
  • +
  • Pawel Jasinski (pawel DOT jasinski AT gmail DOT com)
  • +
  • Phus Lu (phus DOT lu AT gmail DOT com)
  • +
  • Robert Buchholz (rbu AT goodpoint DOT de)
  • +
  • Robert Jordens (jordens AT gmail DOT com)
  • +
  • Ryan Cox (ryan DOT a DOT cox AT gmail DOT com)
  • +
  • Ryan Kelly (ryan AT rfk DOT id DOT au)
  • +
  • Scott Maxwell (scott AT codecobblers DOT com)
  • +
  • Scott Sadler (github AT mashi DOT org)
  • +
  • Simon Knight (simon DOT knight AT gmail DOT com)
  • +
  • Stefan Friesel (sf AT cloudcontrol DOT de)
  • +
  • Stefan van der Walt (stefan AT sun DOT ac DOT za)
  • +
  • Stephen Diehl (stephen DOT m DOT diehl AT gmail DOT com)
  • +
  • Sylvain Corlay (scorlay AT bloomberg DOT net)
  • +
  • Thomas Kluyver (takowl AT gmail DOT com)
  • +
  • Thomas Spura (tomspur AT fedoraproject DOT org)
  • +
  • Tigger Bear (Tigger AT Tiggers-Mac-mini DOT local)
  • +
  • Torsten Landschoff (torsten DOT landschoff AT dynamore DOT de)
  • +
  • Vadim Markovtsev (v DOT markovtsev AT samsung DOT com)
  • +
  • Yannick Hold (yannickhold AT gmail DOT com)
  • +
  • Zbigniew Jędrzejewski-Szmek (zbyszek AT in DOT waw DOT pl)
  • +
  • hugo shi (hugoshi AT bleb2 DOT (none))
  • +
  • jdgleeson (jdgleeson AT mac DOT com)
  • +
  • kyledj (kyle AT bucebuce DOT com)
  • +
  • spez (steve AT hipmunk DOT com)
  • +
  • stu (stuart DOT axon AT jpcreative DOT co DOT uk)
  • +
  • xantares (xantares AT fujitsu-l64 DOT (none))
  • +
+

as reported by:

+
git log --all --format='- %aN (%aE)' | sort -u | sed 's/@/ AT /1' | sed -e 's/\.\([^ ]\)/ DOT \1/g'
+

with some adjustments.

+
+

Not in git log

+
    +
  • Brandon Craig-Rhodes (brandon AT rhodesmill DOT org)
  • +
  • Eugene Chernyshov (chernyshov DOT eugene AT gmail DOT com)
  • +
  • Craig Austin (craig DOT austin AT gmail DOT com)
  • +
+
+
+

gevent_zeromq, now zmq.green

+
    +
  • Travis Cline (travis DOT cline AT gmail DOT com)
  • +
  • Ryan Kelly (ryan AT rfk DOT id DOT au)
  • +
  • Zachary Voase (z AT zacharyvoase DOT com)
  • +
+ + +
+
+ +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/toolz-0.12.0.dist-info/AUTHORS.html b/py311/lib/python3.11/site-packages/toolz-0.12.0.dist-info/AUTHORS.html new file mode 100644 index 0000000..975f82f --- /dev/null +++ b/py311/lib/python3.11/site-packages/toolz-0.12.0.dist-info/AUTHORS.html @@ -0,0 +1,373 @@ + + + + + + + + + +Hans Dembinski's blog – authors + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ + + +

Matthew Rocklin @mrocklin

+

John Jacobsen @eigenhombre

+

Erik Welch @eriknw

+

John Crichton @jcrichton

+

Han Semaj @microamp

+

Graeme Coupar @obmarg

+

Leonid Shvechikov @shvechikov

+

Lars Buitinck @larsmans

+

José Ricardo @josericardo

+

Tom Prince @tomprince

+

Bart van Merriënboer @bartvm

+

Nikolaos-Digenis Karagiannis @digenis

+

Antonio Lima @themiurgo

+

Joe Jevnik @llllllllll

+

Rory Kirchner @roryk

+

Steven Cutting @steven_cutting

+

Aric Coady @coady

+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/py311/lib/python3.11/site-packages/werkzeug/debug/shared/ICON_LICENSE.html b/py311/lib/python3.11/site-packages/werkzeug/debug/shared/ICON_LICENSE.html new file mode 100644 index 0000000..bbaea5a --- /dev/null +++ b/py311/lib/python3.11/site-packages/werkzeug/debug/shared/ICON_LICENSE.html @@ -0,0 +1,359 @@ + + + + + + + + + +Hans Dembinski's blog – icon_license + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + + + +
+ + + +

Silk icon set 1.3 by Mark James

+

http://www.famfamfam.com/lab/icons/silk/

+

License: CC-BY-2.5 or CC-BY-3.0

+ + + +
+ +
+ + + + \ No newline at end of file diff --git a/search.json b/search.json new file mode 100644 index 0000000..67187d9 --- /dev/null +++ b/search.json @@ -0,0 +1,1164 @@ +[ + { + "objectID": "2024_DOEPy_iminuit/talk.html", + "href": "2024_DOEPy_iminuit/talk.html", + "title": "iminuit: A foundational flexible library for fitting", + "section": "", + "text": "Author Hans Dembinski, TU Dortmund, Germany\nAbstract I will start with a general introduction into the theory of fitting and why you may want to use iminuit. I will show a few difficult nuts to crack with iminuit, and generally, how you get the best performance for a particular fit. Fitting speed in research matters: if an analysis runs quickly, one can do more checks and variations, and it makes computationally-intensive methods like the bootstrap for error estimation feasible." + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#about-me", + "href": "2024_DOEPy_iminuit/talk.html#about-me", + "title": "iminuit: A foundational flexible library for fitting", + "section": "About me", + "text": "About me\n\nParticle physicist working with the LHCb experiment\nFormerly astroparticle physicist working with Pierre Auger Observatory and IceCube\nStatistics convener in LHCb (2019-2021, 2024-now)\nInterests\n\nStatistics\nHigh-performance computing\nAPI design: “Make Interfaces Easy to Use Correctly and Hard to Use Incorrectly” Scott Meyers\n\nMemberships\n\nBoost project\nScikit-HEP project\n\nAuthor/maintainer\n\nBoost.Histogram (C++)\niminuit\nresample\njacobi\nnumba-stats\nChromo\npyhepmc\n\nContributor to scipy and matplotlib" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#introduction", + "href": "2024_DOEPy_iminuit/talk.html#introduction", + "title": "iminuit: A foundational flexible library for fitting", + "section": "Introduction", + "text": "Introduction\n\nFitting?\n\nAdjust a statistical model to data set, find model parameters which match data best\nAI community calls this “learning”\n\nDataset: samples \\(\\{ \\vec x_i \\}\\)\n\nSample can be one-dimensional or multidimensional\nValues can be continuous (e.g. candidate mass) or discrete (e.g. bin counts)\n\nModel: Probability density or probability mass function \\(f(\\vec x; \\vec p)\\) which depends on unknown parameters \\(\\vec p\\)\nConjecture: maximum-likelihood estimate (MLE) \\(\\hat{\\vec p} = \\text{argmax}_{\\vec p} L(\\vec p)\\) is optimal, with \\(L(\\vec p) = \\prod_i f(\\vec x_i; \\vec p)\\)\n\nProven: Asymptotically optimal, unbiased with minimum variance\n\nNo proof for finite data sets, but unchallenged\n\nUniversal: MLE can be computed for any problem\nEquivalent: minimize negative log-likelihood (NLL), \\(\\hat{\\vec p} = \\text{argmin}_{\\vec p}(-\\ln L(\\vec p))\\)\n\nLimiting case of maximum-likelihood fit: least-squares fit aka chi-square fit\n\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.stats import norm\n\ndist = norm(0, 1)\nx1 = dist.rvs(size=100, random_state=1)\nplt.plot(x1, np.zeros_like(x1), \"|\", ms=15, label=\"data\")\nw, xe, *_ = plt.hist(x1, label=\"histogram\", facecolor=\"0.8\")\nxm = np.linspace(-3, 3, 1000)\nplt.plot(xm, dist.pdf(xm) * len(x1) * (xe[1]-xe[0]), label=\"model (scaled)\")\nplt.xlabel(\"x\")\nplt.ylabel(\"density\")\nplt.legend();\n\n\n\n\n\n# model pdf\ndef model1(x, mu, sigma):\n return norm.pdf(x, mu, sigma)\n\n# negative log-likelihood\ndef nll1(mu, sigma):\n logL = np.log(model1(x1, mu, sigma))\n return -np.sum(logL)\n\n\nmu = np.linspace(-1, 1, 10)\nsigma = np.linspace(0.5, 1.5, 10)\nmu2, sigma2 = np.meshgrid(mu, sigma)\nz = np.vectorize(nll1)(mu2, sigma2)\n\nplt.figure(figsize=(10, 4))\nax = plt.subplot(121, projection=\"3d\")\nax.view_init(elev=20, azim=40, roll=0)\nax.plot_surface(mu2, sigma2, z)\nax.set(xlabel=\"mu\", ylabel=\"sigma\")\nplt.sca(ax)\nplt.subplots_adjust(left=0.1, right=1);\n\nax = plt.subplot(122)\nplt.pcolormesh(mu2, sigma2, z)\nplt.colorbar()\nax.set(xlabel=\"mu\", ylabel=\"sigma\");" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#iminuit", + "href": "2024_DOEPy_iminuit/talk.html#iminuit", + "title": "iminuit: A foundational flexible library for fitting", + "section": "", + "text": "Python frontend for C++ Minuit2 library maintained by CERN ROOT team\nMinuit2 is a minimizer developed for negative likelihoods and chi-square functions\nComputes parameter errors via HESSE or MINOS methods\nYou can write your own cost function in plain Python\nJupyter notebook friendly object displays\n\n\nfrom iminuit import Minuit\n\nm = Minuit(nll1, mu=0, sigma=1)\nm.errordef = Minuit.LIKELIHOOD\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 129.7\nNfcn = 32\n\n\nEDM = 3.42e-06 (Goal: 0.0001)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nmu\n0.06\n0.09\n\n\n\n\n\n\n\n1\nsigma\n0.89\n0.06\n\n\n\n\n\n\n\n\n\n\n\n\nmu\nsigma\n\n\nmu\n0.00784\n0.000\n\n\nsigma\n0.000\n0.00392\n\n\n\n\n\n\nBundled extensions\n\nLibrary of common cost functions for various fitting problems\n\nUnbinnedNLL: maximum-likelihood estimation on unbinned data\nBinnedNLL: maximum-likelihood estimation on binned data\nExtendedUnbinnedNLL and ExtendedBinnedNLL: fit shape and amplitude\nTemplate: fit non-parametric models\nNormalConstraint: penalty terms aka soft constraints\nLeastSquares\n\nWrapper for scipy minimizers that handle e.g. constrained problems\nInteractive fitting widget for Jupyter notebooks\n\n\n\nfrom iminuit import Minuit\nfrom iminuit.cost import UnbinnedNLL\n\nc = UnbinnedNLL(x1, model1)\n\nm = Minuit(c, mu=0, sigma=1)\nm.limits[\"mu\"] = (-2, 2)\nm.limits[\"sigma\"] = (0.01, 1.5)\nm.interactive()\n\n\n\n\n\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 259.4\nNfcn = 32\n\n\nEDM = 2.83e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nmu\n0.06\n0.09\n\n\n-2\n2\n\n\n\n1\nsigma\n0.89\n0.06\n\n\n0.01\n1.5\n\n\n\n\n\n\n\n\nmu\nsigma\n\n\nmu\n0.00784\n0.000\n\n\nsigma\n0.000\n0.00392\n\n\n\n\n\n\n \n \n \n \n 2024-01-31T20:41:27.598494\n image/svg+xml\n \n \n Matplotlib v3.7.1, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\niminuit computes parameter errors with the Hesse method and the Minos method\nHesse method produces symmetric error intervals and covariance matrix\nMinos method produces asymmetric error intervals and 2D contours\nNeither method is superior: use Hesse unless you have strongly asymmetric errors\n\n\n# 1, 2, 3 sigma contours\nm.draw_mncontour(\"mu\", \"sigma\", cl=(1, 2, 3));" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#iminuit-vs.-other-packages", + "href": "2024_DOEPy_iminuit/talk.html#iminuit-vs.-other-packages", + "title": "iminuit: A foundational flexible library for fitting", + "section": "iminuit vs. other packages", + "text": "iminuit vs. other packages\n\npyhf\n\nBetter choice when you need to compute limits and preserve the likelihood\n\nzfit and RooFit\n\nHelp you build statistical models: automatic normalization & convolutions\n\nThis has to be done “by hand” in iminuit\n\nGood for standard tasks, but restrict your freedom\n\niminuit\n\nHas very good documentation\nWell-designed API\nIs very flexible and open\nIs very fast with numba" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#fast-fits-with-iminuit-numba", + "href": "2024_DOEPy_iminuit/talk.html#fast-fits-with-iminuit-numba", + "title": "iminuit: A foundational flexible library for fitting", + "section": "Fast fits with iminuit + numba", + "text": "Fast fits with iminuit + numba\n\nFast fits are important when…\n\nYou need an unbinned fit on large sample\nYou have to repeat the same fit many times on different data\n\nE.g. fitting many kinematic bins, bootstrapping uncertainties, studying systematic variations\n\n\nPerformance insights\n\nTime spend in minimizer is negligible\nBottleneck is evaluating cost function\nBottleneck inside cost function is usually evaluation the pdf\nCost function is trivially parallelizable: log(pdf) computed independently for each data point\n\nExploit SIMD instructions and parallel computing on multiple cores\n\n\n\n\ndef make_data(size, seed=1):\n rng = np.random.default_rng(seed)\n s = rng.normal(0.5, 0.1, size=size // 2)\n b = rng.exponential(1, size=2 * size)\n b = b[b < 1]\n b = b[: size // 2]\n x = np.append(s, b)\n return x\n\nx2 = make_data(1_000_000)\nplt.hist(x2, bins=100);\n\n\n\n\n\nfrom numba_stats import norm, truncexpon\n\n\ndef model2(x, z, mu, sigma, slope):\n b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)\n s = norm.pdf(x, mu, sigma)\n return (1 - z) * b + z * s\n\n\ndef nll2(z, mu, sigma, slope):\n p = model2(x2, z, mu, sigma, slope)\n return -np.sum(np.log(p))\n\n\nm = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -2.108e+05\nNfcn = 163\n\n\nEDM = 2.8e-07 (Goal: 0.0002)\ntime = 6.4 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.4997\n0.0013\n\n\n\n\n\n\n\n1\nmu\n499.98e-3\n0.29e-3\n\n\n\n\n\n\n\n2\nsigma\n99.59e-3\n0.29e-3\n\n\n\n\n\n\n\n3\nslope\n0.997\n0.008\n\n\n\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n1.77e-06\n-0.02e-6 (-0.049)\n0.21e-6 (0.548)\n-2.4e-6 (-0.229)\n\n\nmu\n-0.02e-6 (-0.049)\n8.36e-08\n-0 (-0.060)\n-0.51e-6 (-0.226)\n\n\nsigma\n0.21e-6 (0.548)\n-0 (-0.060)\n8.18e-08\n-0.37e-6 (-0.164)\n\n\nslope\n-2.4e-6 (-0.229)\n-0.51e-6 (-0.226)\n-0.37e-6 (-0.164)\n6.12e-05\n\n\n\n\n\n\nimport numba as nb\n\n@nb.njit(inline=\"always\")\ndef model2(x, z, mu, sigma, slope):\n b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)\n s = norm.pdf(x, mu, sigma)\n return (1 - z) * b + z * s\n\n\n@nb.njit(parallel=True, fastmath=True)\ndef nll2(z, mu, sigma, slope):\n p = model2(x2, z, mu, sigma, slope)\n return -np.sum(np.log(p))\n\n\nm = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)\nm.migrad()\n\nOMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.\n\n\n\n\n\nMigrad\n\n\nFCN = -2.108e+05\nNfcn = 163\n\n\nEDM = 2.8e-07 (Goal: 0.0002)\ntime = 9.0 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.4997\n0.0013\n\n\n\n\n\n\n\n1\nmu\n499.98e-3\n0.29e-3\n\n\n\n\n\n\n\n2\nsigma\n99.59e-3\n0.29e-3\n\n\n\n\n\n\n\n3\nslope\n0.997\n0.008\n\n\n\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n1.77e-06\n-0.02e-6 (-0.049)\n0.21e-6 (0.548)\n-2.4e-6 (-0.229)\n\n\nmu\n-0.02e-6 (-0.049)\n8.36e-08\n-0 (-0.060)\n-0.51e-6 (-0.226)\n\n\nsigma\n0.21e-6 (0.548)\n-0 (-0.060)\n8.18e-08\n-0.37e-6 (-0.164)\n\n\nslope\n-2.4e-6 (-0.229)\n-0.51e-6 (-0.226)\n-0.37e-6 (-0.164)\n6.12e-05\n\n\n\n\n\n\nm = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -2.108e+05\nNfcn = 163\n\n\nEDM = 2.8e-07 (Goal: 0.0002)\ntime = 1.1 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.4997\n0.0013\n\n\n\n\n\n\n\n1\nmu\n499.98e-3\n0.29e-3\n\n\n\n\n\n\n\n2\nsigma\n99.59e-3\n0.29e-3\n\n\n\n\n\n\n\n3\nslope\n0.997\n0.008\n\n\n\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n1.77e-06\n-0.02e-6 (-0.049)\n0.21e-6 (0.548)\n-2.4e-6 (-0.229)\n\n\nmu\n-0.02e-6 (-0.049)\n8.36e-08\n-0 (-0.060)\n-0.51e-6 (-0.226)\n\n\nsigma\n0.21e-6 (0.548)\n-0 (-0.060)\n8.18e-08\n-0.37e-6 (-0.164)\n\n\nslope\n-2.4e-6 (-0.229)\n-0.51e-6 (-0.226)\n-0.37e-6 (-0.164)\n6.12e-05\n\n\n\n\n\nPlots from the iminuit benchmarks" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#binned-vs.-unbinned-fits", + "href": "2024_DOEPy_iminuit/talk.html#binned-vs.-unbinned-fits", + "title": "iminuit: A foundational flexible library for fitting", + "section": "Binned vs. unbinned fits", + "text": "Binned vs. unbinned fits\n\nIdea: don’t fit original data sample, fit histogram of data sample\n\nHistogram: lossy compression of original data\nAnother way to speed up fitting\n\nFit results based on histograms not more biased, but may have larger variance\nLoss can be made negligible with sufficiently fine binning\nBinned fits require the modeling cdf instead of pdf\nFurther benefit: iminuit cost functions compute chi-square goodness-of-fit test statistic for free, based on S. Baker and RD. Cousins, Nucl.Instrum.Meth. 221 (1984) 437-442\n\n\nn = 100_000\ntruth = np.array((1.0, 1.0, 1.0, 0.1, 1.0))\nrng = np.random.default_rng(1)\ns = rng.normal(truth[2], truth[3], size=int(n * truth[0]))\nb = rng.exponential(truth[4], size=int(n * truth[1]))\nx3 = np.append(s, b)\nx3 = x3[(x3 > 0) & (x3 < 2)]\n\nw, xe = np.histogram(x3, bins=200) # for more complex histograms use boost-histogram\nplt.stairs(w, xe);\n\n\n\n\n\nfrom numba_stats import norm, expon\nfrom iminuit.cost import ExtendedBinnedNLL\n\ndef model3(xe, s, b, mu, sigma, tau):\n return s * n * norm.cdf(xe, mu, sigma) + b * n * expon.cdf(xe, 0, tau)\n\n\nc = ExtendedBinnedNLL(w, xe, model3)\nm = Minuit(c, *truth)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 190.9 (χ²/ndof = 1.0)\nNfcn = 110\n\n\nEDM = 2.26e-06 (Goal: 0.0002)\ntime = 0.5 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\ns\n0.995\n0.004\n\n\n\n\n\n\n\n1\nb\n1.008\n0.005\n\n\n\n\n\n\n\n2\nmu\n999.3e-3\n0.4e-3\n\n\n\n\n\n\n\n3\nsigma\n99.22e-3\n0.34e-3\n\n\n\n\n\n\n\n4\ntau\n1.002\n0.007\n\n\n\n\n\n\n\n\n\n\n\n\ns\nb\nmu\nsigma\ntau\n\n\ns\n1.32e-05\n-0.005e-3 (-0.284)\n-0.04e-6 (-0.029)\n0.30e-6 (0.248)\n-0.004e-3 (-0.161)\n\n\nb\n-0.005e-3 (-0.284)\n2.37e-05\n0\n-0.47e-6 (-0.286)\n0.020e-3 (0.587)\n\n\nmu\n-0.04e-6 (-0.029)\n0\n1.46e-07\n-0.01e-6 (-0.046)\n-0.14e-6 (-0.054)\n\n\nsigma\n0.30e-6 (0.248)\n-0.47e-6 (-0.286)\n-0.01e-6 (-0.046)\n1.12e-07\n-0.37e-6 (-0.159)\n\n\ntau\n-0.004e-3 (-0.161)\n0.020e-3 (0.587)\n-0.14e-6 (-0.054)\n-0.37e-6 (-0.159)\n4.77e-05\n\n\n\n\n\n\n \n \n \n \n 2024-01-31T20:41:46.336920\n image/svg+xml\n \n \n Matplotlib v3.7.1, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nFull study can be reviewed in iminuit docs\n100k data points\nNormal signal + truncated exponential background\nBins: infinite (unbinned likelihood), 200, 100, 50, 20, 10, 5\nStudy bias and variance of fit results\n\nRepeatedly fit random data sets of equal size" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#template-fits", + "href": "2024_DOEPy_iminuit/talk.html#template-fits", + "title": "iminuit: A foundational flexible library for fitting", + "section": "Template fits", + "text": "Template fits\n\nSometimes PDF/CDF not known in parametric form, but can be Monte-Carlo simulated\nIdea: fit a histogram of simulation output (template) to data histogram\nAdditional source of uncertainty from finite size of simulation\nCorrect error propagation with Barlow-Beeston method, Barlow and Beeston, Comput.Phys.Commun. 77 (1993) 219-228\niminuit.cost.Template\n\nExtension to weighted data and simuation, H. Dembinski, A. Abdelmotteleb, Eur.Phys.J.C 82 (2022) 11, 1043\nApproximation to Barlow-Beeston method for unweighted data\nFaster computation\nAllows you to mix parametric models and templates\n\n\n\ndef generate(rng, nmc, truth, bins):\n xe = np.linspace(0, 2, bins + 1)\n b = np.diff(truncexpon.cdf(xe, 0, 2, 0, 1))\n s = np.diff(norm.cdf(xe, 1, 0.1))\n n = rng.poisson(b * truth[0]) + rng.poisson(s * truth[1])\n t = np.array([rng.poisson(b * nmc), rng.poisson(s * nmc)])\n return xe, n, t\n\nrng = np.random.default_rng(1)\ntruth = 750, 250\nxe, n, t = generate(rng, 200, truth, 15)\n\n_, ax = plt.subplots(1, 2, figsize=(10, 4))\n\nax[0].stairs(n, xe, color=\"k\", label=\"data\")\nax[0].legend();\nax[1].stairs(t[0], xe, label=\"background\")\nax[1].stairs(t[1], xe, label=\"signal\")\nax[1].legend(title=\"templates\");\n\n\n\n\n\nfrom iminuit.cost import Template\n\nc = Template(n, xe, t)\nm = Minuit(c, 1, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 8.511 (χ²/ndof = 0.7)\nNfcn = 124\n\n\nEDM = 3.22e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n760\n60\n\n\n0\n\n\n\n\n1\nx1\n249\n34\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\n\n\nx0\n3.98e+03\n-0.6e3 (-0.274)\n\n\nx1\n-0.6e3 (-0.274)\n1.13e+03\n\n\n\n\n\n\n \n \n \n \n 2024-01-31T20:41:46.828903\n image/svg+xml\n \n \n Matplotlib v3.7.1, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nxe, n, t = generate(rng, 1_000_000, truth, 15)\n\nc = Template(n, xe, t)\nm = Minuit(c, 1, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 12.32 (χ²/ndof = 0.9)\nNfcn = 110\n\n\nEDM = 5.76e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n782\n31\n\n\n0\n\n\n\n\n1\nx1\n238\n20\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\n\n\nx0\n960\n-0.2e3 (-0.281)\n\n\nx1\n-0.2e3 (-0.281)\n416\n\n\n\n\n\n\n \n \n \n \n 2024-01-31T20:41:47.026893\n image/svg+xml\n \n \n Matplotlib v3.7.1, https://matplotlib.org/" + }, + { + "objectID": "2024_DOEPy_iminuit/talk.html#summary", + "href": "2024_DOEPy_iminuit/talk.html#summary", + "title": "iminuit: A foundational flexible library for fitting", + "section": "Summary", + "text": "Summary\n\niminuit is a versatile, fast, easy-to-use library for fitting\nBuiltin cost functions cover most common use-cases\nEmphasis is put on having excellent documentation and being bug-free\nDue to its focussed open design\n\nPreferred by statistics experts (e.g. Glen Cowan)\nExcellent for teaching\n\nExcellent performance in combination with numba and numba-stats\nEasy installation: binary wheels for all major platforms (Python-3.12 will be added in next release)" + }, + { + "objectID": "index.html", + "href": "index.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "The Wilson score interval and weighted histograms\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nC++ exceptions in high-performance code\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nPower consumption in sleep mode\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nSimple parallelization in Jupyter Notebooks\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nTransform LaTeX to Unicode\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nLook-elsewhere effect\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nFitting mixture models with COWs\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nRooFit with Chebyshev and Bernstein polynomials\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nInteractive plotting in Jupyter with matplotlib\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nNon-prompt fraction of KS and Lambda particles from hyperon decays\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nInelastic cross-sections of proton-proton and pion-proton collisions\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nBenchmark of building an array with numba\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nComparison of asymptotically chisquare-distributed test statistics\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nRegression example with a neural network\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nTemplate fits with distortion correction\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nFrom a fixed-target to center-of-mass frame and back\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nApproximate PDF for the invariant mass distribution of combinatorial background\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nError propagation with Sympy\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nIntroduction into Machine Learning\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nRender LaTeX to SVG with matplotlib backend\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nTracking efficiency for two-body decay\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nStatistical issues in naive calibration analyses\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nSolving the duffing oscillator\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nVisual cross-section for particle production\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nLeave-one-out cross-validation\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nFitting weighted histograms with SPD method\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nHow to make a plot with legend entries that are hyperlinks\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nRatio bias\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nMCMC demo\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nNew Fit Result Display\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nWhich data structure is faster: array of structs or struct of arrays?\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nLHCb Startertalk on Uncertainties\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\np-value computation and conversion: one-sided, two-sided?\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nNumerically stable calculation of invariant mass\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nAll decays into two Kaons\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nApplying the SPD approximation to negative weights\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nCoverage of HESSE and MINOS intervals\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nFast deep sets with FLAX\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nUnbiasedness of EML fit for a mixture model with fixed component pdfs\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nParticle correlations in proton-proton collisions\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nNumba-accelerated Jackknife\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nHow to use the logaddexp special function in fits\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nChance of seeing a one-sigma deviation in random splits\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nUncertainty of efficiency from fitted decay yields\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\nError propagation for a ratio\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\nNo matching items" + }, + { + "objectID": "py311/lib/python3.11/site-packages/toolz-0.12.0.dist-info/AUTHORS.html", + "href": "py311/lib/python3.11/site-packages/toolz-0.12.0.dist-info/AUTHORS.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "Matthew Rocklin @mrocklin\nJohn Jacobsen @eigenhombre\nErik Welch @eriknw\nJohn Crichton @jcrichton\nHan Semaj @microamp\nGraeme Coupar @obmarg\nLeonid Shvechikov @shvechikov\nLars Buitinck @larsmans\nJosé Ricardo @josericardo\nTom Prince @tomprince\nBart van Merriënboer @bartvm\nNikolaos-Digenis Karagiannis @digenis\nAntonio Lima @themiurgo\nJoe Jevnik @llllllllll\nRory Kirchner @roryk\nSteven Cutting @steven_cutting\nAric Coady @coady" + }, + { + "objectID": "py311/lib/python3.11/site-packages/flax/core/flax_functional_engine.html", + "href": "py311/lib/python3.11/site-packages/flax/core/flax_functional_engine.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "import functools\nimport jax\nfrom jax import numpy as jnp, random, lax\nimport numpy as np\n\n\nfrom flax import linen as nn, struct\n\n\nfrom flax.core import Scope, init, apply, Array, lift, unfreeze\n\n\ndef dense(scope: Scope, inputs: Array, features: int, bias: bool = True,\n kernel_init=nn.linear.default_kernel_init,\n bias_init=nn.initializers.zeros_init()):\n kernel = scope.param('kernel', kernel_init, (inputs.shape[-1], features))\n y = jnp.dot(inputs, kernel)\n if bias:\n y += scope.param('bias', bias_init, (features,))\n return y\n\nmodel_fn = functools.partial(dense, features=3)\n\nx = jnp.ones((1, 2))\ny, params = init(model_fn)(random.key(0), x)\nprint(params)\n\ndef mlp(scope: Scope, inputs: Array, features: int):\n hidden = scope.child(dense, 'hidden')(inputs, features)\n hidden = nn.relu(hidden)\n return dense(scope.push('out'), hidden, 1)\n\ninit(mlp)(random.key(0), x, features=3)\n\nFrozenDict({'params': {'kernel': DeviceArray([[ 0.15374057, -0.6807397 , -1.3350962 ],\n [ 0.59940743, -0.69430196, -0.7663768 ]], dtype=float32), 'bias': DeviceArray([0., 0., 0.], dtype=float32)}})\n\n\n(DeviceArray([[0.17045607]], dtype=float32),\n FrozenDict({'params': {'hidden': {'bias': DeviceArray([0., 0., 0.], dtype=float32), 'kernel': DeviceArray([[-0.22119394, 0.22075175, -0.0925657 ],\n [ 0.40571952, 0.27750877, 1.0542233 ]], dtype=float32)}, 'out': {'kernel': DeviceArray([[ 0.21448377],\n [-0.01530595],\n [ 0.14402702]], dtype=float32), 'bias': DeviceArray([0.], dtype=float32)}}}))\n\n\n\n@struct.dataclass\nclass Embedding:\n table: np.ndarray\n\n def lookup(self, indices):\n return self.table[indices]\n\n def attend(self, query):\n return jnp.dot(query, self.table.T)\n\n# all the embedding module does is provide a convenient initializers\n\ndef embedding(scope: Scope, num_embeddings: int, features: int, init_fn=nn.linear.default_embed_init) -> Embedding:\n table = scope.param('table', init_fn, (num_embeddings, features))\n return Embedding(table)\n\nembedding, _ = init(embedding)(random.key(0), num_embeddings=2, features=3)\nprint(embedding.table)\nprint(embedding.lookup(1))\nprint(embedding.attend(jnp.ones((1, 3,))))\n\n[[ 0.11575121 -0.51936364 -1.113899 ]\n [ 0.45569834 -0.5300623 -0.5873911 ]]\n[ 0.45569834 -0.5300623 -0.5873911 ]\n[[-1.5175114 -0.6617551]]\n\n\n\ndef lstm(scope, carry, inputs,\n gate_fn=nn.activation.sigmoid, activation_fn=nn.activation.tanh,\n kernel_init=nn.linear.default_kernel_init,\n recurrent_kernel_init=nn.initializers.orthogonal(),\n bias_init=nn.initializers.zeros_init()):\n r\"\"\"A long short-term memory (LSTM) cell.\n\n the mathematical definition of the cell is as follows\n .. math::\n \\begin{array}{ll}\n i = \\sigma(W_{ii} x + W_{hi} h + b_{hi}) \\\\\n f = \\sigma(W_{if} x + W_{hf} h + b_{hf}) \\\\\n g = \\tanh(W_{ig} x + W_{hg} h + b_{hg}) \\\\\n o = \\sigma(W_{io} x + W_{ho} h + b_{ho}) \\\\\n c' = f * c + i * g \\\\\n h' = o * \\tanh(c') \\\\\n \\end{array}\n where x is the input, h is the output of the previous time step, and c is\n the memory.\n\n Args:\n carry: the hidden state of the LSTM cell,\n initialized using `LSTMCell.initialize_carry`.\n inputs: an ndarray with the input for the current time step.\n All dimensions except the final are considered batch dimensions.\n gate_fn: activation function used for gates (default: sigmoid)\n activation_fn: activation function used for output and memory update\n (default: tanh).\n kernel_init: initializer function for the kernels that transform\n the input (default: lecun_normal).\n recurrent_kernel_init: initializer function for the kernels that transform\n the hidden state (default: orthogonal).\n bias_init: initializer for the bias parameters (default: zeros_init())\n Returns:\n A tuple with the new carry and the output.\n \"\"\"\n c, h = carry\n hidden_features = h.shape[-1]\n # input and recurrent layers are summed so only one needs a bias.\n dense_h = lambda name: scope.child(dense, name)(\n h, features=hidden_features, bias=True,\n kernel_init=recurrent_kernel_init, bias_init=bias_init)\n dense_i = lambda name: scope.child(dense, name)(\n inputs, features=hidden_features, bias=False,\n kernel_init=kernel_init)\n i = gate_fn(dense_i(name='ii') + dense_h(name='hi'))\n f = gate_fn(dense_i(name='if') + dense_h(name='hf'))\n g = activation_fn(dense_i(name='ig') + dense_h(name='hg'))\n o = gate_fn(dense_i(name='io') + dense_h(name='ho'))\n new_c = f * c + i * g\n new_h = o * activation_fn(new_c)\n return (new_c, new_h), new_h\n\ndef lstm_init_carry(batch_dims, size, init_fn=jnp.zeros):\n shape = batch_dims + (size,)\n return init_fn(shape), init_fn(shape)\n\nx = jnp.ones((1, 2))\ncarry = lstm_init_carry((1,), 3)\ny, variables = init(lstm)(random.key(0), carry, x)\njax.tree_util.tree_map(np.shape, (y, variables))\n\n((((1, 3), (1, 3)), (1, 3)),\n FrozenDict({'params': {'hf': {'bias': (3,), 'kernel': (3, 3)}, 'hg': {'bias': (3,), 'kernel': (3, 3)}, 'hi': {'bias': (3,), 'kernel': (3, 3)}, 'ho': {'bias': (3,), 'kernel': (3, 3)}, 'if': {'kernel': (2, 3)}, 'ig': {'kernel': (2, 3)}, 'ii': {'kernel': (2, 3)}, 'io': {'kernel': (2, 3)}}}))\n\n\n\ndef simple_scan(scope: Scope, xs):\n init_carry = lstm_init_carry(xs.shape[:1], xs.shape[-1])\n# cell = scope.child(lstm, 'cell')\n# ys = []\n# for i in range(xs.shape[1]):\n# x = xs[:, i]\n# init_carry, y = cell(init_carry, x)\n# ys.append(y)\n# return init_carry, ys\n lstm_scan = lift.scan(lstm, in_axes=1, out_axes=1, variable_broadcast='params', split_rngs={'params': False})\n return lstm_scan(scope, init_carry, xs)\n\nkey1, key2 = random.split(random.key(0), 2)\nxs = random.uniform(key1, (1, 5, 2))\n\n\ny, init_variables = init(simple_scan)(key2, xs)\n\nprint('initialized parameter shapes:\\n', jax.tree_util.tree_map(jnp.shape, unfreeze(init_variables)))\n\ninitialized parameter shapes:\n {'params': {'hf': {'bias': (2,), 'kernel': (2, 2)}, 'hg': {'bias': (2,), 'kernel': (2, 2)}, 'hi': {'bias': (2,), 'kernel': (2, 2)}, 'ho': {'bias': (2,), 'kernel': (2, 2)}, 'if': {'kernel': (2, 2)}, 'ig': {'kernel': (2, 2)}, 'ii': {'kernel': (2, 2)}, 'io': {'kernel': (2, 2)}}}\n\n\n\ny = apply(simple_scan)(init_variables, xs)[0]\nprint('output:\\n', y)\n\noutput:\n (DeviceArray([[-0.35626447, 0.25178757]], dtype=float32), DeviceArray([[-0.17885922, 0.13063088]], dtype=float32))" + }, + { + "objectID": "py311/lib/python3.11/site-packages/Markdown-3.5.2.dist-info/LICENSE.html", + "href": "py311/lib/python3.11/site-packages/Markdown-3.5.2.dist-info/LICENSE.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "Copyright 2007, 2008 The Python Markdown Project (v. 1.7 and later) Copyright 2004, 2005, 2006 Yuri Takhteyev (v. 0.2-1.6b) Copyright 2004 Manfred Stienstra (the original version)\nAll rights reserved.\nRedistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:\n\nRedistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.\nRedistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.\nNeither the name of the Python Markdown Project nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.\n\nTHIS SOFTWARE IS PROVIDED BY THE PYTHON MARKDOWN PROJECT ‘’AS IS’’ AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL ANY CONTRIBUTORS TO THE PYTHON MARKDOWN PROJECT BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE." + }, + { + "objectID": "py311/lib/python3.11/site-packages/orbax/checkpoint/orbax_checkpoint.html", + "href": "py311/lib/python3.11/site-packages/orbax/checkpoint/orbax_checkpoint.html", + "title": "Checkpointing with Orbax", + "section": "", + "text": "Take a look at https://orbax.readthedocs.io/en/latest/ for additional documentation on Orbax checkpointing APIs.\n\n!pip install orbax-checkpoint\n\n\nimport time\nfrom collections import namedtuple\nimport jax\nfrom jax.sharding import Mesh, PartitionSpec\nfrom jax.experimental.pjit import pjit\nimport numpy as np\nimport os\nimport portpicker\nfrom etils import epath\nimport orbax.checkpoint\n\n\nport = portpicker.pick_unused_port()\njax.distributed.initialize(f'localhost:{port}', num_processes=1, process_id=0)\n\n\ndevices = np.asarray(jax.devices())\nmesh = Mesh(devices, ('data',))\naxes = PartitionSpec('data',)\nsharding = jax.sharding.NamedSharding(mesh, axes)\n\n\ndirectory = epath.Path('checkpoint_data')\nif directory.exists():\n directory.rmtree()\ndirectory.mkdir(parents=True)\n\n\nprint(directory)\n\n\nA Basic Example\n\nbasic_dir = directory / 'basic'\nif not basic_dir.exists():\n basic_dir.mkdir()\n\n\noptions = orbax.checkpoint.CheckpointManagerOptions(save_interval_steps=1, max_to_keep=3)\nmngr = orbax.checkpoint.CheckpointManager(\n basic_dir, orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler()), options)\n\nThe CheckpointManager is constructed with Checkpointer and CheckpointHandler objects. We will discuss these further below, but at a high level, the Checkpointer controls the manner in which the object is saved while the CheckpointHandler deals with type-specific logic and provides extra options for customization.\nFirst, we’ll need to perform some setup to create a train state that mimics in a very basic form how a real model might look. We use jax.Array for this is example, but it is also possible to use scalars or numpy arrays, assuming they are replicated or not sharded.\n\ndef create_initial_state():\n state = {\n 'layer_0': {\n 'bias': np.zeros(16),\n 'kernel': np.arange(16),\n },\n 'layer_1': {\n 'bias': np.zeros(8),\n 'kernel': np.arange(8),\n },\n }\n\n create_sharded_array = pjit(lambda x: x, in_shardings=None, out_shardings=sharding)\n with Mesh(mesh.devices, mesh.axis_names):\n state = jax.tree_map(create_sharded_array, state)\n\n state['step'] = 0\n return state\n\ndef state_shape(state):\n return jax.eval_shape(lambda: state)\n\nHere’s our mock training step. At every step, we save a checkpoint. Since we specified max_to_keep=3 in our options, we expect to only have the latest 3 checkpoints at the end of training.\n\nstate = create_initial_state()\n\ndef train(step, state):\n # do some training, modify state\n state['step'] = step\n\n mngr.save(step, state)\n\n return state\n\nfor step in range(5):\n state = train(step, state)\n\nprint(f'Steps: {mngr.all_steps()}')\n\nAfter saving, we can restore the latest step. Since this CheckpointManager is only managing a single item, the arguments provided to restore should just match the state (see the below cell). See ‘Multi-Object Checkpointing’ for further details on how these arguments are used when there are multiple items.\n\ndummy_state = state_shape(state)\nrestore_args = jax.tree_util.tree_map(\n lambda _:\n orbax.checkpoint.ArrayRestoreArgs(\n restore_type=jax.Array,\n mesh=mesh, \n mesh_axes=axes), \n state)\nrestore_args['step'] = orbax.checkpoint.RestoreArgs(restore_type=int)\nmngr.restore(mngr.latest_step(), items=dummy_state, \n restore_kwargs={'restore_args': restore_args})\n\nWe can achieve the same result by just using Checkpointer.\n\nckptr = orbax.checkpoint.PyTreeCheckpointer()\n\ndef _maybe_get_sharding(arr):\n if isinstance(arr, jax.Array):\n return arr.sharding\n else:\n return None\n\nshardings = jax.tree_util.tree_map(_maybe_get_sharding, state)\n# restore_args can also be constructed \"automatically\" from a target PyTree.\nrestore_args = orbax.checkpoint.checkpoint_utils.construct_restore_args(\n state, shardings)\nprint(restore_args)\nprint()\n# CheckpointManager saved the checkpoint under /<directory>/<step>/default.\n# 'default' is used as the subdirectory name when the CheckpointManager has a\n# single item. See below for information on how to use multiple items or how\n# to customize this name.\nckpt_path = mngr.directory / str(mngr.latest_step()) / 'default'\nckptr.restore(ckpt_path, item=dummy_state, restore_args=restore_args)\n\n\n\nTracking Metrics\nWhen saving checkpoints across many steps, we are often interested in keeping only the best n checkpoints based on some metric.\n\nckpt_metrics_dir = directory / 'ckpt_with_metrics'\nif not ckpt_metrics_dir.exists():\n ckpt_metrics_dir.mkdir()\n\n\noptions = orbax.checkpoint.CheckpointManagerOptions(\n max_to_keep=3, best_fn=lambda metrics: metrics['loss'], best_mode='min')\nmngr = orbax.checkpoint.CheckpointManager(\n ckpt_metrics_dir, orbax.checkpoint.AsyncCheckpointer(orbax.checkpoint.PyTreeCheckpointHandler()),\n options)\n\n\ndef get_metrics(step):\n return {'accuracy': 1.0, 'loss': step * 1.5}\n\n\ndef train(step, state):\n # do some training, modify state\n metrics = get_metrics(step)\n state['step'] = step\n state_save_args = jax.tree_map(lambda _: orbax.checkpoint.SaveArgs(), state)\n mngr.save(step, state, metrics=metrics)\n return state\n\n\nstate = create_initial_state()\nfor step in range(5):\n state = train(step, state)\n\nmngr.wait_until_finished()\nprint(f'Steps: {mngr.all_steps()}')\n\nNow that we track metrics, we will only keep the best checkpoints saved, while deleting the rest. Since our loss is getting progressively worse, and best_mode=‘min’, we will keep the first checkpoints, rather than the most recent ones. The metrics may be an arbitrary PyTree; it is up to you to define how it is interpreted.\n\n\nMulti-Object Checkpointing\nIn the following example, we will show how to checkpoint multiple different objects at once using CheckpointManager.\n\nmulti_dir = directory / 'multi'\nif not multi_dir.exists():\n multi_dir.mkdir()\n\n\n# Save every 3 steps.\noptions = orbax.checkpoint.CheckpointManagerOptions(save_interval_steps=3)\nmngr = orbax.checkpoint.CheckpointManager(\n multi_dir, {\n 'state': orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler()),\n 'metadata': orbax.checkpoint.Checkpointer(orbax.checkpoint.JsonCheckpointHandler())\n }, options)\n\nWe can save multiple objects simply by specifying a Checkpointer/CheckpointHandler combination for each of them. While Checkpointers can typically be used with any CheckpointHandler, you’ll need to ensure that your object can be saved and restored by the given CheckpointHandler.\n\nNUM_STEPS = 5\n\ndef train(step, state):\n # do some training, modify state\n state['step'] = step\n metadata['timestamp'] = time.time()\n\n # save with default arguments for all params except 'step', which is\n # aggregated and saved using msgpack.\n state_save_args = jax.tree_map(lambda _: orbax.checkpoint.SaveArgs(), state)\n state_save_args['step'] = orbax.checkpoint.SaveArgs(aggregate=True)\n\n # with `force` a save will be performed even if it would not ordinarily do so,\n # based on the step number.\n force = False\n if step == NUM_STEPS - 1:\n force = True\n save_performed = mngr.save(\n step,\n items={\n 'state': state,\n 'metadata': metadata\n },\n # save_kwargs must be a dict with the same keys as items.\n # not all keys in items have to be provided, in which case default kwargs\n # are used each value must be a dict with keyword args passed to the\n # underlying CheckpointHandler for that item (see CheckpointManager\n # object construction)\n save_kwargs={'state': {\n 'save_args': state_save_args\n }}, \n force=force)\n print(f'Save performed: {save_performed}')\n\n return state\n\n\nstate = create_initial_state()\nmetadata = {\n 'version': 1.1,\n 'exp_name': 'my_test_exp',\n 'timestamp': 0,\n}\nfor step in range(NUM_STEPS):\n state = train(step, state)\n\nprint(f'Steps: {mngr.all_steps()}')\n\nWe also have extra arguments to customize saving for the step parameter within the train state. Because this is only an integer, using the default storage mechanism, Tensorstore might be somewhat overkill. It would be more efficient to store it, along with any other similarly small parameters, into a single file using msgpack). The parameters passed here should match the optional arguments for the provided CheckpointHandler.\nLet’s try restoring the latest step. In the items argument, make sure to provide a key for each of the items you want to restore. The values of this items dictionary are what gets provided to CheckpointHandler.restore as the argument of item. As a result, it may be None if the CheckpointHandler does not depend on the value of item.\nThe restore_kwargs argument should be a dictionary with matching top-level keys, but keys can be omitted if no arguments are needed. The values of the restore_kwargs dictionary are provided as keyword args to the matching CheckpointHandler.restore. For example, if a CheckpointHandler subclass called FooBarCheckpointHandler takes kwargs foo and bar (in addition to standard args like path and item, the restore_kwargs for CheckpointManager would be:\nrestore_kwargs = {'foobar_item': {'foo': ???, 'bar': ???}}\n\nmngr.restore(mngr.latest_step(), \n # Safe to provide None for `items` values because they are not\n # really needed in this case. `restore_args` is needed though.\n items={'state': None, 'metadata': None}, \n restore_kwargs={'state': {'restore_args': restore_args}})\n\nIf we skip providing ‘metadata’ in items, it will not be returned in the result.\n\nmngr.restore(mngr.latest_step(), \n items={'state': None}, \n restore_kwargs={'state': {'restore_args': restore_args}})\n\n\n\nAsynchronous Checkpointing\nYou may be wondering what the point of Checkpointer is, and why it is separate from CheckpointHandler. The reason for this is that Checkpointer may have different subclasses, each of which handles certain common logic that we would not want to reimplement for every different CheckpointHandler.\nThis logic may include ensuring save operation atomicity and managing a background thread for asynchronous saving.\n\nasync_dir = directory / 'async'\nif not async_dir.exists():\n async_dir.mkdir()\n\n\nmngr = orbax.checkpoint.CheckpointManager(\n async_dir, {\n 'state': orbax.checkpoint.AsyncCheckpointer(orbax.checkpoint.PyTreeCheckpointHandler()),\n 'metadata': orbax.checkpoint.Checkpointer(orbax.checkpoint.JsonCheckpointHandler())\n })\n\nWith the above configuration, our metadata will be saved synchronously, but our state will be saved in a background thread. After calling save, all files for the state may not have been written yet. In the meantime, we may continue training.\nHowever, we need to call wait_until_finished before ending our training program to block for any outstanding save operations. Calling save again will do this automatically - you cannot have multiple saves for multiple steps running concurrently.\n\ndef train(step, state):\n # do some training, modify state\n state['step'] = step\n metadata['timestamp'] = time.time()\n\n mngr.save(\n step,\n items={\n 'state': state,\n 'metadata': metadata\n })\n\n return state\n\n\nstate = create_initial_state()\nmetadata = {\n 'version': 1.1,\n 'exp_name': 'my_test_exp',\n 'timestamp': 0,\n}\nfor step in range(NUM_STEPS):\n state = train(step, state)\n\nmngr.wait_until_finished()\nprint(f'Steps: {mngr.all_steps()}')\n\n\n\nCheckpointer\nCheckpointer allows you to save an object to a specified directory without providing any of the structure or extra features that CheckpointManager does.\n\nstate = create_initial_state()\nckptr = orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler())\n\nexisting_checkpoint_dir = multi_dir / '0' / 'state'\nrestore_args = jax.tree_map(lambda _: orbax.checkpoint.ArrayRestoreArgs(mesh=mesh, mesh_axes=axes), state)\nrestore_args['step'] = orbax.checkpoint.RestoreArgs()\nrestored = ckptr.restore(existing_checkpoint_dir, restore_args=restore_args)\n\nfor k, v in orbax.checkpoint.utils.to_flat_dict(restored, sep='/').items():\n print(f'{k}: {v}')\n\nAs shown in this example, this object may be useful for restoring a pre-existing checkpoint without requiring a CheckpointManager.\n\n\nCheckpointHandler\nImportant: CheckpointHandler should not be used independently of Checkpointer or CheckpointManager.\nAs shown above, PyTreeCheckpointHandler provides support for most standard use-cases, where a PyTree consisting of jax.Array, scalars, or numpy arrays can be saved using a combination of Tensorstore and msgpack.\nHowever, we may also wish to provide custom support for a novel type or storage medium. Below is an example of how CheckpointHandler can be overridden to support a sharded file concept.\n\nfrom typing import Any, List, Mapping, Optional\nimport asyncio\nfrom concurrent import futures\nimport datetime\n\n\nexecutor = futures.ThreadPoolExecutor(max_workers=2)\ndef ctime():\n return datetime.datetime.now(\n tz=datetime.timezone.utc\n ).isoformat(sep=' ', timespec='milliseconds')\n\n\nclass ShardedFile:\n\n def __init__(self):\n self.devices = [d for d in jax.devices() if d.host_id == jax.process_index()]\n\n def get_device_shard(self, device):\n return f'Data associated with device: {device}... '\n\n def get(self):\n data = f'Process {jax.process_index()}: '\n for d in self.devices:\n data += self.get_device_shard(d)\n return data\n\nclass ShardedFileWriter:\n\n async def copy(self):\n await asyncio.sleep(1)\n self.local_file = self.sharded_file.get()\n print(f'{ctime()}: done copy')\n\n def commit(self):\n path = epath.Path(self.path) / str(jax.process_index())\n\n def _write():\n time.sleep(5)\n path.write_text(self.local_file)\n print(f'{ctime()}: done commit')\n\n future = executor.submit(_write)\n print(f'{ctime()}: started commit')\n return future\n \n def __init__(self, path: str, sharded_file: ShardedFile):\n self.path = path\n self.sharded_file = sharded_file\n\n\nclass ShardedFileCheckpointHandler(orbax.checkpoint.async_checkpoint_handler.AsyncCheckpointHandler):\n\n async def async_save(self, directory: epath.Path, item: ShardedFile) -> List[futures.Future]:\n \"\"\"Saves the given item.\n\n Args:\n directory: save location directory.\n item: nested dictionary.\n \"\"\"\n fw = ShardedFileWriter(os.fspath(directory), item)\n await fw.copy()\n return [fw.commit()]\n\n def save(self, directory: epath.Path, item: Any, *args, **kwargs):\n async def async_save(*args, **kwargs):\n commit_futures = await self.async_save(*args, **kwargs)\n # Futures are already running, so sequential waiting is equivalent to\n # concurrent waiting.\n for future in commit_futures:\n future.result() # Block on result.\n asyncio.run(async_save(directory, item, *args, **kwargs))\n orbax.checkpoint.utils.sync_global_devices('ShardedFileCheckpointHandler:save')\n\n def restore(self,\n directory: epath.Path,\n item: Optional[bytes] = None) -> str:\n del item\n path = directory / str(jax.process_index())\n return path.read_text()\n\n def structure(self, directory: epath.Path) -> int:\n return len(list(directory.iterdir()))\n\n\nhandler_dir = directory / 'sync_handler'\nasync_handler_dir = directory / 'async_handler'\n\nfile = ShardedFile()\n\n\ncheckpointer = orbax.checkpoint.Checkpointer(ShardedFileCheckpointHandler())\ncheckpointer.save(handler_dir, file)\nprint(f'{ctime()}: done save')\n\n\ncheckpointer = orbax.checkpoint.AsyncCheckpointer(ShardedFileCheckpointHandler())\ncheckpointer.save(async_handler_dir, file)\nprint(f'{ctime()}: processing save')\n# do something else\ncheckpointer.wait_until_finished()\nprint(f'{ctime()}: done save')\n\n\nexecutor.shutdown()\n\n\n\nTypeHandler\nTypeHandler as a concept exists in conjunction with PyTreeCheckpointHandler to provide additional customization options for advanced users with custom types they wish to save as part of a PyTree.\nBy default, Orbax includes handler implementations for jax.Array, np.ndarray, scalars, strings, and others. These are simply the leaf types supported by PyTreeCheckpointHandler.\nBy implementing a subclass of TypeHandler and registering a type using register_type_handler, we can add support for a novel type.\n\nclass Foo():\n # Realistically we would use a dataclass for this, but this is just for\n # illustration purposes.\n def __init__(self, a, b, c):\n self.a = a\n self.b = b\n self.c = c\n\n def __str__(self):\n return f'{self.a};{self.b};{self.c}'\n\n\nfrom orbax.checkpoint.value_metadata import Metadata\nfrom orbax.checkpoint.type_handlers import ParamInfo\nfrom typing import Any, Tuple\n\nclass FooHandler(orbax.checkpoint.type_handlers.TypeHandler):\n\n def typestr(self) -> str:\n return 'foo'\n\n def metadata(self, value: Foo) -> Metadata:\n return Metadata()\n\n async def open(self, info: ParamInfo) -> Tuple[Metadata, Any]:\n return Metadata(), None\n\n async def serialize(\n self,\n value: Foo,\n info: ParamInfo,\n args: Optional[orbax.checkpoint.SaveArgs] = None) -> List[orbax.checkpoint.future.Future]:\n # A more sophisticated implementation would make this write asynchronous.\n (info.path / 'data.txt').write_text(str(value))\n return []\n\n async def deserialize(\n self,\n file: Any,\n info: ParamInfo,\n args: Optional[orbax.checkpoint.RestoreArgs] = None) -> Foo:\n del file\n entries = (info.path / 'data.txt').read_text().split(';')\n assert len(entries) == 3\n return Foo(*entries)\n\n\ntype_handler_dir = directory / 'type_handler'\norbax.checkpoint.type_handlers.register_type_handler(Foo, FooHandler(), override=True)\n\n\nfoo_tree = {\n 'one_foo': Foo(2, 4, 6),\n 'two_foo': Foo(1, 2, 3),\n}\nckptr = orbax.checkpoint.Checkpointer(orbax.checkpoint.PyTreeCheckpointHandler())\nckptr.save(type_handler_dir, foo_tree)\n\n\nrestore_args = jax.tree_util.tree_map(lambda _: orbax.checkpoint.RestoreArgs(restore_type=Foo), foo_tree)\nrestored = ckptr.restore(type_handler_dir, restore_args=restore_args)\njax.tree_util.tree_map(str, restored)\n\n\n\nTransformations\nA key component of the Orbax checkpointing library is PyTree transformations. While this functionality is designed to be as flexible as possible, it can be used to support:\n\nPartial restoration of checkpoints where some keys can be dropped and replaced with randomly initialized values.\nCheckpoint version compatibility where newer checkpoints may have different structures than old ones.\nMappings over keys, including one-to-one, many-to-one, one-to-many, and many-to-many transformations.\n\nThe transformations library is discussed in detail here, so we will avoid discussing all possible features and will instead focus on concrete examples.\nLet’s start with a simple example first.\n\nfrom orbax.checkpoint.transform_utils import Transform\n\n\noriginal_tree = {\n 'a': 1,\n 'b': {\n 'c': 5,\n 'd': [0, 1, 2, 3]\n },\n 'f': 2,\n}\ntransformations = {\n 'a1': Transform(original_key='a'), # rename\n 'b': {\n # doubled original\n 'c': Transform(value_fn=lambda v: v * 2)\n # drop b/d\n },\n # one to many mapping\n 'x': Transform(multi_value_fn=lambda _, kv: kv['b']['d'][0]),\n 'y': Transform(multi_value_fn=lambda _, kv: kv['b']['d'][1:]),\n # many to one mapping\n 'z': Transform(multi_value_fn=lambda _, kv: kv['a'] * 2 + sum(kv['b']['d'])),\n}\nnew_tree = { # defines the structure of the result\n 'a1': ...,\n 'b': {\n 'c': ...,\n },\n 'x': ...,\n 'y': ...,\n 'z': ...,\n # 'f' defined in original_tree and new_tree, but not in transforms. Value\n # carried over from original_tree.\n 'f': ...,\n # This value matters since it is not present in original_tree or\n # transformations, so the value here will simply be preserved in the result.\n 'g': 5,\n}\n\norbax.checkpoint.apply_transformations(original_tree, transformations, new_tree)\n\nAn important rule of thumb to remember is that the output of apply_transformations will always match the structure of new_tree. This provides an easy way to know exactly what your result will look like after applying transformations.\nOften, users have to deal with very large PyTrees and it would become very burdensome to specify transformations for large numbers of keys. Our library provides two solutions to this: regexes and implicit transformations.\nImplicit transformations have been alluded to in our first example, but let’s focus on them specifically.\nIn the following example, transformations is an empty dictionary, so we rely exclusively on implicit transformations. Key/value pairs present in new_tree but not in original_tree simply remain in place in the result, while a key present in original_tree but not in new_tree will be dropped from the result.\n\noriginal_tree = {\n 'a': 1,\n 'b': {\n 'c': 5,\n 'd': 6,\n 'e': 7,\n },\n 'f': 2,\n}\ntransformations = {}\nnew_tree = {\n 'a': ...,\n 'b': {\n 'c': ...,\n },\n 'f': ...,\n 'g': {\n 'h': 3,\n 'i': 4,\n },\n}\n\norbax.checkpoint.apply_transformations(original_tree, transformations, new_tree)\n\nWe can also change the default_to_original argument to customize the behavior when keys are unspecified in the transformations tree. Setting default_to_original=False means that unspecified keys will be taken from new_tree, not original_tree.\nThis can be useful if we just want to take a few values from our original checkpoint, while using the rest from our new state.\n\noriginal_tree = {\n 'a': 1,\n 'b': {\n 'c': 5,\n 'd': 6,\n 'e': 7,\n },\n 'f': 2,\n}\ntransformations = {'a': Transform(value_fn=lambda x: x*10)}\nnew_tree = {\n 'a': 11,\n 'b': {\n 'c': 12,\n 'd': 13,\n 'e': 14,\n },\n 'f': 15,\n}\norbax.checkpoint.apply_transformations(original_tree, transformations, new_tree, default_to_original=False)\n\nReturning to our other feature: support for regexes. Real model states often represent a conceptual parameter (a single layer, perhaps) with multiple actual key/value pairs. In this case, it can be useful to use a regex to refer to the parameter in question.\nIn the following example, we have one model (conceputally a pretrained checkpoint) with two layers, and another model (conceputally our in-memory model state) with four layers. We would like to insert the two layers of the checkpoint as the middle two layers of the new state, while leaving the bottom and top layers randomly initialized.\n\nimport flax.linen as nn\nfrom orbax.checkpoint import test_utils\n\n\nfrom flax.training.train_state import TrainState\nimport optax\nfrom jax import numpy as jnp\n\ndef init_flax_model(model):\n params = model.init(jax.random.PRNGKey(0), jnp.ones([8, 8]))\n tx = optax.adamw(learning_rate=0.001)\n state = TrainState.create(apply_fn=model.apply, params=params, tx=tx)\n return jax.tree_util.tree_map(np.asarray, state)\n\nclass SmallModel(nn.Module):\n @nn.compact\n def __call__(self, x):\n x = x.reshape((x.shape[0], -1))\n x = nn.Dense(features=8)(x)\n x = nn.sigmoid(x)\n x = nn.Dense(features=8)(x)\n return x\n\nold_state = init_flax_model(SmallModel())\n# multiply by 100 to represent \"training\"\nold_state = jax.tree_util.tree_map(lambda x: x * 100, old_state)\n\nclass LargeModel(nn.Module):\n @nn.compact\n def __call__(self, x):\n x = x.reshape((x.shape[0], -1)) # flatten\n x = nn.Dense(features=8)(x)\n x = nn.sigmoid(x)\n x = nn.Dense(features=4)(x)\n x = nn.sigmoid(x)\n x = nn.Dense(features=4)(x)\n x = nn.sigmoid(x)\n x = nn.Dense(features=2)(x)\n return x\n\nnew_state = init_flax_model(LargeModel())\n\n\ntransformations = {\n # NewModel layer_0 is a newly inserted layer, thus use_fallback=True.\n # The \"fallback\" tree in this case is new_tree.\n # Since the layer_0 has the same name in old and new, we need to provide\n # an indication that the value of layer_0 should come from new_tree rather\n # than original_tree.\n r'(.*)Dense_0(.*)': Transform(use_fallback=True),\n # OriginalModel layer 0 maps to NewModel layer 1\n r'(.*)Dense_1(.*)': Transform(original_key=r'\\1Dense_0\\2'),\n # OriginalModel layer 1 maps to NewModel layer 2\n r'(.*)Dense_2(.*)': Transform(original_key=r'\\1Dense_1\\2')\n} # Note: NewModel layer 3 is newly added.\nrestored_state = orbax.checkpoint.apply_transformations(old_state, transformations, new_state)\nprint(restored_state.params['params'])\n\nWe can see in the result that layers 0 and 3 are still small, while layers 1 and 2 have large values, coming from the original checkpoint." + }, + { + "objectID": "unpublished/p-values_and_parameters_with_limits.html", + "href": "unpublished/p-values_and_parameters_with_limits.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "from iminuit import Minuit\nfrom iminuit.cost import ExtendedBinnedNLL\nimport numpy as np\nimport numba as nb\nfrom matplotlib import pyplot as plt\nfrom progressbar import progressbar\nfrom pyik.mplext import plot_hist\nfrom IPython.display import display\nfrom scipy.stats import norm\n\n\n# numba currently does not support scipy, so we cannot access\n# scipy.stats.norm.cdf in a JIT'ed function. As a workaround,\n# we wrap special functions from scipy to implement the needed\n# functions here.\ndef get(name, narg):\n from numba.extending import get_cython_function_address\n import ctypes\n addr = get_cython_function_address(\"scipy.special.cython_special\", name)\n functype = ctypes.CFUNCTYPE(ctypes.c_double, *([ctypes.c_double] * narg))\n return functype(addr)\n\n\nerf = get(\"__pyx_fuse_1erf\", 1)\n\n\n@nb.njit\ndef norm_cdf(x, mu, sigma):\n z = (x - mu) / (sigma * np.sqrt(2))\n r = np.empty_like(x)\n for i, zi in enumerate(z):\n r[i] = 0.5 * (1.0 + erf(zi))\n return r\n\n\ndef model(x, sig, bkg):\n return sig * norm_cdf(x, 0.5, 0.1) + bkg * x\n\n\nbkg = 200\nxe = np.linspace(0, 1, 21)\nn = np.diff(model(xe, 20, bkg))\n\nnll = ExtendedBinnedNLL(n, xe, model)\n\nm1 = Minuit(nll, 1, 1)\nm1.limits = (0, None)\ndisplay(m1.migrad())\nm2 = Minuit(nll, 1, 1)\nm2.limits[1] = (0, None)\ndisplay(m2.migrad())\nm3 = Minuit(nll, 0, 1)\nm3.fixed[0] = True\nm3.limits = (0, None)\ndisplay(m3.migrad())\nexp1 = m3.fval - m1.fval, m1.values[0], m1.errors[0]\nexp2 = m3.fval - m2.fval, m2.values[0], m2.errors[0]\n\n\n\n\nFCN = 3.861e-06\nNfcn = 91\n\n\nEDM = 3.86e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nsig\n20\n12\n\n\n0\n\n\n\n\n1\nbkg\n200\n18\n\n\n0\n\n\n\n\n\n\n\n\n\nsig\nbkg\n\n\nsig\n138\n-118 (-0.564)\n\n\nbkg\n-118 (-0.564)\n318\n\n\n\n\n\n\n\n\nFCN = 2.091e-06\nNfcn = 99\n\n\nEDM = 2.09e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nsig\n20\n12\n\n\n\n\n\n\n\n1\nbkg\n200\n18\n\n\n0\n\n\n\n\n\n\n\n\n\nsig\nbkg\n\n\nsig\n138\n-118 (-0.564)\n\n\nbkg\n-118 (-0.564)\n318\n\n\n\n\n\n\n\n\nFCN = 3.132\nNfcn = 42\n\n\nEDM = 2.31e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nsig\n0.0\n0.1\n\n\n0\n\nyes\n\n\n1\nbkg\n220\n15\n\n\n0\n\n\n\n\n\n\n\n\n\nsig\nbkg\n\n\nsig\n0\n0\n\n\nbkg\n0\n220\n\n\n\n\n\n\nplot_hist(xe, n)\n\n\n\n\n\nprint(f\"with limit : dQ = {exp1[0]:.3g} \"\n f\"significance of signal amplitude = {exp1[1] / exp1[2]:.1f}σ\")\nprint(f\"without limit: dQ = {exp2[0]:.3g} \"\n f\"significance of signal amplitude = {exp2[1] / exp2[2]:.1f}σ\")\n\nwith limit : dQ = 3.13 significance of signal amplitude = 1.7σ\nwithout limit: dQ = 3.13 significance of signal amplitude = 1.7σ\n\n\n\nrng = np.random.default_rng(1)\nvbkg = np.diff(model(xe, 0, bkg))\nr1 = []\nr2 = []\nfor itoy in progressbar(range(50000)):\n nll.n = rng.poisson(vbkg)\n m1 = Minuit(nll, 0, bkg)\n m1.limits = (0, None)\n m1.migrad()\n m2 = Minuit(nll, 0, bkg)\n m2.limits[1] = (0, None)\n m2.migrad()\n m3 = Minuit(nll, 0, bkg)\n m3.fixed[0] = True\n m3.limits = (0, None)\n m3.migrad()\n if m3.valid:\n if m2.valid:\n r1.append((m3.fval - m1.fval, m1.values[0]))\n if m2.valid:\n r2.append((m3.fval - m2.fval, m2.values[0]))\nr1 = np.array(r1)\nr2 = np.array(r2)\n\n100% (50000 of 50000) |##################| Elapsed Time: 0:01:40 Time: 0:01:40\n\n\n\nplt.figure()\nplt.title(\"dQ distribution\")\nplt.hist(r1[:,0], bins=50, alpha=0.5, density=True, label=\"with limits\")\nplt.hist(r2[:,0], bins=50, alpha=0.5, density=True, label=\"without limits\")\nplt.legend()\nplt.axvline(exp1[0], color=\"C0\")\nplt.axvline(exp2[0], color=\"C1\")\nplt.semilogy()\n\nplt.figure()\nplt.title(\"signal amplitude\")\nplt.hist(r1[:,1], bins=50, alpha=0.5, density=True, label=\"with limits\")\nplt.hist(r2[:,1], bins=50, alpha=0.5, density=True, label=\"without limits\")\nplt.legend()\nplt.axvline(exp1[1], color=\"C0\")\nplt.axvline(exp2[1], color=\"C1\")\nplt.semilogy();\n\n\n\n\n\n\n\n\ndef pvalue2sigma(p):\n return norm.ppf(1 - p)\n\n\np1 = np.sum(r1[:,1] > exp1[1])/len(r1)\np2 = np.sum(r2[:,1] > exp2[1])/len(r2)\nprint(f\"p-value based on amplitude with limit {p1:.3f} significance = {pvalue2sigma(p1):.1f}σ\")\nprint(f\"p-value based on amplitude without limit {p2:.3f} significance = {pvalue2sigma(p2):.1f}σ\")\n\np-value based on amplitude with limit 0.031 significance = 1.9σ\np-value based on amplitude without limit 0.031 significance = 1.9σ\n\n\n\np1 = np.sum(r1[:,0] > exp1[0])/len(r1)\np2 = np.sum(r2[:,0] > exp1[0])/len(r2)\nprint(f\"p-value based on dQ with limit {p1:.3f} significance = {pvalue2sigma(p1):.1f}σ\")\nprint(f\"p-value based on dQ without limit {p2:.3f} significance = {pvalue2sigma(p2):.1f}σ\")\n\np-value based on dQ with limit 0.037 significance = 1.8σ\np-value based on dQ without limit 0.077 significance = 1.4σ\n\n\nThe p-value computed from the fitted amplitude should agree with the p-value computed from dQ. This is only the case if the fit imposes a limit on the signal amplitude." + }, + { + "objectID": "unpublished/finance.html", + "href": "unpublished/finance.html", + "title": "ETF analysis", + "section": "", + "text": "import yfinance as yf\nluxor = yf.download(\"X010.DE\", period=\"max\")\nishares = yf.download(\"EUNL.DE\", period=\"max\")\n\n[*********************100%***********************] 1 of 1 completed\n[*********************100%***********************] 1 of 1 completed\nfrom matplotlib import pyplot as plt\n\nplt.plot(luxor.index, luxor[\"Open\"], label=\"luxor\")\nplt.plot(ishares.index, ishares[\"Open\"], label=\"ishares\")\nplt.legend()\nplt.ylabel(\"price\")\nplt.xlabel(\"date\")\n\nText(0.5, 0, 'date')\ndiff = (luxor[\"Open\"] - ishares[\"Open\"]) / luxor[\"Open\"]\nplt.plot(diff.index, diff)\nplt.ylim(0, 0.15)\nplt.xlabel(\"date\")\nplt.ylabel(\"(Luxor - iShares) / Luxor\")\n\nText(0, 0.5, '(Luxor - iShares) / Luxor')" + }, + { + "objectID": "unpublished/finance.html#monthly-delta", + "href": "unpublished/finance.html#monthly-delta", + "title": "ETF analysis", + "section": "Monthly delta", + "text": "Monthly delta\n\nfrom datetime import date, timedelta\n\nstart = date(luxor.index[0].year, luxor.index[0].month, 1)\nend = date(luxor.index[-1].year, luxor.index[-1].month, 31)\ndt = timedelta(days=1)\n\nmonths = {}\n\nfor i, v in zip(luxor.index, luxor[\"Open\"]):\n months.setdefault((i.year, i.month), []).append(\n (i.day, v)\n )\n\n\nlen(months)\n\n173\n\n\n\nm2 = []\nfor pairs in months.values():\n days, values = np.transpose(pairs)\n m = np.mean(values)\n m2.append(\n (days, values / m - 1)\n )\n\nfor days, vals in m2:\n plt.plot(days, vals)\n\nplt.ylim(-0.1, 0.1)\n\n(-0.1, 0.1)\n\n\n\n\n\n\nimport numpy as np\n\ndata = {}\n\nfor days, vals in m2:\n for d, v in zip(days.astype(int), vals):\n data.setdefault(d, []).append(v)\n\nstats = {}\nfor d, vals in data.items():\n stats[d] = (np.mean(vals), np.std(vals), np.median(vals))\n\n\nplt.errorbar(stats.keys(), [v[0] for v in stats.values()], [v[1] for v in stats.values()], fmt=\"o\", label=\"mean\")\nplt.plot(stats.keys(), [v[2] for v in stats.values()], \"s\", mfc=\"none\", label=\"median\", zorder=5)\nplt.legend()\nplt.axhline(0, ls=\"--\", color=\"0.5\")\nplt.xlabel(\"day in month\")\nplt.ylabel(\"deviation from average price per month\")\n\nText(0, 0.5, 'deviation from average price per month')\n\n\n\n\n\n\nluxor_info = yf.Ticker(\"X010.DE\").info\nishares_info = yf.Ticker(\"EUNL.DE\").info\n\n\nluxor_info[\"averageDailyVolume10Day\"]\n\n57938\n\n\n\nishares_info[\"averageDailyVolume10Day\"]\n\n464867" + }, + { + "objectID": "unpublished/plot_random_table_results.html", + "href": "unpublished/plot_random_table_results.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "import numpy as np\nimport matplotlib.pyplot as plt\n\n\ndata = \"\"\"\n method ntot 4 64 256 1024\nboyett 10 87.4±0.1μs 96.7±0.08μs 180±0.8μs 472±0.7μs\nboyett 100 919±2μs 864±2μs 962±3μs 1.26±0ms\nboyett 1000 9.04±0.01ms 8.12±0.01ms 7.99±0.03ms 8.15±0.03ms\nboyett 10000 99.0±0.4ms 89.9±0.2ms 86.9±0.3ms 86.4±0.6ms\n boyett2 10 17.0±0.1ms failed failed failed\n boyett2 100 22.9±0.2ms 23.6±0.05ms 24.4±0.2ms failed\n boyett2 1000 85.4±0.09ms 93.0±0.4ms 97.6±0.5ms 102±0.8ms\n boyett2 10000 722±1ms 798±2ms 846±4ms 904±4ms\npatefield 10 70.6±0.2μs 852±10μs 2.22±0ms 5.44±0ms\npatefield 100 98.8±0.08μs 3.65±0.03ms 13.1±0.2ms 41.9±0.4ms\npatefield 1000 122±0.2μs 4.53±0.01ms 19.5±0.01ms 71.9±0.3ms\npatefield 10000 156±0.2μs 5.39±0ms 22.6±0.02ms 90.7±0.3ms\npatefield2 10 112±2μs 1.17±0ms 2.94±0ms 6.48±0.02ms\npatefield2 100 157±2μs 6.26±0.01ms 22.0±0.02ms 71.9±0.4ms\npatefield2 1000 166±0.9μs 7.13±0.05ms 32.1±0.1ms 118±1ms\npatefield2 10000 201±0.8μs 7.53±0.02ms 32.6±0.2ms 137±0.1ms\n\"\"\"\n\n\nd = {}\nncell = None\nntot = []\nfor line in data.split(\"\\n\"):\n if not line or line.isspace():\n continue\n method, ntoti, *times = [x for x in line.split() if x]\n if ncell is None:\n ncell = [int(x) for x in times]\n continue\n else:\n ntoti = int(ntoti)\n if ntoti not in ntot:\n ntot.append(ntoti)\n times = [(float(x.split(\"±\")[0]) * (1e-3 if x.endswith(\"μs\") else 1)) if x != \"failed\" else np.nan for x in times]\n d.setdefault(method, {})[ntoti] = times\n\ntimes = {\n key: np.empty((len(ntot), len(ncell))) for key in d\n}\n\nntot = np.sort(ntot)\nncell = np.array(ncell)\n\nfor key in d:\n v = times[key]\n for i, ntoti in enumerate(ntot):\n v[i] = d[key][ntoti]\n\n\nfig, ax = plt.subplots(1, 2, figsize=(10, 4), sharey=True)\n\nplt.sca(ax[0])\nfor key, val in times.items():\n plt.plot(ntot, val[:, 0], label=key)\nplt.loglog()\nplt.legend(title=\"ncell = 4\")\nplt.ylabel(\"run time / ms\")\nplt.xlabel(\"total number of entries\")\n\nplt.sca(ax[1])\nfor key, val in times.items():\n plt.plot(ncell, val[-1], label=key)\nplt.loglog()\nplt.legend(title=\"ntot = 10000\")\nplt.xlabel(\"total number of cells\")\n\nText(0.5, 0, 'total number of cells')\n\n\n\n\n\n\nfrom scipy.optimize import least_squares\n\ndef patefield(ntot, ncell, par):\n return par[0] * np.outer(np.log(ntot) ** par[1], ncell)\n\ndef boyett(ntot, ncell, par):\n return par * np.outer(ntot, np.ones_like(ncell))\n\nmodel = {}\nr1 = least_squares(lambda par: np.log(patefield(ntot, ncell, par) / times[\"patefield\"]).ravel(), (1.0, 1.0), bounds=np.transpose([(0, np.inf), (-2, 2)]))\nassert r1.success\nprint(r1)\nmodel[\"patefield\"] = lambda ntot, ncell: patefield(ntot, ncell, r1.x)\n\nr2 = least_squares(lambda par: np.log(boyett(ntot, ncell, par) / times[\"boyett\"]).ravel(), 2.0, bounds=(0, np.inf))\nassert r2.success\nmodel[\"boyett\"] = lambda ntot, ncell: boyett(ntot, ncell, r2.x[0])\n\nfig, ax = plt.subplots(1, 2, figsize=(10, 4), sharey=True)\n\nplt.sca(ax[0])\nicell = 0\nfor key, val in times.items():\n l, = plt.plot(ntot, val[:, icell], \"o\", label=key)\n if key in model:\n m = model[key]\n plt.plot(ntot, m(ntot, ncell)[:, icell], \"-\", color=l.get_color())\nplt.loglog()\nplt.legend(title=f\"ncell = {ncell[icell]}\")\nplt.ylabel(\"run time / ms\")\nplt.xlabel(\"total number of entries\")\n\nplt.sca(ax[1])\nintot = -1\nfor key, val in times.items():\n l, = plt.plot(ncell, val[intot], \"o\", label=key)\n if key in model:\n m = model[key]\n plt.plot(ncell, m(ntot, ncell)[intot], \"-\", color=l.get_color())\nplt.loglog()\nplt.legend(title=f\"ntot = {ntot[intot]}\")\nplt.xlabel(\"total number of cells\")\n\n active_mask: array([0, 0])\n cost: 1.3946620974325732\n fun: array([-4.03239230e-01, -1.21206891e-01, 3.07411523e-01, 7.97434019e-01,\n 2.43799745e-01, -5.92996375e-01, -4.84587076e-01, -2.60966312e-01,\n 6.07956704e-01, -2.33910748e-01, -3.07308913e-01, -2.25876351e-01,\n 7.70147778e-01, 2.91843448e-04, -4.68183168e-02, -5.01314065e-02])\n grad: array([-1.91174556e-06, -5.01863655e-08])\n jac: array([[276.76918167, 0.83403245],\n [276.76918168, 0.83403245],\n [276.76918167, 0.83403244],\n [276.76918168, 0.83403245],\n [276.76918167, 1.52717962],\n [276.76918168, 1.52717963],\n [276.76918167, 1.52717962],\n [276.76918168, 1.52717963],\n [276.76918166, 1.93264473],\n [276.76918167, 1.93264474],\n [276.76918168, 1.93264474],\n [276.76918167, 1.93264474],\n [276.76918167, 2.2203268 ],\n [276.76918168, 2.2203268 ],\n [276.76918167, 2.2203268 ],\n [276.76918167, 2.2203268 ]])\n message: '`ftol` termination condition is satisfied.'\n nfev: 18\n njev: 16\n optimality: 1.911745556526512e-06\n status: 2\n success: True\n x: array([0.00361311, 1.41832278])\n\n\nText(0.5, 0, 'total number of cells')\n\n\n\n\n\n\nfor ntoti in ntot:\n for ncelli in ncell:\n print(ntoti, ncelli, model[\"boyett\"](ntoti, ncelli) / ntoti / (model[\"patefield\"](ntoti, ncelli) / (np.log(ntoti) * ncelli)))\n\n10 4 [[2.05157943]]\n10 64 [[2.05157943]]\n10 256 [[2.05157943]]\n10 1024 [[2.05157943]]\n100 4 [[1.53518469]]\n100 64 [[1.53518469]]\n100 256 [[1.53518469]]\n100 1024 [[1.53518469]]\n1000 4 [[1.29567967]]\n1000 64 [[1.29567967]]\n1000 256 [[1.29567967]]\n1000 1024 [[1.29567967]]\n10000 4 [[1.14876958]]\n10000 64 [[1.14876958]]\n10000 256 [[1.14876958]]\n10000 1024 [[1.14876958]]" + }, + { + "objectID": "posts/Uncertainty of efficiency computed from fitted yields.html", + "href": "posts/Uncertainty of efficiency computed from fitted yields.html", + "title": "Uncertainty of efficiency from fitted decay yields", + "section": "", + "text": "We derive an approximate formula to compute the uncertainty of the efficiency computed from fitted yields of some decays. The formula is suitable to draw error bars in plots.\n#!pip install numpy iminuit numba_stats matplotlib scipy sympy joblib\nimport numpy as np\nfrom iminuit import Minuit\nfrom iminuit.cost import ExtendedBinnedNLL, NormalConstraint\nfrom numba_stats import bernstein, norm\nfrom matplotlib import pyplot as plt\nimport joblib\nfrom sympy import *" + }, + { + "objectID": "posts/Uncertainty of efficiency computed from fitted yields.html#introduction", + "href": "posts/Uncertainty of efficiency computed from fitted yields.html#introduction", + "title": "Uncertainty of efficiency from fitted decay yields", + "section": "Introduction", + "text": "Introduction\nThe efficiency of a selection is\n\\[\n\\epsilon = \\frac{k}{n},\n\\]\nwhere \\(k\\) is the number of events which pass and \\(n\\) is the total number of events.\nFor the sake of computing confidence intervals for an efficiency computed from a sample, \\(\\epsilon\\) is typically assumed to be a binomial proportion. This means that \\(k\\) is binomially distributed for fixed \\(n\\) in repeated experiments.\nIn practice, this assumption is usually incorrect. The efficiency is usually computed from a total sample of events whose number \\(n\\) is Poisson distributed, since experiments that involve random processes are typically run for a fixed time rather than until a fixed number of events has occured. This produces a larger variance for \\(\\epsilon\\) compared to what is expected from the binomial model.\nWe demonstrate this numerically.\n\nrng = np.random.default_rng(1)\n\nx = np.linspace(0, 5, 100)\n\n# sigmoid\np = 1.0 / (1.0 + np.exp(-x))\n\nplt.plot(x, p, color=\"k\", label=r\"$\\epsilon(x) = 1/(1 + e^{-x})$\")\n\nntry = 10000\nfor i, n in enumerate((5, 10, 100)):\n eff1 = []\n eff2 = []\n for pi in p:\n y1 = rng.binomial(n, pi, size=ntry) / n\n\n ni = rng.poisson(n, size=ntry)\n with np.errstate(invalid=\"ignore\"):\n y2 = rng.binomial(ni, pi) / ni\n\n for y, eff in [(y1, eff1), (y2, eff2)]:\n# q = np.nanquantile(y, [0.16, 1-0.16])\n m = np.nanmean(y)\n s = np.nanstd(y)\n eff.append((m - s, m + s))\n eff1 = np.array(eff1)\n eff2 = np.array(eff2)\n\n col = f\"C{i}\"\n for eff, ls, label in ((eff1, \":\", f\"n={n} fixed\"), \n (eff2, \"--\", f\"poisson n with $\\\\langle n \\\\rangle = {n}$ \")):\n plt.plot(x, eff[:, 0], color=col, ls=ls, label=label)\n plt.plot(x, eff[:, 1], color=col, ls=ls)\n\n print(np.mean(np.diff(eff2, axis=1) / np.diff(eff1, axis=1)))\n\nplt.legend()\nplt.xlabel(r\"$x$\")\nplt.ylabel(r\"$\\epsilon$\");\n\n1.1395442804276614\n1.0638370186185993\n1.0046267505795776\n\n\n\n\n\nShown is an sigmoidal efficiency curve and a normal approximation interval computed from the standard deviation of the sample estimate \\(\\hat \\epsilon\\). For small \\(n\\), the interval is wider if \\(n\\) is Poisson distributed.\nWe compute the variance of \\(\\epsilon\\) for both cases explicitly. In the following, it will be important to clearly distinguish sample values from their population values. We will use \\(\\hat x\\) for the sample value, which has a population value \\(x\\). The sample value of the efficiency then is \\(\\hat \\epsilon = \\hat k / \\hat n\\)." + }, + { + "objectID": "posts/Uncertainty of efficiency computed from fitted yields.html#no-background-n-fixed", + "href": "posts/Uncertainty of efficiency computed from fitted yields.html#no-background-n-fixed", + "title": "Uncertainty of efficiency from fitted decay yields", + "section": "No background, n fixed", + "text": "No background, n fixed\nIf \\(\\hat n = n\\) is fixed, the variance of \\(\\hat \\epsilon\\) can be computed from the variance of the binomial distribution \\(\\text{var}(\\hat k) = n \\epsilon (1 - \\epsilon)\\). We expand the formula for \\(\\epsilon\\) in a Taylor series in the variable \\(k = \\epsilon n\\) around the sample value \\(\\hat k\\) and obtain\n\\[\n\\epsilon = \\sum_{i=0}^{\\infty} \\frac {(k - \\hat k) ^i}{i!} \\frac{\\partial^i \\epsilon}{\\partial {k}^i}(\\hat k) = \\hat \\epsilon + \\frac {k - \\hat k} n.\n\\]\nThe series naturally break off after the first order. Thanks to that, we can compute the variance of \\(\\hat \\epsilon\\) exactly,\n\\[\n\\text{var}(\\hat\\epsilon) = \\text{E}\\big((\\hat \\epsilon - \\epsilon)^2\\big) = \\text{E}\\left(\\frac{(\\hat k - k)^2}{n^2}\\right) = \\frac{\\text{E}\\big((\\hat k - k)^2\\big)}{n^2} = \\frac{\\text{var}(\\hat k)}{n^2} = \\frac{\\epsilon (1 - \\epsilon)}{n}\n\\]\nThe same result is obtain with the standard error propagation formula, which is exact in this case, since \\(\\epsilon\\) is a linear function of \\(k\\). We performed the calculation from first principles instead of using this formula, because it is instructive for the next step, where \\(\\epsilon\\) becomes a non-linear function." + }, + { + "objectID": "posts/Uncertainty of efficiency computed from fitted yields.html#no-background-n-poisson-distributed", + "href": "posts/Uncertainty of efficiency computed from fitted yields.html#no-background-n-poisson-distributed", + "title": "Uncertainty of efficiency from fitted decay yields", + "section": "No background, n Poisson distributed", + "text": "No background, n Poisson distributed\nTo simplify the calculation, we use that the efficiency can be written in terms of two independently Poisson-distributed numbers, the number of successes \\(n_1\\) and failures \\(n_2\\),\n\\[\n\\epsilon = \\frac{n_1}{n_1 + n_2} = \\frac{n_1}{n},\n\\]\nwith \\(n = n_1 + n_2\\).\nIn a particular sample, the sample values \\(\\hat n_k\\) will deviate from the expectation values \\(n_k\\) by random offsets \\(d_k = \\hat n_k - n_k\\). This has an effect on the estimator \\(\\hat \\epsilon(\\hat n_1, \\hat n_2) = \\hat n_1 / (\\hat n_1 + \\hat n_2)\\), that we can evaluate by expanding the formula around the true values in a two-dimensional Taylor series with \\(\\hat n_k = n_k\\), treating the integer values \\(\\hat n_k\\) like real numbers,\n\\[\n\\hat \\epsilon = \\sum_{i=0}^\\infty\\sum_{j=0}^\\infty \\frac {d_1^i \\,d_2^j}{i! \\, j!} \\frac{\\partial^{i+j} \\hat \\epsilon}{\\partial \\hat n_1^i \\partial \\hat n_2^j} (n_1, n_2)\n\\]\n\ndef bivariate_series(expr, x, y, dx, dy, order_max):\n return sum(\n sum(\n dx ** i * dy ** j / \n (factorial(i) * factorial(j)) *\n diff(expr, x, i, y, j)\n for i in range(order_max + 1)\n if (i + j) <= order_max\n ) for j in range(order_max + 1)\n )\n\n\nn1, n2, n, epsilon = symbols(\"n_1 n_2 n epsilon\", real=True, nonnegative=True)\nd1, d2 = symbols(\"d_1 d_2\", real=True)\n\neps = n1 / (n1 + n2)\n\ndelta_eps = (bivariate_series(eps, n1, n2, d1, d2, 7) - eps)\ndelta_eps = delta_eps.subs([(n1, epsilon * n), (n2, (1-epsilon) * n)])\ndelta_eps = delta_eps.as_poly([d1, d2])\ndelta_eps\n\n\\(\\displaystyle \\operatorname{Poly}{\\left( \\frac{1 - \\epsilon}{n^{7}} d_{1}^{7} + \\frac{6 - 7 \\epsilon}{n^{7}} d_{1}^{6}d_{2} + \\frac{\\epsilon - 1}{n^{6}} d_{1}^{6} + \\frac{15 - 21 \\epsilon}{n^{7}} d_{1}^{5}d_{2}^{2} + \\frac{6 \\epsilon - 5}{n^{6}} d_{1}^{5}d_{2} + \\frac{1 - \\epsilon}{n^{5}} d_{1}^{5} + \\frac{20 - 35 \\epsilon}{n^{7}} d_{1}^{4}d_{2}^{3} + \\frac{15 \\epsilon - 10}{n^{6}} d_{1}^{4}d_{2}^{2} + \\frac{4 - 5 \\epsilon}{n^{5}} d_{1}^{4}d_{2} + \\frac{\\epsilon - 1}{n^{4}} d_{1}^{4} + \\frac{15 - 35 \\epsilon}{n^{7}} d_{1}^{3}d_{2}^{4} + \\frac{20 \\epsilon - 10}{n^{6}} d_{1}^{3}d_{2}^{3} + \\frac{6 - 10 \\epsilon}{n^{5}} d_{1}^{3}d_{2}^{2} + \\frac{4 \\epsilon - 3}{n^{4}} d_{1}^{3}d_{2} + \\frac{1 - \\epsilon}{n^{3}} d_{1}^{3} + \\frac{6 - 21 \\epsilon}{n^{7}} d_{1}^{2}d_{2}^{5} + \\frac{15 \\epsilon - 5}{n^{6}} d_{1}^{2}d_{2}^{4} + \\frac{4 - 10 \\epsilon}{n^{5}} d_{1}^{2}d_{2}^{3} + \\frac{6 \\epsilon - 3}{n^{4}} d_{1}^{2}d_{2}^{2} + \\frac{2 - 3 \\epsilon}{n^{3}} d_{1}^{2}d_{2} + \\frac{\\epsilon - 1}{n^{2}} d_{1}^{2} + \\frac{1 - 7 \\epsilon}{n^{7}} d_{1}d_{2}^{6} + \\frac{6 \\epsilon - 1}{n^{6}} d_{1}d_{2}^{5} + \\frac{1 - 5 \\epsilon}{n^{5}} d_{1}d_{2}^{4} + \\frac{4 \\epsilon - 1}{n^{4}} d_{1}d_{2}^{3} + \\frac{1 - 3 \\epsilon}{n^{3}} d_{1}d_{2}^{2} + \\frac{2 \\epsilon - 1}{n^{2}} d_{1}d_{2} + \\frac{1 - \\epsilon}{n} d_{1} - \\frac{\\epsilon}{n^{7}} d_{2}^{7} + \\frac{\\epsilon}{n^{6}} d_{2}^{6} - \\frac{\\epsilon}{n^{5}} d_{2}^{5} + \\frac{\\epsilon}{n^{4}} d_{2}^{4} - \\frac{\\epsilon}{n^{3}} d_{2}^{3} + \\frac{\\epsilon}{n^{2}} d_{2}^{2} - \\frac{\\epsilon}{n} d_{2}, d_{1}, d_{2}, domain=\\mathbb{Z}\\left(n, \\epsilon\\right) \\right)}\\)\n\n\nTo compute the variance \\(\\text{var}(\\hat \\epsilon)\\) we need to square this and compute the expection on both sides. We use that \\(\\text{E}(d_1^i d_2^j) = \\text{E}(d_1^i) \\, \\text{E}(d_2^j)\\), since \\(\\hat n_1\\) and \\(\\hat n_2\\) are independently sampled. The expectation value \\(\\text{E}(d_k^j)\\) is the \\(j\\)-th central moment of the Poisson distribution with \\(\\lambda = n_k\\). These moments are Touchard polynomials, which are conveniently computed with a recursion formula.\n\n# compute mapping d_k ** i -> E[d_k ** i]\n\ndef central_moments(n):\n x = symbols(\"x\")\n r = [1]\n for ni in range(n):\n # recursion for Touchard polynomial\n m = x * sum(binomial(ni, k) * r[k] for k in range(ni))\n r.append(m.expand())\n\n return [lambdify([x], ri) for ri in r]\n\nreplacements = central_moments(14)\n\nNow we compute \\(\\text{var}(\\hat \\epsilon) = \\text{E}\\big((\\hat \\epsilon - \\epsilon)^2\\big)\\). Since SymPy does not know how to compute an expectation and that it is a linear operator, we need to resort to advanced expression manipulation in the following steps. We use that the expectation is linear and \\(\\text{E}(n_i n_j) = \\text{E}(n_i) \\, \\text{E}(n_j)\\) for independent \\(n_i\\) and \\(n_j\\).\n\ndef expectation(expr, args, args2, replacements):\n poly = expr.as_poly(args)\n expr2 = []\n for powers, coef in poly.terms():\n x = prod(\n 1 if pi == 0 else replacements[pi](argi)\n for pi, argi in zip(powers, args2)\n )\n expr2.append(coef * x)\n return sum(expr2)\n\n\n# replace powers of d_k with the corresponding central moments\nexpr2 = expectation(delta_eps ** 2, (d1, d2), (epsilon * n, (1-epsilon)*n), replacements)\nexpr2 = expr2.as_poly(1/n)\nexpr2\n\n\\(\\displaystyle \\operatorname{Poly}{\\left( \\left(- \\epsilon^{2} + \\epsilon\\right) \\frac{1}{n^{13}} + \\left(- 4081 \\epsilon^{2} + 4081 \\epsilon\\right) \\frac{1}{n^{12}} + \\left(- 233058 \\epsilon^{2} + 233058 \\epsilon\\right) \\frac{1}{n^{11}} + \\left(- 1413895 \\epsilon^{2} + 1413895 \\epsilon\\right) \\frac{1}{n^{10}} + \\left(- 1356075 \\epsilon^{2} + 1356075 \\epsilon\\right) \\frac{1}{n^{9}} + \\left(- 130095 \\epsilon^{2} + 130095 \\epsilon\\right) \\frac{1}{n^{8}} + \\left(720 \\epsilon^{2} - 720 \\epsilon\\right) \\frac{1}{n^{7}} + \\left(118 \\epsilon^{2} - 118 \\epsilon\\right) \\frac{1}{n^{6}} + \\left(- 24 \\epsilon^{2} + 24 \\epsilon\\right) \\frac{1}{n^{5}} + \\left(- 6 \\epsilon^{2} + 6 \\epsilon\\right) \\frac{1}{n^{4}} + \\left(- 2 \\epsilon^{2} + 2 \\epsilon\\right) \\frac{1}{n^{3}} + \\left(- \\epsilon^{2} + \\epsilon\\right) \\frac{1}{n^{2}} + \\left(- \\epsilon^{2} + \\epsilon\\right) \\frac{1}{n}, \\frac{1}{n}, domain=\\mathbb{Z}\\left[\\epsilon\\right] \\right)}\\)\n\n\nWe keep only terms up to \\(O(1/n^2)\\).\n\nexpr3 = []\nfor (p, ), coef in expr2.terms():\n if p <= 2:\n expr3.append(coef * n ** -p)\neps_var = sum(expr3).subs(epsilon, n1/n).subs(n, n1 + n2).simplify()\neps_var\n\n\\(\\displaystyle \\frac{n_{1} n_{2} \\left(n_{1} + n_{2} + 1\\right)}{\\left(n_{1} + n_{2}\\right)^{4}}\\)\n\n\nUp to the extra factor \\((n + 1)/n\\), this is the variance of the binomial distribution we derived previously.\nThis formula is accurate for \\(n > 10\\). To get accurate results at even smaller \\(n\\), we need to include to higher orders.\n\nexpr3 = []\nfor (p, ), coef in expr2.terms():\n if p <= 4:\n expr3.append(coef * n ** -p)\neps_var = sum(expr3).subs(epsilon, n1/n).subs(n, n1 + n2).simplify()\neps_var\n\n\\(\\displaystyle \\frac{n_{1} n_{2} \\left(2 n_{1} + 2 n_{2} + \\left(n_{1} + n_{2}\\right)^{3} + \\left(n_{1} + n_{2}\\right)^{2} + 6\\right)}{\\left(n_{1} + n_{2}\\right)^{6}}\\)\n\n\nThis formula is accurate down to \\(n \\ge 5\\).\nThe values \\(n_k\\) that appear in the formula are the unknown expectation values and need to be replaced by their sample estimates \\(\\hat n_k\\) in practice. This is an application of the plug-in principle on which also bootstrap theory is based. There is a small bias involved in doing this, but it is negligible here.\n\neps_var_py = lambdify((n1, n2), eps_var)\n\n\nrng = np.random.default_rng(1)\neps = np.linspace(0, 1, 100)\ndata = {}\nfor n in (5, 10, 20, 50, 100, 1000):\n s1 = []\n s2 = []\n for epsi in eps:\n n1 = n * epsi\n n2 = n * (1 - epsi)\n k1 = rng.poisson(n1, size=10000)\n k2 = rng.poisson(n2, size=10000)\n ma = k1 + k2 > 0\n k1 = k1[ma]\n k2 = k2[ma]\n s1.append(np.std(k1 / (k1 + k2)))\n # it is important to average the variance instead of the standard deviation\n s2.append(np.mean(eps_var_py(k1, k2)) ** 0.5)\n data[n] = np.array((eps, s1, s2))\n\n\nplt.figure(figsize=(10, 4))\nfor n, (eps, s1, s2) in data.items():\n l, = plt.plot(eps, s1, \":\", label=f\"MC, {n=}\")\n plt.plot(eps, s2, \"--\", color=l.get_color(), label=f\"formula, {n=}\")\n print(f\"{n=} median deviation {np.median(s1[s2 > 0] / s2[s2 > 0]) - 1:.2f}\")\nplt.subplots_adjust(right=0.6)\nplt.xlabel(r\"$\\epsilon$\")\nplt.ylabel(r\"$\\sigma(\\epsilon)$\")\nplt.figlegend(loc=(0.69, 0.2));\n\nn=5 median deviation 0.00\nn=10 median deviation -0.01\nn=20 median deviation -0.00\nn=50 median deviation -0.00\nn=100 median deviation 0.00\nn=1000 median deviation 0.00\n\n\n\n\n\n\nWilson interval\nTo-do." + }, + { + "objectID": "posts/Uncertainty of efficiency computed from fitted yields.html#with-background", + "href": "posts/Uncertainty of efficiency computed from fitted yields.html#with-background", + "title": "Uncertainty of efficiency from fitted decay yields", + "section": "With background", + "text": "With background\nWe now assume that the estimates \\(\\hat n_1\\) and \\(\\hat n_2\\) cannot obtained by direct counting, but need to be estimated from a fit that inclues a signal and background component. If the fit is unbiased, the average contribution from the background is zero, which we will assume, but it enlargens the uncertainty of the estimate.\nWe can model this by adding a random contribution from the background to our estimate,\n\\[\n\\hat n = n + z_1 \\sigma_\\text{poisson} + z_2 \\sigma_\\text{bkg} = n + z_1 \\sqrt{n} + z_2 \\sigma_\\text{bkg},\n\\]\nwhere \\(z_1\\) and \\(z_2\\) are independent random variables with zero mean and unit variance.\nThe expected variance of \\(\\hat n\\) then is\n\\[\n\\sigma^2(\\hat n) = n + \\sigma^2_\\text{bkg}.\n\\]\nIn practice, estimating \\(\\sigma^2_\\text{bkg}\\) directly is difficult, but fortunately, the left-hand side \\(\\sigma^2(\\hat n)\\) can be estimated from the fit (MINUITs Hesse error). Thus we can compute an estimate for the unknown \\(\\sigma^2_\\text{bkg}\\) with the formula,\n\\[\n\\sigma^2_\\text{bkg} = \\sigma^2(\\hat n) - n.\n\\]\nIn practice, we have to replace the expectation values \\(\\sigma^2(\\hat n)\\) and \\(n\\) with the corresponding estimates from the fit.\nWe insert this into the formula for \\(\\hat \\epsilon\\) and get:\n\\[\n\\hat \\epsilon = \\frac{\\hat n_1}{\\hat n_1 + \\hat n_2} = \\frac{n_1 + z_1 \\sqrt{n_1} + z_2 \\sigma_\\text{bkg,1}}{n_1 + n_2 + z_1 \\sqrt{n_1} + z_2 \\sigma_\\text{bkg,1} + z_3 \\sqrt{n_2} + z_4 \\sigma_\\text{bkg,2}}\n\\]\nAs before, the variance of \\(\\hat \\epsilon\\) is computed via standard error propagation, using that the \\(z_i\\) are independent and standard normal. This reflects the assumption that the background in both fits is independently sampled, which in practice is usually not exactly true. If the correlation can be estimated, it can be trivially included in the calculation, but I will not do this here.\n\\[\n\\sigma^2(\\hat \\epsilon) = \\left[\\left(\\frac{\\text{d}\\hat \\epsilon}{\\text{d}z_1}\\right)^2 + \\left(\\frac{\\text{d}\\hat \\epsilon}{\\text{d}z_2}\\right)^2 +\n\\left(\\frac{\\text{d}\\hat \\epsilon}{\\text{d}z_3}\\right)^2 +\n\\left(\\frac{\\text{d}\\hat \\epsilon}{\\text{d}z_4}\\right)^2\n\\right]_{z_1 = 0,\\, z_2=0,\\,z_3=0,\\,z_4=0}.\n\\]\n\n(n1, n2, sigma1, sigma2,\n sigma_bkg1, sigma_bkg2,\n z1, z2, z3, z4) = symbols(\n \"n_1 n_2 sigma_1 sigma_2 \"\n \"sigma_bkg_1 sigma_bkg_2 \"\n \"z_1 z_2 z_3 z_4\", real=True)\n\ng = (n1 + z1 * sqrt(n1) + z2 * sigma_bkg1) / \\\n (n1 + n2\n + z1 * sqrt(n1) + z2 * sigma_bkg1 \n + z3 * sqrt(n2) + z4 * sigma_bkg2); g\n\n\\(\\displaystyle \\frac{\\sqrt{n_{1}} z_{1} + n_{1} + \\sigma_{bkg 1} z_{2}}{\\sqrt{n_{1}} z_{1} + n_{1} + \\sqrt{n_{2}} z_{3} + n_{2} + \\sigma_{bkg 1} z_{2} + \\sigma_{bkg 2} z_{4}}\\)\n\n\n\ng = g.subs([(sigma_bkg1, sqrt(sigma1 ** 2 - n1)), \n (sigma_bkg2, sqrt(sigma2 ** 2 - n2))]); g\n\n\\(\\displaystyle \\frac{\\sqrt{n_{1}} z_{1} + n_{1} + z_{2} \\sqrt{- n_{1} + \\sigma_{1}^{2}}}{\\sqrt{n_{1}} z_{1} + n_{1} + \\sqrt{n_{2}} z_{3} + n_{2} + z_{2} \\sqrt{- n_{1} + \\sigma_{1}^{2}} + z_{4} \\sqrt{- n_{2} + \\sigma_{2}^{2}}}\\)\n\n\n\ng_var = sum(diff(g, z) ** 2 for z in (z1, z2, z3, z4))\ng_var = g_var.subs([(z1, 0), (z2, 0), (z3, 0), (z4, 0)])\ng_var = g_var.simplify(); g_var\n\n\\(\\displaystyle \\frac{n_{1}^{2} \\sigma_{2}^{2} + n_{2}^{2} \\sigma_{1}^{2}}{\\left(n_{1} + n_{2}\\right)^{4}}\\)\n\n\nWe thus obtain\n\\[\n\\sigma(\\hat \\epsilon) = \\frac{\\sqrt{\nn_1^2 \\, \\sigma^2(\\hat n_2) +\nn_2^2 \\, \\sigma^2(\\hat n_1)\n}}{n^2},\n\\]\nwith \\(n = n_1 + n_2\\). In practice, \\(n_1\\), \\(\\sigma^2(\\hat n_1)\\), … need to be replaced with their fit estimates.\nAs a check, I show how one can recover the previous formula, which is a special case, by setting \\(\\sigma^2(\\hat n_k) = n_k\\) (which means no background contribution),\n\\[\n\\sigma^2(\\hat \\epsilon) = \\frac{\nn_1^2 \\, \\sigma^2(\\hat n_2) +\nn_2^2 \\, \\sigma^2(\\hat n_1)\n}{n^4} =\n\\frac{\nn_1^2 \\, n_2 +\nn_2^2 \\, n_1\n}{n^4} =\n\\frac{\nn_1 \\, n_2 \\, (n_1 + n_2)\n}{n^4} =\n\\frac{\nn_1 \\, n_2\n}{n^3}.\n\\]\nWe again check the formula against Monte-Carlo simulations.\n\ng_var_py = lambdify([n1, sigma1, n2, sigma2], g_var)\n\n\ndef model(x, b0, b1, b2, s, mu, sigma):\n nb = bernstein.scaled_cdf(x, (b0, b1, b2), -1, 1)\n ns = s * norm.cdf(x, mu, sigma)\n return ns + nb\n\n@joblib.delayed\ndef run(eps, n):\n rng = np.random.default_rng(1)\n\n t1 = []\n t2 = []\n for itry in range(100):\n sigma = 0.1\n results = []\n for p in (eps, 1 - eps):\n s = rng.normal(0, sigma, size=rng.poisson(p * n))\n b = rng.uniform(-1, 1, size=n // 5)\n x = np.append(s, b)\n w, xe = np.histogram(x, bins=50, range=(-1, 1))\n cost = ExtendedBinnedNLL(w, xe, model)\n cost += NormalConstraint((\"mu\", \"sigma\"), (0, sigma), (1, 1))\n s_start = eps * n\n b_start = n\n m = Minuit(cost, \n b0=b_start, b1=b_start, b2=b_start,\n s=s_start + 1, mu=0, sigma=sigma)\n m.limits[:4] = (0, None)\n m.limits[\"mu\"] = (-0.5, 0.5)\n m.limits[\"sigma\"] = (0.02, 1)\n m.migrad()\n if m.valid and m.accurate:\n results.append((m.values[\"s\"], m.errors[\"s\"]))\n else:\n print(n, f\"{eps:.2f}\", itry, \"signal\" if p == eps else \"background\",\n \"fit failed\")\n if len(results) < 2:\n continue\n\n (n_pass, sigma_pass), (n_fail, sigma_fail) = results\n\n t1i = n_pass / (n_pass + n_fail)\n t2i = g_var_py(n_pass, sigma_pass, n_fail, sigma_fail)\n t1.append(t1i)\n t2.append(t2i)\n\n s1 = np.std(t1)\n s2 = np.mean(t2) ** 0.5\n return s1, s2\n\ndata = {}\neps = np.linspace(0, 1, 20)\nwith joblib.Parallel(-1) as pool:\n for n in (50, 100, 1000):\n s1, s2 = np.transpose(pool(run(epsi, n) for epsi in eps))\n data[n] = (eps, s1, s2)\n\n\nplt.figure(figsize=(10, 4))\nfor n, (eps, s1, s2) in data.items():\n l, = plt.plot(eps, s1, \":\", label=f\"MC, {n=}\")\n plt.plot(eps, s2, \"--\", color=l.get_color(), label=f\"formula, {n=}\")\n print(f\"{n=} median deviation {np.median(s1[s2 > 0] / s2[s2 > 0]) - 1:.2f}\")\nplt.subplots_adjust(right=0.6)\nplt.xlabel(r\"$\\epsilon$\")\nplt.ylabel(r\"$\\sigma(\\epsilon)$\")\nplt.figlegend(loc=(0.69, 0.2));\n\nn=50 median deviation 0.02\nn=100 median deviation -0.03\nn=1000 median deviation 0.04\n\n\n\n\n\nAgain, the derived formula does reasonably well with deviations around 5 % for \\(n > 50\\). This is usually sufficient to calculate error bars for efficiency plots. There should be some bias for small \\(n\\) as before, but it is difficult to see here due to other larger random scatter, since I generated less toys to keep the computation time on my laptop reasonable.\nFinally, I compare the uncertainty estimates for various levels of background. The estimate with zero background corresponds to the estimate which one obtains with the naive formula.\n\nplt.figure(figsize=(14, 4))\neps = np.linspace(0, 1, 100)\nfor n in (10, 50, 100):\n n_pass = n * eps\n n_fail = n * (1 - eps)\n n_bkg = 0\n col = None\n for i, n_bkg in enumerate((0, 1, 5, 10)):\n s2 = g_var_py(n_pass, (n_pass + n_bkg) ** 0.5, n_fail, (n_fail + n_bkg) ** 0.5) ** 0.5\n col = plt.plot(eps, s2, \"--\" if i > 0 else \":\", color=col,\n lw=0.5 * i + 1.5, label=f\"{n=} {n_bkg=}\")[0].get_color()\nplt.subplots_adjust(right=0.5)\nplt.figlegend(loc=(0.64, 0.2))\nplt.xlabel(r\"$\\epsilon$\")\nplt.ylabel(r\"$\\sigma(\\hat \\epsilon)$\")\nplt.semilogy();\n\n\n\n\nThe error of using a formula that neglects the background contribution is largest when the efficiency is close to zero or one.\nThese are frequently the interesting cases. For example, the tracking efficency of the LHCb experiment is estimated from fits to decays in calibration samples and the efficiency is around 96 %, so that the difference in the formulas may impact the estimated uncertainty.\n\nFormula for fits to selected and total events\nSometimes it is not easy to split the two samples into failed and passed cases, one has instead fits to the passed and total cases.\nIn this case, we need to replace \\(n_2\\) and \\(\\sigma^2(\\hat n_2)\\) in the main formula for \\(\\sigma(\\hat \\epsilon)\\), since we then only have \\(n = n_1 + n_2\\), \\(\\sigma^2(\\hat n)\\), \\(n_1\\), and \\(\\sigma^2(\\hat n_1)\\) available.\nSince both estimates \\(\\hat n_1\\) and \\(\\hat n_2\\) are completely independent (again assuming independent background), their sum \\(\\hat n = \\hat n_1 + \\hat n_1\\) must have the variance \\(\\sigma^2(\\hat n) = \\sigma^2(\\hat n_1) + \\sigma^2(\\hat n_2)\\).\nThus the formula for this case becomes\n\\[\n\\sigma(\\hat \\epsilon) = \\frac{\\sqrt{\nn_1^2 \\, (\\sigma^2(\\hat n) - \\sigma^2(\\hat n_1)) +\n(n - n_1)^2 \\, \\sigma^2(\\hat n_1)\n}}{n^2}.\n\\]" + }, + { + "objectID": "posts/logaddexp.html", + "href": "posts/logaddexp.html", + "title": "How to use the logaddexp special function in fits", + "section": "", + "text": "The logaddexp special function was made to perform this calculation more accurately: \\[\ny = \\ln(\\exp(a) + \\exp(b)).\n\\] This calculation occurs when we fit a mixed PDF with two components to a data sample, and both components can be modeled with PDFs from the exponential family. In that case, one can compute \\(a\\) and \\(b\\) efficiently, which represent the log-probability-density. This is not artificial, the normal, student’s t, and exponential distributions are from the exponential family.\nBecause of the finite dynamic range of floating point numbers and round-off, this computation can become inaccurate or yield infinity. One can avoid these issues by first determining whether \\(a\\) or \\(b\\) is larger and then by factoring it out of the equation. The logaddexp special function implements this strategy.\nWe explore this based on a toy example of a signal peak sampled from the student’s t distribution over an exponential background.\n\nfrom iminuit import Minuit\nfrom iminuit.cost import ExtendedUnbinnedNLL\nfrom numba_stats import t, truncexpon, expon\nimport numpy as np\nimport numba as nb\nimport matplotlib.pyplot as plt\n\nThis is our toy data sample that we wish to fit.\n\nrng = np.random.default_rng(1)\n\nx = 0.1 * rng.standard_t(2, size=1000) + 1\nx = np.append(x, rng.exponential(size=2000))\nx = x[(0 < x) & (x < 2)]\n\nplt.hist(x, bins=100);\n\n\n\n\nAs a reference, we first run the standard fit without logaddexp.\n\ndef model(x, ns, df, mu, sigma, nb, tau):\n return ns + nb, ns * t.pdf(x, df, mu, sigma) + nb * truncexpon.pdf(\n x, 0, 2, 0, tau)\n\n\ncost = ExtendedUnbinnedNLL(x, model)\n\nm = Minuit(cost, ns=1000, nb=2000, df=2, mu=1, sigma=0.1, tau=1)\nm.limits[\"ns\", \"nb\", \"sigma\", \"tau\"] = (0, None)\nm.limits[\"df\"] = (1, None)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -3.516e+04\nNfcn = 170\n\n\nEDM = 8.76e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nns\n930\n50\n\n\n0\n\n\n\n\n1\ndf\n3.9\n1.0\n\n\n1\n\n\n\n\n2\nmu\n0.997\n0.005\n\n\n\n\n\n\n\n3\nsigma\n0.103\n0.006\n\n\n0\n\n\n\n\n4\nnb\n1.80e3\n0.06e3\n\n\n0\n\n\n\n\n5\ntau\n1.04\n0.05\n\n\n0\n\n\n\n\n\n\n\n\n\nns\ndf\nmu\nsigma\nnb\ntau\n\n\nns\n2.35e+03\n-15.2 (-0.296)\n-9.210e-3 (-0.036)\n111.09e-3 (0.354)\n-1.4e3 (-0.517)\n-0.8988 (-0.341)\n\n\ndf\n-15.2 (-0.296)\n1.12\n0.051e-3 (0.009)\n2.40e-3 (0.350)\n15.2 (0.253)\n0.0098 (0.170)\n\n\nmu\n-9.210e-3 (-0.036)\n0.051e-3 (0.009)\n2.78e-05\n-0.001e-3 (-0.024)\n9.206e-3 (0.031)\n-0.023e-3 (-0.082)\n\n\nsigma\n111.09e-3 (0.354)\n2.40e-3 (0.350)\n-0.001e-3 (-0.024)\n4.2e-05\n-111.06e-3 (-0.302)\n-0.07e-3 (-0.196)\n\n\nnb\n-1.4e3 (-0.517)\n15.2 (0.253)\n9.206e-3 (0.031)\n-111.06e-3 (-0.302)\n3.22e+03\n0.8986 (0.291)\n\n\ntau\n-0.8988 (-0.341)\n0.0098 (0.170)\n-0.023e-3 (-0.082)\n-0.07e-3 (-0.196)\n0.8986 (0.291)\n0.00296\n\n\n\n\n\n\n \n \n \n \n 2024-05-12T21:35:43.174007\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nCalling the logpdf methods for the two component PDFs is computationally more efficient than the pdf methods.\n\n%%timeit\nt.logpdf(x, 2, 0, 1)\n\n27.8 µs ± 491 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n\n\n\n%%timeit\nt.pdf(x, 2, 0, 1)\n\n43.8 µs ± 3.19 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n\n\n\n%%timeit\ntruncexpon.logpdf(x, 0, 2, 0, 1)\n\n8.12 µs ± 1.14 µs per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n\n\n\n%%timeit\ntruncexpon.pdf(x, 0, 2, 0, 1)\n\n21.4 µs ± 382 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n\n\nNow we run the fit again using the logpdf methods of the component distributions and logaddexp. Our model now returns the log-probability and we notify the ExtendedUnbinnedNLL cost function of that fact with the keyword log=True.\n\ndef model_log(x, ns, df, mu, sigma, nb, tau):\n return ns + nb, np.logaddexp(\n np.log(ns) + t.logpdf(x, df, mu, sigma),\n np.log(nb) + truncexpon.logpdf(x, 0, 2, 0, tau))\n\n\ncost_log = ExtendedUnbinnedNLL(x[(0 < x) & (x < 2)], model_log, log=True)\n\nm_log = Minuit(cost_log, ns=1000, nb=2000, df=2, mu=1, sigma=0.1, tau=1)\nm_log.limits[\"ns\", \"nb\", \"sigma\", \"tau\"] = (0, None)\nm_log.limits[\"df\"] = (1, None)\nm_log.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -3.516e+04\nNfcn = 170\n\n\nEDM = 8.76e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nns\n930\n50\n\n\n0\n\n\n\n\n1\ndf\n3.9\n1.0\n\n\n1\n\n\n\n\n2\nmu\n0.997\n0.005\n\n\n\n\n\n\n\n3\nsigma\n0.103\n0.006\n\n\n0\n\n\n\n\n4\nnb\n1.80e3\n0.06e3\n\n\n0\n\n\n\n\n5\ntau\n1.04\n0.05\n\n\n0\n\n\n\n\n\n\n\n\n\nns\ndf\nmu\nsigma\nnb\ntau\n\n\nns\n2.35e+03\n-15.2 (-0.296)\n-9.210e-3 (-0.036)\n111.09e-3 (0.354)\n-1.4e3 (-0.517)\n-0.8988 (-0.341)\n\n\ndf\n-15.2 (-0.296)\n1.12\n0.051e-3 (0.009)\n2.40e-3 (0.350)\n15.2 (0.253)\n0.0098 (0.170)\n\n\nmu\n-9.210e-3 (-0.036)\n0.051e-3 (0.009)\n2.78e-05\n-0.001e-3 (-0.024)\n9.206e-3 (0.031)\n-0.023e-3 (-0.082)\n\n\nsigma\n111.09e-3 (0.354)\n2.40e-3 (0.350)\n-0.001e-3 (-0.024)\n4.2e-05\n-111.06e-3 (-0.302)\n-0.07e-3 (-0.196)\n\n\nnb\n-1.4e3 (-0.517)\n15.2 (0.253)\n9.206e-3 (0.031)\n-111.06e-3 (-0.302)\n3.22e+03\n0.8986 (0.291)\n\n\ntau\n-0.8988 (-0.341)\n0.0098 (0.170)\n-0.023e-3 (-0.082)\n-0.07e-3 (-0.196)\n0.8986 (0.291)\n0.00296\n\n\n\n\n\n\n \n \n \n \n 2024-05-12T21:43:43.385557\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nAs expected, the results are identical up to round-off error. However, the computation should now be more numerically stable. Despite the fact that the logpdf is faster to compute, computing the log-likelihood is a bit slower. It is possible that the branch inside logaddexp slows down the computation.\n\n%%timeit\ncost(*m.values)\n\n113 µs ± 13.5 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n\n\n\n%%timeit\ncost_log(*m_log.values)\n\n140 µs ± 1.5 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n\n\nIt is difficult to show that the second computation is more numerically stable. I make an attempt here by running the fit many times with variable starting values. This should lead to a valid more often if the computation is numerically stable. However, both versions of the fit perform equally well in this test.\n\nrng = np.random.default_rng(1)\nvalids = np.empty((100, 2), dtype=bool)\nfor i in range(len(valids)):\n m.reset()\n m_log.reset()\n\n ns = rng.exponential(1000)\n df = 1 + rng.exponential(1)\n mu = rng.normal(1, 0.25)\n sigma = rng.exponential(0.1) + 0.01\n nb = rng.exponential(2000)\n tau = rng.exponential(1)\n\n m.values = (ns, df, mu, sigma, nb, tau)\n m.migrad()\n\n m_log.values = (ns, df, mu, sigma, nb, tau)\n m_log.migrad()\n\n valids[i, 0] = m.valid\n valids[i, 1] = m_log.valid\nprint(np.mean(valids, axis=0))\n\n[0.99 0.99]\n\n\nThe reason for this is that ExtendedUnbinnedNLL has some tricks (mostly the safe_log) which make sure that the simpler, less numerically stable, calculation also works most of the time." + }, + { + "objectID": "posts/Correlation.html", + "href": "posts/Correlation.html", + "title": "Particle correlations in proton-proton collisions", + "section": "", + "text": "Here, I run simulations of proton-proton collisions with Chromo to investigate two-particle correlations in pseudorapidity space. With Chromo, it is very easy to study the output of different event generators.\nI generate collisions at 13 TeV center-of-mass energy. For each event, I form all unique pairs of final-state charged particles, these are effectively long-lived prompt charged particles. Prompt here means that they either have no ancestors or that their ancestors are short-lived. I then fill the pseudorapidity pairs into a two-dimensional histogram.\nWe can see interesting patterns in these plots when different generators are compared. SIBYLL-2.3d shows a stronger concentration around mid-rapidity and has more pairs which are close in rapidity than the other generators.\n\nimport impy.models as im\nfrom impy.kinematics import CenterOfMass\nfrom impy.constants import TeV\nimport boost_histogram as bh\nfrom matplotlib import pyplot as plt\nimport numba as nb\nimport numpy as np\n\n\nekin = CenterOfMass(13 * TeV, \"proton\", \"proton\")\n\n@nb.njit\ndef pairs(eta, seed):\n r1 = []\n r2 = []\n np.random.seed(seed)\n for i, etai in enumerate(eta):\n for j in range(i):\n etaj = eta[j]\n # avoid structures that arise from implementation-specific\n # ordering of particles in HEPEVT record\n if np.random.uniform(0, 1) < 0.5:\n r1.append(etai)\n r2.append(etaj)\n else:\n r1.append(etaj)\n r2.append(etai)\n return np.array(r1), np.array(r2)\n\nnevent = 1000\n\nhistograms = {}\nfor Model in (im.Sibyll21, im.Sibyll23d, im.QGSJetII04, im.EposLHC, im.Pythia6):\n h = bh.Histogram(bh.axis.Regular(300, -15, 15), bh.axis.Regular(300, -15, 15))\n for i, ev in enumerate(Model(ekin)(nevent)):\n fsc = ev.final_state_charged()\n with np.errstate(invalid=\"ignore\", divide=\"ignore\"):\n eta1, eta2 = pairs(fsc.eta, i)\n h.fill(eta1, eta2)\n histograms[Model] = h\n\n ====================================================\n | |\n | S I B Y L L 2.1 |\n | |\n | HADRONIC INTERACTION MONTE CARLO |\n | BY |\n | Ralph ENGEL |\n | R.S. FLETCHER, T.K. GAISSER |\n | P. LIPARI, T. STANEV |\n | |\n | Publication to be cited when using this program: |\n | R. Engel et al., Proc. 26th ICRC, 1 (1999) 415 |\n | |\n | last modified: 28. Sept. 2001 by R. Engel |\n ====================================================\n\n\n Table: J, sqs, PT_cut, SIG_tot, SIG_inel, B_el, rho, <n_s>, <n_h>\n ------------------------------------------------------------------------\n 1 1.000E+01 1.45 38.33 30.88 10.83 -0.185 1.964 0.003\n 1 1.259E+01 1.49 38.27 31.16 11.10 -0.127 1.949 0.006\n 1 1.585E+01 1.54 38.44 31.54 11.36 -0.078 1.934 0.012\n 1 1.995E+01 1.59 38.84 32.05 11.63 -0.037 1.918 0.019\n 1 2.512E+01 1.64 39.46 32.67 11.89 -0.004 1.901 0.029\n 1 3.162E+01 1.69 40.29 33.41 12.16 0.022 1.884 0.043\n 1 3.981E+01 1.75 41.28 34.24 12.42 0.044 1.864 0.060\n 1 5.012E+01 1.81 42.39 35.13 12.69 0.061 1.844 0.082\n 1 6.310E+01 1.87 43.59 36.05 12.95 0.075 1.821 0.109\n 1 7.943E+01 1.93 44.92 37.06 13.22 0.085 1.797 0.141\n 1 1.000E+02 2.00 46.35 38.14 13.48 0.094 1.771 0.180\n 1 1.259E+02 2.07 47.84 39.28 13.73 0.101 1.742 0.226\n 1 1.585E+02 2.14 49.39 40.48 13.98 0.106 1.711 0.279\n 1 1.995E+02 2.22 51.05 41.74 14.24 0.110 1.677 0.342\n 1 2.512E+02 2.30 52.84 43.09 14.50 0.113 1.641 0.413\n 1 3.162E+02 2.38 54.79 44.56 14.77 0.116 1.603 0.495\n 1 3.981E+02 2.46 56.96 46.16 15.05 0.118 1.563 0.587\n 1 5.012E+02 2.55 59.39 47.94 15.33 0.119 1.522 0.691\n 1 6.310E+02 2.65 62.15 49.92 15.63 0.120 1.479 0.808\n 1 7.943E+02 2.74 65.29 52.14 15.95 0.121 1.435 0.938\n 1 1.000E+03 2.84 68.88 54.63 16.28 0.122 1.391 1.081\n 1 1.259E+03 2.95 72.53 57.06 16.61 0.123 1.347 1.240\n 1 1.585E+03 3.06 76.46 59.64 16.96 0.123 1.303 1.416\n 1 1.995E+03 3.17 80.67 62.36 17.32 0.123 1.259 1.609\n 1 2.512E+03 3.29 85.17 65.23 17.70 0.124 1.216 1.822\n 1 3.162E+03 3.41 89.95 68.24 18.10 0.124 1.175 2.055\n 1 3.981E+03 3.54 95.02 71.38 18.51 0.124 1.135 2.312\n 1 5.012E+03 3.67 100.35 74.64 18.95 0.124 1.096 2.594\n 1 6.310E+03 3.81 105.95 78.03 19.40 0.124 1.059 2.903\n 1 7.943E+03 3.95 111.81 81.53 19.88 0.124 1.024 3.244\n 1 1.000E+04 4.09 117.90 85.14 20.38 0.125 0.990 3.618\n 1 1.259E+04 4.25 124.23 88.85 20.90 0.125 0.959 4.030\n 1 1.585E+04 4.40 130.77 92.65 21.44 0.125 0.928 4.481\n 1 1.995E+04 4.57 137.51 96.55 22.01 0.125 0.898 4.957\n 1 2.512E+04 4.74 144.44 100.54 22.60 0.125 0.865 5.394\n 1 3.162E+04 4.91 151.56 104.61 23.21 0.125 0.825 5.707\n 1 3.981E+04 5.09 158.85 108.75 23.85 0.125 0.779 5.909\n 1 5.012E+04 5.28 166.31 112.98 24.50 0.125 0.733 6.076\n 1 6.310E+04 5.47 173.92 117.27 25.17 0.125 0.690 6.243\n 1 7.943E+04 5.67 181.69 121.64 25.87 0.125 0.649 6.417\n 1 1.000E+05 5.88 189.60 126.07 26.58 0.125 0.611 6.601\n 1 1.259E+05 6.10 197.66 130.57 27.31 0.125 0.576 6.799\n 1 1.585E+05 6.32 205.84 135.12 28.06 0.125 0.543 7.013\n 1 1.995E+05 6.55 214.16 139.74 28.82 0.125 0.512 7.247\n 1 2.512E+05 6.78 222.60 144.41 29.60 0.125 0.484 7.501\n 1 3.162E+05 7.03 231.15 149.14 30.39 0.125 0.457 7.763\n 1 3.981E+05 7.28 239.82 153.93 31.21 0.125 0.432 7.999\n 1 5.012E+05 7.54 248.60 158.77 32.03 0.125 0.408 8.168\n 1 6.310E+05 7.81 257.49 163.65 32.87 0.125 0.385 8.278\n 1 7.943E+05 8.08 266.48 168.59 33.72 0.125 0.364 8.364\n 1 1.000E+06 8.37 275.56 173.57 34.59 0.125 0.343 8.442\n 1 1.259E+06 8.67 284.75 178.60 35.47 0.125 0.324 8.517\n 1 1.585E+06 8.97 294.03 183.68 36.36 0.125 0.307 8.590\n 1 1.995E+06 9.28 303.40 188.80 37.27 0.125 0.290 8.663\n 1 2.512E+06 9.61 312.87 193.97 38.18 0.125 0.274 8.735\n 1 3.162E+06 9.94 322.42 199.18 39.11 0.125 0.259 8.808\n 1 3.981E+06 10.29 332.06 204.43 40.05 0.125 0.246 8.882\n 1 5.012E+06 10.64 341.79 209.72 41.00 0.125 0.233 8.957\n 1 6.310E+06 11.00 351.59 215.05 41.95 0.125 0.220 9.032\n 1 7.943E+06 11.38 361.49 220.42 42.92 0.125 0.209 9.101\n 1 1.000E+07 11.77 371.46 225.83 43.90 0.125 0.198 9.158\n\n Table: J, sqs, PT_cut, SIG_tot, SIG_inel, B_el, rho, <n_s>, <n_h>\n ------------------------------------------------------------------------\n 2 1.000E+01 1.45 23.12 19.94 10.12 -0.067 1.845 0.009\n 2 1.259E+01 1.49 23.22 20.21 10.27 -0.031 1.827 0.016\n 2 1.585E+01 1.54 23.49 20.57 10.42 -0.001 1.807 0.026\n 2 1.995E+01 1.59 23.87 21.00 10.56 0.024 1.783 0.040\n 2 2.512E+01 1.64 24.33 21.49 10.70 0.045 1.756 0.058\n 2 3.162E+01 1.69 24.89 21.95 10.84 0.061 1.727 0.081\n 2 3.981E+01 1.75 25.53 22.48 10.97 0.074 1.695 0.110\n 2 5.012E+01 1.81 26.25 23.06 11.11 0.085 1.659 0.144\n 2 6.310E+01 1.87 27.04 23.69 11.24 0.093 1.621 0.185\n 2 7.943E+01 1.93 27.89 24.36 11.37 0.100 1.580 0.233\n 2 1.000E+02 2.00 28.80 25.08 11.50 0.105 1.536 0.288\n 2 1.259E+02 2.07 29.72 25.72 11.44 0.109 1.490 0.351\n 2 1.585E+02 2.14 30.77 26.45 11.39 0.113 1.442 0.423\n 2 1.995E+02 2.22 31.96 27.30 11.38 0.115 1.392 0.503\n 2 2.512E+02 2.30 33.35 28.30 11.40 0.117 1.340 0.592\n 2 3.162E+02 2.38 34.99 29.47 11.45 0.119 1.288 0.690\n 2 3.981E+02 2.46 36.93 30.84 11.55 0.120 1.236 0.798\n 2 5.012E+02 2.55 39.22 32.47 11.69 0.121 1.184 0.916\n 2 6.310E+02 2.65 41.93 34.36 11.88 0.122 1.132 1.044\n 2 7.943E+02 2.74 45.10 36.57 12.12 0.123 1.082 1.183\n 2 1.000E+03 2.84 48.81 39.13 12.41 0.123 1.033 1.333\n 2 1.259E+03 2.95 52.13 41.43 12.78 0.123 0.987 1.497\n 2 1.585E+03 3.06 55.69 43.86 13.16 0.124 0.942 1.673\n 2 1.995E+03 3.17 59.50 46.43 13.56 0.124 0.899 1.865\n 2 2.512E+03 3.29 63.55 49.12 13.97 0.124 0.859 2.072\n 2 3.162E+03 3.41 67.83 51.93 14.39 0.124 0.821 2.298\n 2 3.981E+03 3.54 72.35 54.85 14.83 0.124 0.785 2.543\n 2 5.012E+03 3.67 77.09 57.87 15.28 0.124 0.752 2.809\n 2 6.310E+03 3.81 82.06 61.01 15.75 0.124 0.720 3.099\n 2 7.943E+03 3.95 87.24 64.23 16.24 0.125 0.691 3.416\n 2 1.000E+04 4.09 92.61 67.55 16.74 0.125 0.664 3.761\n 2 1.259E+04 4.25 98.18 70.96 17.26 0.125 0.638 4.138\n 2 1.585E+04 4.40 103.94 74.44 17.79 0.125 0.614 4.543\n 2 1.995E+04 4.57 109.87 78.01 18.34 0.125 0.591 4.953\n 2 2.512E+04 4.74 115.96 81.65 18.91 0.125 0.565 5.303\n 2 3.162E+04 4.91 122.21 85.36 19.50 0.125 0.535 5.552\n 2 3.981E+04 5.09 128.62 89.15 20.10 0.125 0.504 5.737\n 2 5.012E+04 5.28 135.18 92.99 20.72 0.125 0.474 5.904\n 2 6.310E+04 5.47 141.88 96.91 21.35 0.125 0.446 6.067\n 2 7.943E+04 5.67 148.71 100.88 22.00 0.125 0.419 6.233\n 2 1.000E+05 5.88 155.68 104.91 22.66 0.125 0.394 6.404\n 2 1.259E+05 6.10 162.78 109.01 23.34 0.125 0.371 6.584\n 2 1.585E+05 6.32 170.01 113.15 24.03 0.125 0.349 6.776\n 2 1.995E+05 6.55 177.35 117.36 24.73 0.125 0.329 6.981\n 2 2.512E+05 6.78 184.81 121.61 25.45 0.125 0.311 7.200\n 2 3.162E+05 7.03 192.39 125.92 26.18 0.125 0.293 7.426\n 2 3.981E+05 7.28 200.08 130.28 26.93 0.125 0.277 7.632\n 2 5.012E+05 7.54 207.88 134.69 27.68 0.125 0.261 7.792\n 2 6.310E+05 7.81 215.78 139.15 28.45 0.125 0.246 7.909\n 2 7.943E+05 8.08 223.78 143.66 29.23 0.125 0.232 8.004\n 2 1.000E+06 8.37 231.89 148.21 30.02 0.125 0.219 8.090\n 2 1.259E+06 8.67 240.09 152.81 30.83 0.125 0.207 8.172\n 2 1.585E+06 8.97 248.39 157.46 31.64 0.125 0.196 8.249\n 2 1.995E+06 9.28 256.79 162.15 32.47 0.125 0.185 8.324\n 2 2.512E+06 9.61 265.28 166.88 33.30 0.125 0.175 8.398\n 2 3.162E+06 9.94 273.86 171.66 34.15 0.125 0.165 8.470\n 2 3.981E+06 10.29 282.53 176.48 35.01 0.125 0.156 8.542\n 2 5.012E+06 10.64 291.29 181.34 35.88 0.125 0.148 8.615\n 2 6.310E+06 11.00 300.14 186.25 36.76 0.125 0.140 8.687\n 2 7.943E+06 11.38 309.08 191.19 37.64 0.125 0.133 8.756\n 2 1.000E+07 11.77 318.10 196.18 38.54 0.125 0.126 8.819\n\n Table: J, sqs, PT_cut, SIG_tot, SIG_inel, B_el, rho, <n_s>, <n_h>\n ------------------------------------------------------------------------\n 3 1.000E+01 1.45 18.30 16.13 8.84 -0.025 1.845 0.009\n 3 1.259E+01 1.49 18.83 16.69 9.14 0.009 1.827 0.016\n 3 1.585E+01 1.54 19.41 17.26 9.40 0.036 1.807 0.026\n 3 1.995E+01 1.59 20.02 17.83 9.64 0.056 1.783 0.040\n 3 2.512E+01 1.64 20.63 18.41 9.85 0.072 1.756 0.058\n 3 3.162E+01 1.69 21.32 18.99 10.05 0.084 1.727 0.081\n 3 3.981E+01 1.75 22.04 19.60 10.23 0.093 1.695 0.110\n 3 5.012E+01 1.81 22.81 20.23 10.40 0.101 1.659 0.144\n 3 6.310E+01 1.87 23.61 20.89 10.56 0.106 1.621 0.185\n 3 7.943E+01 1.93 24.46 21.57 10.72 0.111 1.580 0.233\n 3 1.000E+02 2.00 25.34 22.28 10.87 0.114 1.536 0.288\n 3 1.259E+02 2.07 26.25 23.01 11.06 0.116 1.490 0.351\n 3 1.585E+02 2.14 27.21 23.75 11.21 0.118 1.442 0.423\n 3 1.995E+02 2.22 28.23 24.53 11.32 0.120 1.392 0.503\n 3 2.512E+02 2.30 29.37 25.37 11.42 0.121 1.340 0.592\n 3 3.162E+02 2.38 30.68 26.34 11.51 0.122 1.288 0.690\n 3 3.981E+02 2.46 32.25 27.48 11.61 0.123 1.236 0.798\n 3 5.012E+02 2.55 34.17 28.86 11.73 0.123 1.184 0.916\n 3 6.310E+02 2.65 36.58 30.57 11.90 0.123 1.132 1.044\n 3 7.943E+02 2.74 39.60 32.72 12.12 0.124 1.082 1.183\n 3 1.000E+03 2.84 43.42 35.40 12.41 0.124 1.033 1.333\n 3 1.259E+03 2.95 46.38 37.52 12.78 0.124 0.987 1.497\n 3 1.585E+03 3.06 49.56 39.77 13.16 0.124 0.942 1.673\n 3 1.995E+03 3.17 52.95 42.14 13.56 0.124 0.899 1.865\n 3 2.512E+03 3.29 56.56 44.63 13.97 0.124 0.859 2.072\n 3 3.162E+03 3.41 60.38 47.23 14.39 0.125 0.821 2.298\n 3 3.981E+03 3.54 64.41 49.94 14.83 0.125 0.785 2.543\n 3 5.012E+03 3.67 68.64 52.75 15.28 0.125 0.752 2.809\n 3 6.310E+03 3.81 73.06 55.67 15.75 0.125 0.720 3.099\n 3 7.943E+03 3.95 77.67 58.67 16.24 0.125 0.691 3.416\n 3 1.000E+04 4.09 82.46 61.77 16.74 0.125 0.664 3.761\n 3 1.259E+04 4.25 87.43 64.95 17.26 0.125 0.638 4.138\n 3 1.585E+04 4.40 92.55 68.21 17.79 0.125 0.614 4.543\n 3 1.995E+04 4.57 97.83 71.55 18.34 0.125 0.591 4.953\n 3 2.512E+04 4.74 103.26 74.97 18.91 0.125 0.565 5.303\n 3 3.162E+04 4.91 108.83 78.45 19.50 0.125 0.535 5.552\n 3 3.981E+04 5.09 114.54 82.01 20.10 0.125 0.504 5.737\n 3 5.012E+04 5.28 120.37 85.63 20.72 0.125 0.474 5.904\n 3 6.310E+04 5.47 126.34 89.31 21.35 0.125 0.446 6.067\n 3 7.943E+04 5.67 132.43 93.05 22.00 0.125 0.419 6.233\n 3 1.000E+05 5.88 138.64 96.85 22.66 0.125 0.394 6.404\n 3 1.259E+05 6.10 144.96 100.72 23.34 0.125 0.371 6.584\n 3 1.585E+05 6.32 151.39 104.63 24.03 0.125 0.349 6.776\n 3 1.995E+05 6.55 157.93 108.61 24.73 0.125 0.329 6.981\n 3 2.512E+05 6.78 164.58 112.63 25.45 0.125 0.311 7.200\n 3 3.162E+05 7.03 171.33 116.71 26.18 0.125 0.293 7.426\n 3 3.981E+05 7.28 178.17 120.84 26.93 0.125 0.277 7.632\n 3 5.012E+05 7.54 185.12 125.01 27.68 0.125 0.261 7.792\n 3 6.310E+05 7.81 192.15 129.24 28.45 0.125 0.246 7.909\n 3 7.943E+05 8.08 199.28 133.52 29.23 0.125 0.232 8.004\n 3 1.000E+06 8.37 206.50 137.84 30.02 0.125 0.219 8.090\n 3 1.259E+06 8.67 213.80 142.20 30.83 0.125 0.207 8.172\n 3 1.585E+06 8.97 221.19 146.62 31.64 0.125 0.196 8.249\n 3 1.995E+06 9.28 228.67 151.07 32.47 0.125 0.185 8.324\n 3 2.512E+06 9.61 236.23 155.58 33.30 0.125 0.175 8.398\n 3 3.162E+06 9.94 243.87 160.12 34.15 0.125 0.165 8.470\n 3 3.981E+06 10.29 251.60 164.71 35.01 0.125 0.156 8.542\n 3 5.012E+06 10.64 259.40 169.34 35.88 0.125 0.148 8.615\n 3 6.310E+06 11.00 267.28 174.01 36.76 0.125 0.140 8.687\n 3 7.943E+06 11.38 275.24 178.72 37.64 0.125 0.133 8.756\n 3 1.000E+07 11.77 283.28 183.48 38.54 0.125 0.126 8.819\n ====================================================\n | |\n | S I B Y L L 2.3d |\n | |\n | HADRONIC INTERACTION MONTE CARLO |\n | BY |\n | Eun-Joo AHN, Felix RIEHN |\n | R. ENGEL, A. FEDYNITCH, R.S. FLETCHER, |\n | T.K. GAISSER, P. LIPARI, T. STANEV |\n | |\n | Publication to be cited when using this program: |\n | Eun-Joo AHN et al., Phys.Rev. D80 (2009) 094003 |\n | F. RIEHN et al., hep-ph: 1912.03300 |\n | last modifications: F. Riehn (05/20/2020) |\n ====================================================\n\n SIG_AIR_INI: initializing target: (i,A) 1 0 air..\n SIG_AIR_INI: initializing target: (i,A) 2 14 nit..\n SIG_AIR_INI: initializing target: (i,A) 3 16 oxy..\n ====================================================\n | |\n | QUARK GLUON STRING JET -II MODEL |\n | |\n | HADRONIC INTERACTION MONTE CARLO |\n | BY |\n | S. OSTAPCHENKO |\n | |\n | e-mail: sergei@tf.phys.ntnu.no |\n | |\n | Version II-04 |\n | |\n | Publication to be cited when using this program: |\n | S.Ostapchenko, PRD 83 (2011) 014018 |\n | |\n | last modification: 09.04.2013 |\n | |\n | Any modification has to be approved by the author|\n ====================================================\n\n DATDIR/Users/hdembinski/Extern/impy/src/impy/iamdata/qgsjet/ \n qgaini: cross sections readout from the file: qgsdat-II-04\n done\n qgaini: nuclear cross sections readout from the file sectnu-II-04\nimpy.models.epos.EposLHC::init_generator(): First initialization\n /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/ 52\ninitializations ...\n###################################################################\n# EPOS LHC K. WERNER, T. PIEROG #\n# Contact: tanguy.pierog@kit.edu #\n###################################################################\n# WARNING: This is a special retuned version !!! #\n# Do not publish results without contacting the authors. #\n###################################################################\nread from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.iniev ...\nread from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.initl ...\nread from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.inirj.lhc ...\nread from /Users/hdembinski/Extern/impy/src/impy/iamdata/epos/epos.inics.lhc ...\n Compute Cross-section (can take a while...)\nseedj: 3862463 0.2000000000000000D+01\n EPOS used with FUSION option\n1 \n ******************************************************************************\n ******************************************************************************\n ** **\n ** **\n ** *......* Welcome to the Lund Monte Carlo! **\n ** *:::!!:::::::::::* **\n ** *::::::!!::::::::::::::* PPP Y Y TTTTT H H III A **\n ** *::::::::!!::::::::::::::::* P P Y Y T H H I A A **\n ** *:::::::::!!:::::::::::::::::* PPP Y T HHHHH I AAAAA **\n ** *:::::::::!!:::::::::::::::::* P Y T H H I A A **\n ** *::::::::!!::::::::::::::::*! P Y T H H III A A **\n ** *::::::!!::::::::::::::* !! **\n ** !! *:::!!:::::::::::* !! This is PYTHIA version 6.428 **\n ** !! !* -><- * !! Last date of change: 5 Sep 2013 **\n ** !! !! !! **\n ** !! !! !! Now is 0 Jan 2000 at 0:00:00 **\n ** !! !! **\n ** !! lh !! Disclaimer: this program comes **\n ** !! !! without any guarantees. Beware **\n ** !! hh !! of errors and use common sense **\n ** !! ll !! when interpreting results. **\n ** !! !! **\n ** !! Copyright T. Sjostrand (2011) **\n ** **\n ** An archive of program versions and documentation is found on the web: **\n ** http://www.thep.lu.se/~torbjorn/Pythia.html **\n ** **\n ** When you cite this program, the official reference is to the 6.4 manual: **\n ** T. Sjostrand, S. Mrenna and P. Skands, JHEP05 (2006) 026 **\n ** (LU TP 06-13, FERMILAB-PUB-06-052-CD-T) [hep-ph/0603175]. **\n ** **\n ** Also remember that the program, to a large extent, represents original **\n ** physics research. Other publications of special relevance to your **\n ** studies may therefore deserve separate mention. **\n ** **\n ** Main author: Torbjorn Sjostrand; Department of Theoretical Physics, **\n ** Lund University, Solvegatan 14A, S-223 62 Lund, Sweden; **\n ** phone: + 46 - 46 - 222 48 16; e-mail: torbjorn@thep.lu.se **\n ** Author: Stephen Mrenna; Computing Division, GDS Group, **\n ** Fermi National Accelerator Laboratory, MS 234, Batavia, IL 60510, USA; **\n ** phone: + 1 - 630 - 840 - 2556; e-mail: mrenna@fnal.gov **\n ** Author: Peter Skands; CERN/PH-TH, CH-1211 Geneva, Switzerland **\n ** phone: + 41 - 22 - 767 24 47; e-mail: peter.skands@cern.ch **\n ** **\n ** **\n ******************************************************************************\n ******************************************************************************\n1****************** PYINIT: initialization of PYTHIA routines *****************\n\n ==============================================================================\n I I\n I PYTHIA will be initialized for a p+ on p+ collider I\n I at 13000.000 GeV center-of-mass energy I\n I I\n ==============================================================================\n\n ******** PYMAXI: summary of differential cross-section maximum search ********\n\n ==========================================================\n I I I\n I ISUB Subprocess name I Maximum value I\n I I I\n ==========================================================\n I I I\n I 92 Single diffractive (XB) I 7.1189D+00 I\n I 93 Single diffractive (AX) I 7.1189D+00 I\n I 94 Double diffractive I 1.0176D+01 I\n I 95 Low-pT scattering I 5.4002D+01 I\n I 96 Semihard QCD 2 -> 2 I 1.2350D+04 I\n I I I\n ==========================================================\n\n ****** PYMULT: initialization of multiple interactions for MSTP(82) = 4 ******\n pT0 = 3.28 GeV gives sigma(parton-parton) = 5.65D+02 mb: accepted\n\n ****** PYMIGN: initialization of multiple interactions for MSTP(82) = 4 ******\n pT0 = 3.28 GeV gives sigma(parton-parton) = 2.38D+02 mb: accepted\n\n ********************** PYINIT: initialization completed **********************\n\n\n\nfor Model, h in histograms.items():\n plt.figure()\n h2 = h[bh.loc(-10):bh.loc(10):bh.rebin(5), bh.loc(-10):bh.loc(10):bh.rebin(5)]\n h3 = h2[:, sum]\n nxy = h2.values() / nevent\n nx = ny = h3.values() / nevent\n corr = nxy / np.outer(nx, ny) ** 0.5\n print(Model.__name__, np.mean(corr))\n plt.pcolormesh(*h2.axes.edges.T, np.ma.masked_invalid(corr).T)\n plt.xlabel(\"eta 1\")\n plt.ylabel(\"eta 2\")\n plt.colorbar()\n plt.clim(0, 0.05)\n plt.title(Model.__name__)\n\nSibyll21 0.020957722916460298\nSibyll23d 0.020768885431597404\nQGSJetII04 0.02160297235716634\nEposLHC 0.021529706058337213\nPythia6 0.022110986939247387" + }, + { + "objectID": "posts/deep_sets/fast_deep_set.html", + "href": "posts/deep_sets/fast_deep_set.html", + "title": "Fast deep sets with FLAX", + "section": "", + "text": "In Zaheer et al., NIPS 2017 Deep Sets are introduced, a new network architecture with interesting applications to particle physics. One of the first applications in particle physics is discussed here: PT. Komiske, EM. Metodiev, J. Thaler, MIT-CTP 5064.\nIn particle physics, we often want to use mappings from events to numbers. For example, we could assign the label 1 if the event contained an interesting signal, and 0 otherwise. Each event contains a variable number of tracks. This is an obstacle for applying normal feed-forward networks, which cannot handle inputs of varying size.\nDeep Sets can learn the mapping from a set of inputs to an output. The size of the set can vary from sample to sample. The architecture assumes that the mapping is invariant to permutations of the elements in the set. This is the case in the aforementioned example application. Although we could introduce an ordering over the set (for example, sort tracks along the momentum), such an ordering is arbitrary.\nMathematically, a Deep Set architecture consists of two feed forward networks \\(\\rho\\) and \\(\\phi\\) and a pooling operation invariant to permutations. One typically uses the sum, but other operations are possible, like the maximum, \\[\ny(\\{x\\}) = \\rho(\\sum_i \\phi_i(x_i)).\n\\]\nDeep Sets are a very promising architecture, but implementing them in JAX / FLAX is a challenge. Here I show a fast implementation, the first that I am aware of, which is able to handle sets of variable length. Other implementations I found on GitHub only work for sets of constant size.\nThe main challenge to solve was to write the code in such a way that JAX can process input and output as tensors of fixed size. This is a requirement for the JIT compiler and to obtain a fast training loop. The compilation generally requires that all shapes are of fixed size, which contradicts the idea of passing sets of variable size to the network.\nMy solution consists of three parts." + }, + { + "objectID": "posts/deep_sets/fast_deep_set.html#efficient-padding", + "href": "posts/deep_sets/fast_deep_set.html#efficient-padding", + "title": "Fast deep sets with FLAX", + "section": "Efficient padding", + "text": "Efficient padding\nJAX wants inputs of fixed size. We could decide to process one set at a time and pad the size of the set to the largest possible set. This would be wasteful in our application, however, because the track length distribution has a long tail. Most events have only few tracks but rarely an event has many tracks. Padding all events up to the largest size would be very inefficient.\nMy solution is to let the model process several events at once. To do this, I concatenate all the sets into a large contiguous block. This block can be padded to a fixed size. If the block is large enough, the sizes of different events average out, and we only need to pad a small fraction of the block size." + }, + { + "objectID": "posts/deep_sets/fast_deep_set.html#handling-the-sum", + "href": "posts/deep_sets/fast_deep_set.html#handling-the-sum", + "title": "Fast deep sets with FLAX", + "section": "Handling the sum", + "text": "Handling the sum\nNext, we need to replace the sum, because it operates on a dynamically sized chunks, and we can’t have those inside a JIT’ed function. My solution is to replace it with a matrix-vector product. All the inputs are in already one large block. We pass those in parallel to the first \\(\\phi\\) network. The output is another large block of latent vectors, one for each element in each set. To sum these over the original sets, I use matrices constructed like this one \\[\n\\begin{pmatrix}\n1 & 1 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{pmatrix},\n\\] which, when applied to a block with six vectors from three sets, computes the sum over the first set with three vectors, the second with two, and the third with one vector. The output are three latent vectors which are further processed in parallel by the second network. The summation matrix is also padded with zeros to match the input block. When finally the loss is computed, we discard the padded values by multiplying them with zeros." + }, + { + "objectID": "posts/deep_sets/fast_deep_set.html#using-sparse-matrices", + "href": "posts/deep_sets/fast_deep_set.html#using-sparse-matrices", + "title": "Fast deep sets with FLAX", + "section": "Using sparse matrices", + "text": "Using sparse matrices\nThe summation matrix is sparse. We use the experimental support for sparse matrices to speed up the computation. In this toy example, I observed a speedup by a factor of two.\nThe code below demonstrates these ideas on a toy example.\n\nimport jax\nimport jax.numpy as jnp\nfrom jax.experimental import sparse\nfrom flax import linen as nn\nimport numpy as np\nimport optax\nfrom typing import Collection, Callable, Optional\n\n# We generate some toy data, 1000 sets each with a size drawn from\n# an exponential distribution. The maximum size is fixed to 400.\n# Each set has elements which are vectors of size two.\nrng = np.random.default_rng(1)\n\ndef generate(rng, size):\n n = rng.exponential(size=size)\n n *= 400 / np.max(n)\n n += 1 # ensure there are no empty sets\n n = n.astype(int)\n\n # The mapping to learn is a non-linear function of the inputs. One can also\n # replace np.mean with np.max or np.sum.\n def true_mapping(x):\n return np.log1p(np.abs(np.sum(x[:, 0] ** 2 + 3 * x[:, 1])))\n\n X = [np.array(rng.normal(size=(ni, 2)), dtype=np.float32) for ni in n]\n y = [true_mapping(x) for x in X]\n\n return X, y\n\ndef preprocess(X, y, padded_X_size, padded_y_size, dtype=np.float32):\n \"\"\"\n Concatenates the input sets and pads inputs and outputs to fixed sizes.\n\n Returns padded arrays, the summation matrix, and a mask to undo the padding.\n \"\"\"\n assert padded_y_size >= len(y)\n n = [len(x) for x in X]\n mask = np.zeros(padded_y_size, dtype=bool)\n mask[:len(y)] = 1\n assert padded_X_size >= np.sum(n)\n y = np.concatenate([y, np.zeros(padded_y_size - len(y), dtype=dtype)])\n X = np.concatenate(X, dtype=dtype)\n X = np.concatenate([X, np.zeros((padded_X_size - len(X),) + X.shape[1:], dtype=dtype)])\n indices = np.empty((np.sum(n), 2), dtype=int)\n a = 0\n for j, b in enumerate(np.cumsum(n)):\n indices[a:b, 0] = j\n indices[a:b, 1] = np.arange(a, b)\n a = b\n sum_matrix = sparse.BCOO(\n (np.ones(len(indices), dtype=np.int8), indices),\n shape=(padded_y_size, padded_X_size),\n indices_sorted=True, unique_indices=True)\n print(f\"X fractional overhead {np.mean(np.sum(sum_matrix.todense(), axis=0) == 0):.2f}\", )\n print(f\"y fractional overhead {np.mean(~mask):.2f}\")\n X = jnp.array(X)\n y = jnp.array(y)\n return X, y, sum_matrix, mask\n\nX_train, y_train = generate(rng, 100)\nX_train, y_train, sum_train, mask_train = preprocess(X_train, y_train, 5_000, 100)\n\nX_test, y_test = generate(rng, 100)\nX_test, y_test, sum_test, mask_test = preprocess(X_test, y_test, sum(len(x) for x in X_test), len(y_test))\n\nX fractional overhead 0.00\ny fractional overhead 0.00\nX fractional overhead 0.00\ny fractional overhead 0.00\n\n\nWe train the model until the loss on the test sample is lower than 0.1. Achieving a better loss is difficult with such a small training sample.\n\nclass MLP(nn.Module):\n nodes: Collection[int]\n nonlin: Callable\n output: Optional[int] = None\n\n @nn.compact\n def __call__(self, x):\n for size in self.nodes:\n x = nn.Dense(size)(x)\n x = self.nonlin(x)\n if self.output is not None:\n return nn.Dense(self.output)(x)\n return x\n\nclass Model(nn.Module):\n phi_nodes: Collection[int]\n rho_nodes: Collection[int]\n\n def setup(self):\n self.phi = MLP(self.phi_nodes, nn.relu)\n self.rho = MLP(self.rho_nodes, nn.relu, 1)\n\n @nn.compact\n def __call__(self, x, sum_matrix):\n x = self.phi(x)\n y = sum_matrix @ x\n y = self.rho(y)\n return y.flatten()\n\nrng_key = jax.random.PRNGKey(0)\n\nmodel = Model((256,) * 6, (256,) * 6)\ntheta = model.init(rng_key, X_train, sum_train)\n# model.tabulate does not work with BCOO\n# print(model.tabulate(rng_key, X, sum_matrix))\n\nopt = optax.adam(learning_rate=1e-3)\nopt_state = opt.init(theta)\n\n@jax.jit\ndef loss_fn(theta, X, y, sum_matrix, mask):\n yp = model.apply(theta, X, sum_matrix)\n return jnp.mean(mask * (y - yp) ** 2)\n\n@jax.jit\ndef step(theta, opt_state, X, y, sum_matrix, mask):\n loss, grad = jax.value_and_grad(loss_fn)(theta, X, y, sum_matrix, mask)\n updates, opt_state = opt.update(grad, opt_state)\n theta = optax.apply_updates(theta, updates)\n return loss, theta, opt_state\n\nbest_loss = np.inf\nbest_theta = theta\ntrain_loss = []\ntest_loss = []\nfor epoch in range(1, 5000):\n loss, theta, opt_state = step(theta, opt_state,\n X_train, y_train, \n sum_train, mask_train)\n train_loss.append(loss)\n\n loss = loss_fn(theta, X_test, y_test, sum_test, mask_test)\n test_loss.append(loss)\n\n if test_loss[-1] < best_loss:\n best_loss = test_loss[-1]\n best_theta = theta\n\n stop = best_loss < 0.1\n if epoch == 1 or stop or epoch % 50 == 0:\n print(f\"epoch = {epoch:5} \"\n f\"loss(train) = {train_loss[-1]:6.3f} \"\n f\"loss(test) = {test_loss[-1]:6.3f}\")\n if stop:\n break\n\nepoch = 1 loss(train) = 23.354 loss(test) = 7.235\nepoch = 50 loss(train) = 2.143 loss(test) = 2.690\nepoch = 100 loss(train) = 0.434 loss(test) = 0.438\nepoch = 149 loss(train) = 0.019 loss(test) = 0.095\n\n\nI plot the training and testing loss as a function of the epoch. Adam does a good job of minimizing the loss without further rate scheduling. I tried several other optimizers, Adam performed best out of those which worked (some optimizers from optax fail on this problem with an error message).\n\nfrom matplotlib import pyplot as plt\n\nplt.figure()\nplt.plot(train_loss, label=\"train\")\nplt.plot(test_loss, label=\"test\")\nplt.legend()\nplt.xlabel(\"epoch\")\nplt.ylabel(\"loss\")\nplt.semilogy();\n\n\n\n\nTo show that the network managed to learn the mapping, I compare the predictions on the test set with the corresponding true values. While there are still sizable residuals, the model has learned to approximate the true mapping. Improving the model further would require a larger training sample.\n\nyp = model.apply(best_theta, X_test, sum_test)\n\nfig, ax = plt.subplots(2, 1, sharex=True, sharey=False)\nax[0].plot(y_test, \".\", label=\"input\")\nax[0].plot(yp, \"o\", mfc=\"none\", label=\"model\")\nax[0].legend()\nres = y_test - yp\nax[1].plot(res, \".\")\nax[1].set_ylabel(\"residual\")\nax[1].set_xlabel(\"set index\")\n\nplt.figure()\nplt.hist(res, bins=20, density=True)\nplt.title(f\"standard deviation = {np.std(res):.2f}\");" + }, + { + "objectID": "posts/SPD with negative weights.html", + "href": "posts/SPD with negative weights.html", + "title": "Applying the SPD approximation to negative weights", + "section": "", + "text": "In Bohm and Zech, NIMA 748 (2014) 1-6, the scaled Poisson distribution is discussed, which is an approximate distribution for sums of weights. This distribution is used in fits to histograms which are filled with weighted samples.\nI demonstrate here that a fit based on the SPD performs better than a least-squares fit, which effectively uses a normal distribution instead of the SPD to describe random fluctuations. I run repeated simulations in which weighted samples are generated. They are filled into a generalized histogram that keeps track of the variance of the weights. The cost function for the fit is simple: all bins in the histogram have the same constant expectation. The cost is computed with the least-squares method and the SPD method, respectively.\nI measure the bias and the variance of the results in repeated toy experiments. The SPD-based cost function yields a smaller bias and the estimated uncertainty better reflects the true variance of the fit results.\n\nimport numpy as np\nimport numba as nb\nimport matplotlib.pyplot as plt\nfrom iminuit import Minuit\nfrom scipy.stats import norm\nimport boost_histogram as bh \n\n\n@nb.njit\ndef cost_lsq(w, wvar, mu):\n r = 0.0\n for i in range(len(w)):\n if wvar[i] > 0:\n r += (w[i] - mu) ** 2 / wvar[i]\n return r\n\n\n@nb.njit\ndef cost_ml(w, wvar, mu):\n # Bohm and Zech, NIMA 748 (2014) 1-6\n r = 0.0\n for i in range(len(w)):\n if wvar[i] > 0 and w[i] * mu > 0:\n s_inv = w[i] / wvar[i]\n mu_prime = mu * s_inv\n n_prime = w[i] * s_inv \n r += mu_prime - n_prime * np.log(mu_prime)\n return 2 * r\n\n\nnmc = 1000\n\ndef run(cost, w_mu, w_sigma_rel, mu_pts, bins):\n expected_mu = w_mu * mu_pts / bins\n rng = np.random.default_rng(seed=1)\n fitted_mu = []\n fitted_mu_var = []\n h = bh.Histogram(bh.axis.Regular(bins, 0, 1), storage=bh.storage.Weight())\n for imc in range(nmc):\n h.reset()\n n = rng.poisson(mu_pts)\n x = rng.uniform(size=n)\n wi = rng.normal(w_mu, w_sigma_rel * w_mu, size=n)\n h.fill(x, weight=wi)\n m = Minuit(lambda mu: cost(h.values(), h.variances(), mu), mu=expected_mu)\n m.limits[\"mu\"] = (0, None)\n m.migrad()\n if m.valid:\n fitted_mu.append(m.values[0])\n fitted_mu_var.append(m.errors[0] ** 2)\n else:\n print(\"error\", imc)\n \n bias = np.mean(fitted_mu) - expected_mu\n sample_var = np.var(fitted_mu, ddof=1)\n mean_fitted_var = np.mean(fitted_mu_var)\n return expected_mu, bias, mean_fitted_var, sample_var\n\nresults = tuple(map(lambda cost: run(cost, 1.0, 0.1, 100, 5), (cost_lsq, cost_ml)))\nresults = np.array(results)\n\nlabels = \"LSQ\", \"ML\"\nfor label, (expected_mu, bias, mean_fitted_var, sample_var ) in zip(labels, results):\n print(label)\n print(f\" relative bias = {bias / expected_mu:.3f}\")\n print(f\" average estimated variance {mean_fitted_var ** 0.5:.3f}, true variance {sample_var ** 0.5:.3f}\")\n\nLSQ\n relative bias = -0.042\n average estimated variance 1.966, true variance 2.158\nML\n relative bias = -0.001\n average estimated variance 2.008, true variance 2.035" + }, + { + "objectID": "posts/Numerically stable calculation of invariant mass.html", + "href": "posts/Numerically stable calculation of invariant mass.html", + "title": "Numerically stable calculation of invariant mass", + "section": "", + "text": "The art of writing numerical stable formulas is to find a mathematically equivalent computation for a formula that produces accurate results even though computation with floating point numbers suffers from finite precision and round-off. We derive here numerically stable formulas for the calculation of the invariant mass of an ancestor particle in particle physics from the momenta and (assumed) masses of its products." + }, + { + "objectID": "posts/Numerically stable calculation of invariant mass.html#two-body-decay", + "href": "posts/Numerically stable calculation of invariant mass.html#two-body-decay", + "title": "Numerically stable calculation of invariant mass", + "section": "Two-body decay", + "text": "Two-body decay\nWe start with a derivation of a numerically stable formula for the 1D case, and then move to the general 3D case. The 1D solution is not applicable in practice, but it is easier to understand one of the two main numerical issues with the naive implementation when we start in 1D.\n\nSolution in 1D\nWe start with the formula for the invariant mass squared of an ancestor particle which decays into two children,\n\\[\nM^2 = m_1^2 + m_2^2 + 2 (E_1 E_2 - \\vec{p}_1 \\vec{p}_2).\n\\]\nFor small \\(m_i\\), we have a catastrophic cancelation in the third term. In floating point precision, the third term can be zero, even if the masses are small but positive. Due to round-off errors, it is even possible that the third term gets negative.\nTo fix this, we rewrite the product of the energies\n\\[\nE_1 E_2 = \\sqrt{|\\vec{p}_1|^2 + m_1^2}\\sqrt{|\\vec{p}_2| + m_2^2} = |\\vec{p}_1| |\\vec{p}_2| \\sqrt{\\left(1 + \\frac{m_1^2}{|\\vec{p}_1|^2}\\right)\\left(1 + \\frac{m_2^2}{|\\vec{p}_2|^2}\\right)} = |\\vec{p}_1| |\\vec{p}_2| \\sqrt{1 + x}\n\\]\nwith\n\\[\nx = \\frac{m_1^2}{|\\vec{p}_1|^2} + \\frac{m_2^2}{|\\vec{p}_2|^2} + \\frac{m_1^2}{|\\vec{p}_1|^2} \\frac{m_2^2}{|\\vec{p}_2|^2}\n\\]\nIn 1D, we have (assuming parallel vectors)\n\\[\nE_1 E_2 - \\vec{p}_1 \\vec{p}_2 = E_1 E_2 - |\\vec{p}_1||\\vec{p}_2| = |\\vec{p}_1||\\vec{p}_2| (\\sqrt{x + 1} - 1)\n\\]\nThe cancellation is now reduced to the expression \\(\\sqrt{x + 1} - 1\\). With a brilliant trick from Stackoverflow, we can rewrite this as\n\\[\n\\sqrt{x + 1} - 1 = \\frac{(\\sqrt{x + 1} - 1)(\\sqrt{x + 1} + 1)}{\\sqrt{x + 1} + 1} = \\frac{x}{\\sqrt{x + 1} + 1}\n\\]\nwhich is numerically stable, as the following plot demonstrates.\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nx = np.geomspace(1e-8, 1e2, 1000).astype(np.float32)\n\nplt.plot(x, np.sqrt(x ** 2 + 1) - 1, label=\"sqrt(x^2 + 1) - 1\")\nplt.plot(x, x ** 2 / (np.sqrt(x ** 2 + 1) + 1), label=\"x^2 / (sqrt(x^2 + 1) + 1)\")\nplt.loglog()\nplt.legend();\n\n\n\n\nThe rewritten formula for \\(M^2\\) produces a value that is always non-negative even in the presence of round-off errors. This is a common issue with naive implementations even in double precision.\nBelow, we compare a naive implementation msq1 with the numerically stable implementation msq2 on random values. The reference is computed with the naive implementation using mpmath on floating point numbers with 100 digits of precision.\n\nimport numpy as np\nimport numba as nb\nimport matplotlib.pyplot as plt\nimport mpmath as mp\n\n# naive implementation\ndef msq1(p1, p2, m1, m2):\n m1_sq = m1 ** 2\n m2_sq = m2 ** 2\n \n # energies of particles 1 and 2\n e1 = np.sqrt(p1 ** 2 + m1_sq)\n e2 = np.sqrt(p2 ** 2 + m2_sq)\n\n # dangerous cancelation in third term\n return m1_sq + m2_sq + 2 * (e1 * e2 - p1 * p2)\n\n\n# numerically stable implementation\n@nb.vectorize\ndef msq2(p1, p2, m1, m2):\n x1 = (m1 / p1) ** 2\n x2 = (m2 / p2) ** 2\n x = x1 + x2 + x1 * x2\n y = 2 * p1 * p2 * x / (np.sqrt(1 + x) + 1)\n return m1 ** 2 + m2 ** 2 + y\n\n\nrng = np.random.default_rng(1)\n\np1 = 10 ** rng.uniform(0, 5, size=10000).astype(np.float32)\np2 = 10 ** rng.uniform(0, 5, size=10000).astype(np.float32)\nm1 = 10 ** rng.uniform(-5, 5, size=10000).astype(np.float32)\nm2 = 10 ** rng.uniform(-5, 5, size=10000).astype(np.float32)\n\n# as reference we use naive formula computed with mpmath\nM0 = []\nwith mp.workdps(100):\n for p1i, p2i, m1i, m2i in zip(p1, p2, m1, m2): \n p1i = mp.mpf(float(p1i))\n p2i = mp.mpf(float(p2i))\n \n m1_sq = mp.mpf(float(m1i)) ** 2\n m2_sq = mp.mpf(float(m2i)) ** 2\n\n e1 = mp.sqrt(p1i ** 2 + m1_sq)\n e2 = mp.sqrt(p2i ** 2 + m2_sq)\n\n M0.append(float(m1_sq + m2_sq + 2 * (e1 * e2 - p1i * p2i)))\n\nM0 = np.array(M0)\n\nM = {f.__name__: f(p1, p2, m1, m2) for f in (msq1, msq2)}\n\nfig, ax = plt.subplots(1, len(M), figsize=(len(M) * 4.5, 4), sharey=True)\nfor (fname, Mi), axi in zip(M.items(), ax): \n axi.hist((Mi - M0) / M0, bins=100)\n axi.set_yscale(\"log\")\n axi.set_title(fname)\nfig.supxlabel(r\"$(M'^2 - M^2) / M^2$\");\n\n\n\n\nThe numerically stable formula retains full accuracy, while the naive formula suffers a catastrophic loss of accuracy for certain input values.\n\n\nSolution in 3D\nIn 3D, the solution is more complex than in the 1D case. We have now\n\\[\nE_1 E_2 - \\vec{p}_1 \\vec{p}_2 = |\\vec{p}_1| |\\vec{p}_2| \\left(\\sqrt{x + 1} - \\frac{\\vec{p}_1 \\vec{p}_2}{|\\vec{p}_1| |\\vec{p}_2|}\\right) = |\\vec{p}_1| |\\vec{p}_2| \\left(\\sqrt{x + 1} - \\cos{\\alpha}\\right)\n\\]\nwhere \\(\\alpha\\) is the angle between \\(\\vec{p}_1\\) and \\(\\vec{p}_2\\). We have again a numerical instability for small \\(x\\) and small \\(\\alpha\\), when \\(\\vec{p}_1\\) and \\(\\vec{p}_2\\) are nearly parallel, which happens often in practice.\nTo avoid the cancelation in the subtraction of two numbers close to 1, we use the same trick as before, but get a different outcome\n\\[\n\\sqrt{x + 1} - \\cos{\\alpha} = \\frac{x + 1 - \\cos^2{\\alpha}}{\\sqrt{x + 1} + \\cos{\\alpha}} = \\frac{x + \\sin^2{\\alpha}}{\\sqrt{x + 1} + \\cos{\\alpha}}.\n\\]\nThis time we only replace the left-hand side with the right-hand side if \\(\\cos{\\alpha} > 0\\), since for other values of the cosine, there is no subtraction happening. The expression is again guaranteed to produce a non-negative value even in the presence of round-off errors.\nSo far so good, but we face an additional issue. The naive computation of \\(\\alpha\\) is also not numerically stable for small \\(\\alpha\\) and we need another trick from Stackoverflow to compute it accurately,\n\\[\n\\alpha = \\rm{arctan2}(|\\vec{p}_1 \\times \\vec{p}_2|, \\vec{p}_1 \\vec{p}_2 )\n\\]\nwhere \\(\\rm{arctan2}\\) refers to the computer function with the same name.\nWe again compare a naive implementation msq1 with the numerically stable version msq2 on random inputs.\n\nimport numpy as np\nimport numba as nb\nimport matplotlib.pyplot as plt\nimport mpmath as mp\n\n# naive implementation\ndef msq1(px1, py1, pz1, px2, py2, pz2, m1, m2):\n p1_sq = px1 ** 2 + py1 ** 2 + pz1 ** 2\n p2_sq = px2 ** 2 + py2 ** 2 + pz2 ** 2\n m1_sq = m1 ** 2\n m2_sq = m2 ** 2\n \n # energies of particles 1 and 2\n e1 = np.sqrt(p1_sq + m1_sq)\n e2 = np.sqrt(p2_sq + m2_sq)\n\n # dangerous cancelation in third term\n return m1_sq + m2_sq + 2 * (e1 * e2 - (px1 * px2 + py1 * py2 + pz1 * pz2))\n\n\n# numerically stable implementation\n@nb.vectorize\ndef msq2(px1, py1, pz1, px2, py2, pz2, m1, m2):\n p1_sq = px1 ** 2 + py1 ** 2 + pz1 ** 2\n p2_sq = px2 ** 2 + py2 ** 2 + pz2 ** 2\n m1_sq = m1 ** 2\n m2_sq = m2 ** 2\n x1 = m1_sq / p1_sq\n x2 = m2_sq / p2_sq\n x = x1 + x2 + x1 * x2\n a = angle(px1, py1, pz1, px2, py2, pz2)\n cos_a = np.cos(a)\n if cos_a >= 0:\n y1 = (x + np.sin(a) ** 2) / (np.sqrt(x + 1) + cos_a) \n else:\n y1 = -cos_a + np.sqrt(x + 1) \n y2 = 2 * np.sqrt(p1_sq * p2_sq)\n return m1_sq + m2_sq + y1 * y2\n\n\n# numerically stable calculation of angle\n@nb.njit\ndef angle(x1, y1, z1, x2, y2, z2):\n # cross product\n cx = y1 * z2 - y2 * z1\n cy = x1 * z2 - x2 * z1\n cz = x1 * y2 - x2 * y1\n \n # norm of cross product\n c = np.sqrt(cx * cx + cy * cy + cz * cz)\n \n # dot product\n d = x1 * x2 + y1 * y2 + z1 * z2\n \n return np.arctan2(c, d)\n\n\nrng = np.random.default_rng(1)\n\ndtype = np.float32\nsize = 10000\npx1 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)\npy1 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)\npz1 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)\npx2 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)\npy2 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)\npz2 = (10 ** rng.uniform(0, 5, size=size) * rng.choice([-1, 1], size=size)).astype(dtype)\nm1 = 10 ** rng.uniform(-5, 5, size=size).astype(dtype)\nm2 = 10 ** rng.uniform(-5, 5, size=size).astype(dtype)\n\n# as reference we use naive formula computed with mpmath\nM0 = []\nwith mp.workdps(100):\n for px1i, py1i, pz1i, px2i, py2i, pz2i, m1i, m2i in zip(px1, py1, pz1, px2, py2, pz2, m1, m2):\n px1i = mp.mpf(float(px1i))\n py1i = mp.mpf(float(py1i))\n pz1i = mp.mpf(float(pz1i))\n px2i = mp.mpf(float(px2i))\n py2i = mp.mpf(float(py2i))\n pz2i = mp.mpf(float(pz2i))\n \n m1_sq = mp.mpf(float(m1i)) ** 2\n m2_sq = mp.mpf(float(m2i)) ** 2\n\n e1 = mp.sqrt(px1i ** 2 + py1i ** 2 + pz1i ** 2 + m1_sq)\n e2 = mp.sqrt(px2i ** 2 + py2i ** 2 + pz2i ** 2 + m2_sq)\n\n M0.append(float(m1_sq + m2_sq + 2 * (e1 * e2 - (px1i * px2i + py1i * py2i + pz1i * pz2i))))\n\nM0 = np.array(M0)\n\nM = {f.__name__: f(px1, py1, pz1, px2, py2, pz2, m1, m2) for f in (msq1, msq2)}\n\nfig, ax = plt.subplots(1, len(M), figsize=(len(M) * 4.5, 4), sharey=True)\nfor (fname, Mi), axi in zip(M.items(), ax):\n d = (Mi - M0) / M0\n axi.hist(d, bins=100)\n axi.set_yscale(\"log\")\n axi.set_title(f\"{fname}\")\nfig.supxlabel(r\"$(M'^2 - M^2) / M^2$\", y=-0.05);\n\n\n\n\nWe again see great loss of accuracy for the naive implementation, while the numerically stable formula produces accurate results." + }, + { + "objectID": "posts/UncertaintiesLecture.html", + "href": "posts/UncertaintiesLecture.html", + "title": "LHCb Startertalk on Uncertainties", + "section": "", + "text": "by Hans Dembinski, TU Dortmund" + }, + { + "objectID": "posts/UncertaintiesLecture.html#what-is-an-uncertainty-really", + "href": "posts/UncertaintiesLecture.html#what-is-an-uncertainty-really", + "title": "LHCb Startertalk on Uncertainties", + "section": "What is an uncertainty, really?", + "text": "What is an uncertainty, really?\nWe write \\(a = 3 \\pm 0.5\\), where 0.5 is the standard error (jargon: “error” = uncertainty).\nWhat does this truely mean?\n\nThe true value given by nature is 3, but when we measure it, it fluctuates (e.g. Heisenberg uncertainty principle).\nI made an experiment and this expresses my belief that the value is most likely 3 and is located within the interval 2.5 to 3.5 with 68 % probability. This is based on the data that I recorded and my initial assumption that the value of \\(a\\) is positive.\nSomeone made an experiment and their estimate for the true value of \\(a\\) based on the data is 3. The interval (2.5, 3.5) contains the true value with 68 % probability.\n\n\n\nIs nonsense, but b) and c) are valid interpretations\n\nIn HEP, we generally mean c), because we compute uncertainties based on the Frequentist school of statistics\n\nfollows from the Bayesian school of statistics\n\nIn b) or c), what does “68% probability” really mean in each context?" + }, + { + "objectID": "posts/UncertaintiesLecture.html#statistical-and-systematic-uncertainties", + "href": "posts/UncertaintiesLecture.html#statistical-and-systematic-uncertainties", + "title": "LHCb Startertalk on Uncertainties", + "section": "Statistical and systematic uncertainties", + "text": "Statistical and systematic uncertainties\n\nEither type can be correlated or uncorrelated\n\n\nStatistical uncertainty\n\nOrigin: We use finite sample instead of infinite distribution\nGoes down as simple size increases\nExample\n\nArithmetic mean \\(\\bar x = \\frac 1N\\sum_i x_i\\)\nVariance of mean $ = {N(N - 1)}_i (x_i - x)^2$\n\nWe have reliable standard recipes to calculate these\nVariance or standard deviation?\n\nStandard deviation \\(\\sigma = \\sqrt{V}\\)\nWe communicate standard deviations, but compute variances\nVariance is more fundamental\n\nSecond central moment of distribution\n\\(V = V_1 + V_2\\), but \\(\\sigma = \\sqrt{\\sigma_1^2 + \\sigma_2^2}\\)\n\n\n\n\n\nSystematic uncertainty\n\nOrigin: Imperfect appartus or technique, deviations of reality from simplified models\nQuantifies potential mistakes in analysis or lack of knowledge\nDoes not go down as sample size increases\nCorrelated examples\n\nReal-life: Measured the length of ten shoes with ruler than has factory tolerance of 0.1 mm\nLHCb: Measured cross-sections with value of luminosity that has uncertainty of 1-2%\n\nUncorrelated examples\n\nReal-life: ???\nLHCb: Measured momentum of particle in LHCb differs from true momentum \\(\\sigma_p/p \\approx 0.5\\!-\\!1\\%\\)\n\nConceptually different, requires different math\nNot well covered in statistics books, but some theoretical results available\nMany misconceptions\n\n\n\nStatistical errors can be systematic errors\n\nDepends on context\nStatistical error of parameter in control study is systematic error in main study\nExample: LHCb performs calibration measurements to determine tracking and PID efficiencies\n\nStatistical errors in calibration measurements are systematic errors in main measurement" + }, + { + "objectID": "posts/UncertaintiesLecture.html#statistical-uncertainties-basic-concepts", + "href": "posts/UncertaintiesLecture.html#statistical-uncertainties-basic-concepts", + "title": "LHCb Startertalk on Uncertainties", + "section": "Statistical uncertainties: Basic concepts", + "text": "Statistical uncertainties: Basic concepts\nWe need to distinguish between true value, estimator, and estimate (or estimated value).\n\nSample \\(\\{ x_i \\}\\) drawn from distribution \\(F(x)\\) consists of random values \\(x_i\\) that are independent and identically distributed (iid)\nSample values occur with probability\n\n\\(P(k) = F(k)\\) if \\(k\\) discrete\n\\(d P(x) = d F(x) = f(x) dx\\) if \\(x\\) continuous\n\n\\(f(x)\\) probability density function (pdf)\n\\(F(x)\\) cumulative density function (cdf)\nWill refer to \\(f(x)\\) and \\(F(x)\\) as the distribution\nBeware lax jargon: distribution != sample\n\n\n# Example: standard normal distribution (\"Gaussian\")\nfrom matplotlib import pyplot as plt\nimport numpy as np\nfrom scipy.stats import norm\nx = np.linspace(-5, 5, 1000)\nfx = norm.pdf(x)\nFx = norm.cdf(x)\nsample = norm.rvs(size=50, random_state=1)\n\n\ndef plot():\n fig, ax = plt.subplots(1, 2, figsize=(14, 5), sharex=True)\n for axi, y, label in zip(ax, (fx, Fx), (\"f(x), pdf\", \"F(x), cdf\")):\n axi.plot(x, y, label=label)\n axi.set_xlabel(\"x\")\n ax[0].plot(sample, np.zeros_like(sample), \"|\", label=\"sample\")\n for axi in ax: axi.legend()\n\n\nplot()\n\n\n\n\n\nDistribution usually depends on parameters that we are interested in\n\nExample: Normal distribution depends on parameters \\(\\mu, \\sigma\\) (location, width)\n\nNotation: \\(f(x; a, b, \\dots)\\), semicolon separates random variable \\(x\\) from parameters \\(a, b, \\dots\\)\nIn general, \\(x\\) can be a vector and parameters can be written as vector, too: \\(f(\\vec x; \\vec a)\\)\n\n\nEstimation\n\nEstimation is statistician speak for “measurement”\nWe have sample \\(\\{ x_i \\}\\) which is distributed according to \\(f(x; a)\\), \\(a\\) is unknown\nEstimator \\(h[x]\\) maps data sample to estimate \\(\\hat a\\) of \\(a\\)\nNotation\n\n\\(h[x]\\) estimator maps sample \\(\\{ x_i \\}\\) to value\n\\(f(x)\\) is function that maps value to value\n\nExamples of estimators of location of distribution\n\nArithmetic mean \\(h_1[x] = \\frac 1N \\sum_i x_i\\)\nSample median \\(h_2[x]\\): Sort \\(x_i\\), pick value in the middle\n\nEstimator can be formula in closed form or an algorithm (recipe)\n\n\n# Example: normal distribution with true value mu=1, sigma=0.5\n# mu is of interest; sigma is not of interest, a nuisance parameter\n\nfrom matplotlib import pyplot as plt\nimport numpy as np\nfrom scipy.stats import norm\n\nmu, sigma = 1, 0.5\ndist = norm(mu, sigma)\nx = np.linspace(-1, 3, 1000)\nsample = dist.rvs(size=20, random_state=1)\n\nest1 = np.mean(sample)\nest2 = np.median(sample)\n\n\ndef plot():\n plt.plot(x, dist.pdf(x))\n plt.xlabel(\"x\")\n plt.plot(sample, np.zeros_like(sample), \"|\")\n plt.axvline(dist.stats()[0], label=\"$\\mu$\", color=\"C3\")\n plt.axvline(est1, ls=\"--\", label=\"mean\", color=\"C2\")\n plt.axvline(est2, ls=\":\", label=\"median\", color=\"C4\")\n plt.legend();\n\n\nplot()\n\n\n\n\n\n\nOptimal estimators\n\nMany kinds of estimators, some better than others\nWe want optimal estimators\nSome desirable properties\n\nAs close to true value as possible (we come back to that)\nEasy to compute and/or easy to apply to any problem\nRobustness against outliers\n\nIn HEP, we mainly use\n\nMaximum-likelihood estimator (MLE)\nPlug-in estimator\nBootstrap estimator (meta estimator)\n\n\n\n\nHow far away is estimate from true value?\n\nDepends on estimator, sample size, distribution\nAlmost magic: accuracy of estimator can be also estimated from sample!\nDefine “close to true value”\n\nIn experiment, we have sample \\(\\{ x_i \\}\\) with size \\(N\\) and obtained estimate \\(\\hat a = h[x]\\)\nSample \\(\\{ x_i \\}\\) is random and so is estimate \\(\\hat a\\)\nIf experiment is identically repeated \\(K\\) times, we get sample \\(\\{ x_i \\}_k\\) and estimate \\(\\hat a_k\\) each time\nQuantify properties of sample of estimates \\(\\{ \\hat a_1, \\hat a_2, \\dots, \\hat a_K \\}\\) to describe accuracy of estimator with bias and variance estimates\n\nBias: \\(\\hat B[\\hat a] = \\frac1K \\sum_i (\\hat a_i - a)\\)\nVariance: \\(\\hat V[\\hat a] = \\frac1K\\sum_i (\\hat a_i - a)^2\\)\n\n\nExample for a biased estimator: if true mean \\(\\mu\\) is unknown, variance estimate \\(\\hat V[x] = \\frac1N\\sum_i (x_i - \\hat\\mu)^2\\) is biased\n\n\nfrom matplotlib import pyplot as plt\nimport numpy as np\nfrom scipy.stats import norm\n\n# true values and distribution\nmu, sigma = 1, 1\ndist = norm(mu, sigma)\n\nN, K = 10, 1000\nvar_1, var_2 = [], []\nfor k in range(K):\n sample = dist.rvs(size=N, random_state=k+1)\n hat_mu = np.mean(sample)\n v1 = np.mean((sample - np.mean(sample)) ** 2)\n v2 = np.mean((sample - mu) ** 2)\n var_1.append(v1)\n var_2.append(v2)\n\n\n# hidden plotting code used in next cell\ndef plot():\n mu, sigma = dist.stats()\n fig, ax = plt.subplots(1, 2, sharex=True, sharey=True, figsize=(14, 5))\n ax[0].hist(var_1, bins=50, label=\"$\\hat V[x; \\hat \\mu]$\");\n ax[1].hist(var_2, bins=50, label=\"$\\hat V[x; \\mu]$\");\n for axi in ax:\n axi.axvline(sigma ** 2, color=\"C3\", label=\"true V[x]\")\n ax[0].axvline(np.mean(var_1), color=\"C1\", ls=\"--\", label=\"mean of $\\hat V[x; \\hat \\mu]$\")\n ax[1].axvline(np.mean(var_2), color=\"C1\", ls=\"--\", label=\"mean of $\\hat V[x; \\mu]$\")\n for axi in ax:\n axi.legend()\n\n\nplot()\n\n\n\n\n\nPerhaps surprising that \\(\\hat V[x; \\hat \\mu]\\) is biased although \\(\\hat \\mu\\) is an unbiased estimate of \\(\\mu\\)\nIn this case, simple well-known bias correction: \\(\\frac N{N-1}\\hat V[x; \\hat \\mu]\\) is unbiased (for derivation see e.g. F. James book (references at end))\n\n\n# Example for variance of estimator: variance of mean and median\nfrom matplotlib import pyplot as plt\nimport numpy as np\nfrom scipy.stats import norm, t\n\nmu, sigma = 0, 1\n\ndist = {\n \"norm\": norm(mu, sigma),\n \"t\": t(2, mu, sigma),\n} \n\nN, K = 10, 1000\ndata = {}\nfor dkey, d in dist.items():\n est = {\"mean\": [], \"median\": []}\n all_samples = []\n for k in range(K):\n sample = d.rvs(size=N, random_state=k+1)\n est[\"mean\"].append(np.mean(sample))\n est[\"median\"].append(np.median(sample))\n all_samples.append(sample)\n data[dkey] = est, all_samples\n\n\ndef plot():\n for dkey, (est, all_samples) in data.items():\n fig, ax = plt.subplots(1, 3, figsize=(14, 4))\n ax[0].hist(np.concatenate(all_samples), bins=50, range=(-10, 10))\n ax[0].set_title(dkey)\n ax[0].set_xlabel(\"x\")\n for axi, (key, esti) in zip(ax[1:], est.items()):\n axi.hist(esti, bins=50)\n axi.set_title(f\"$\\\\langle \\mathrm{{{key}}} \"\n f\"\\\\rangle = {np.mean(esti):.3f}$ \"\n f\"$\\hat V[\\mathrm{{{key}}}] = {np.var(esti):.3f}$\"\"\")\n axi.set_xlabel(f\"{key}\")\n\n\nplot()\n\n\n\n\n\n\n\n\nFor normal distribution\n\nMedian has larger variance than arithmetic mean\n\nFor Student’s t distribution\n\nMedian has smaller variance than arithmetic mean\n\nWhich estimator is least biased and has smallest variance depends on distribution\nGood estimator for any distribution: maximum-likelihood estimator\n\n\n\nPlug-in estimator\n\nPlug-in principle: replace true values in some formula with estimates, e.g.\n\n\\[\nc = g(a, b, \\dots) \\to \\hat c = g(\\hat a, \\hat b, \\dots)\n\\]\n\nJustification (1D, likewise for ND)\n\nFor \\(N\\to \\infty\\), \\(\\Delta a = \\hat a - a \\to 0\\) for any reasonable estimator\n\\(g(\\hat a) = g(a + \\Delta a) = g(a) + O(\\Delta a) \\to g(a)\\) for smooth \\(g\\)\n\nExample: \\(\\sqrt{N}\\) uncertainty estimator for Poisson-distributed count \\(N\\)\n\nVariance of Poisson distribution of counts \\(N\\): \\(V_N = \\lambda\\)\nPlug-in principle: Replace expected count \\(\\lambda\\) with observed count \\(N\\): \\(\\hat V_N = N\\)\nStandard deviation is square-root of variance: \\(\\hat \\sigma_N = \\sqrt{\\hat V_N} = \\sqrt{N}\\)\n\nPlug-in estimators are usually biased (unless g is linear) and may fail to give reasonable answers for very small samples\n\nExtreme case: \\(\\lambda = 0.01\\), sample \\(N = 0 \\to \\sqrt{0} = 0 \\neq 0.1\\)\n\nBe aware of potentially bias when you use plug-in estimator\n\n\n\nBootstrapping estimator bias and variance\n\nBootstrap method is generic way to compute bias and variance of any estimator\nLooks like a magic trick, but well-founded in theory (excellent introductory book by inventor Brad Efron at the end)\n\n\nParametric bootstrap\n\nLHCb jargon: “running toy simulations”\nSituation:\n\nYou have sample \\(\\{ x_i \\}\\) distributed along \\(f(x; a)\\) with unknown parameter \\(a\\)\nYou have estimator \\(\\hat a = h[x]\\) (can be a formula, fit, anything)\nYou want to estimate bias and variance of \\(h[x]\\)\n\nIf we knew true \\(a\\), could simply simulate the experiment many times\n\nGenerate sample \\(\\{ x_i \\}_k\\) from \\(f(x; a)\\) with same size as \\(\\{ x_i \\}\\), compute \\(\\hat a_k\\)\nCompute bias and variance estimates from sample \\(\\{ \\hat a_k \\}\\) as shown before\n\nDon’t know \\(f(x; a)\\), but its plug-in estimate \\(\\hat f(x) = f(x; \\hat a)\\)!\nSo do the same but sample from \\(\\hat f(x)\\)\nThis actually works, and very well!\n\n\n# parametric bootstrap to estimate variance of mean\nimport numpy as np\nfrom scipy.stats import norm\n\nmu, sigma = 1, 1\nN = 10\nsample = norm(mu, sigma).rvs(N, random_state=1)\n\n# in general, we use a fit here to get parameters of distribution\nmu_est = np.mean(sample)\nsigma_est = np.std(sample, ddof=1)\ndist = norm(mu_est, sigma_est)\n\nmu_est_b = []\nfor k in range(1000):\n b = dist.rvs(N, random_state=k)\n mu_est_b.append(np.mean(b))\n \nprint(f\"Bias = {np.mean(mu_est_b) - mu_est:.2f} \"\n f\"std.dev. = {np.std(mu_est_b, ddof=1):.2f} \"\n f\"(expected: {sigma / N ** 0.5:.2f})\")\n\nBias = -0.01 std.dev. = 0.39 (expected: 0.32)\n\n\n\n\nNon-parametric bootstrap\n\nTaking previous idea one step further\nSampling does not require \\(\\hat f(x)\\), but \\(\\hat F(x)\\) (the cdf)\n\\(\\hat F(x)\\) can be estimated directly from sample \\(\\{ x_i \\}\\) (empirical cdf)\nAccuracy a bit worse than parametric bootstrap, but even easier to use\n\n\n# bootstrap bias and variance of arithmetic mean\nimport numpy as np\nfrom resample.bootstrap import resample\nfrom scipy.stats import norm\n\nmu, sigma = 1, 1\nN = 10\nsample = norm(mu, sigma).rvs(N, random_state=1)\n\nmu_est_b = []\nfor replicas in resample(sample, size=1000):\n mu_est_b.append(np.mean(replicas))\n \nprint(f\"Bias = {np.mean(mu_est_b) - mu_est:.2f} \"\n f\"std.dev. = {np.std(mu_est_b, ddof=1):.2f} \"\n f\"(expected: {sigma / N ** 0.5:.2f})\")\n\nBias = 0.00 std.dev. = 0.38 (expected: 0.32)\n\n\n\n# Bonus: bootstrap bias and variance of np.var\nimport numpy as np\nfrom resample.bootstrap import bias, variance\nfrom scipy.stats import norm\n\nmu, sigma = 1, 1\nN = 10\nsample = norm(mu, sigma).rvs(N, random_state=1)\n\nfor ddof in (0, 1):\n def fn(s):\n return np.var(s, ddof=ddof)\n \n print(f\"Bias = {bias(fn, sample, random_state=1):.2f} \"\n f\"std.dev. = {np.sqrt(variance(fn, sample, random_state=1)):.2f}\")\n\nBias = -0.13 std.dev. = 0.53\nBias = 0.01 std.dev. = 0.59\n\n\n\nUse np.var(s, ddof=1) if you want unbiased estimate\n\n\n\n\nStandard intervals, coverage and coverage probability\n\nStandard interval for parameter \\(a\\): \\(\\hat a \\pm \\sqrt{\\hat V[\\hat a]}\\)\nNot a statement about true value of \\(a\\), just a quantification of the properties of the estimate!\nBut what does it mean then?\nQuadrature of circle: How to make a statement about the true value without using the true value?\nFrequentist school of statistics: construct intervals such that true value is covered by interval in fraction \\(P_\\text{cov}\\) of identically repeated experiments\nFormal way to do this: Neyman construction\n\nWorks perfectly for continuous distributions, but not for discrete (Poisson, Binomial, …)\nRarely used in practice, too cumbersome\n\nCoverage important and useful property of intervals\nJargon: “This interval has (proper) coverage”\n\nMeaning: actual coverage probability in identically repeated experiments is equal to expected \\(P_\\text{cov}\\)\n\nFor \\(N \\to \\infty\\), \\(\\hat a \\pm \\sqrt{\\hat V[\\hat a]}\\) interval has coverage probability of 68%\n\nAnd if well-constructed, also for finite \\(N\\)\n\nStandard errors should form interval with 68 % coverage probability\n\nIf coverage is too small/large, adjust interval estimator\n\nExample: Coverage of the commonly used Poisson interval \\(N \\pm \\sqrt{N}\\)?\n\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.stats import poisson\n\nN = [0.5, 10., 45.]\nK = 100000\n\ndata = []\nfor Ni in N:\n n = poisson(Ni).rvs(size=K, random_state=int(Ni))\n sigma_est = np.sqrt(n)\n inside = np.abs(Ni - n) < sigma_est\n cov_prob = np.mean(inside)\n data.append((Ni, n, sigma_est, inside, cov_prob))\n\n\ndef plot():\n fig, ax = plt.subplots(1, len(N), sharex=True, figsize=(14, 4))\n for (Ni, n, sigma_est, inside, cov_prob), axi in zip(data, ax):\n axi.axhline(Ni, color=\"0.5\", zorder=0)\n i = np.arange(50)\n n = n[:50]\n sigma_est = sigma_est[:50]\n inside = inside[:50]\n for m, c in zip((~inside, inside), (\"C0\", \"C1\")):\n axi.errorbar(i[m], n[m], sigma_est[m], fmt=\"o\", color=c)\n axi.set_title(f\"P_cov = {cov_prob:.2f}\")\n fig.supxlabel(\"repeated experiments\");\n\n\nplot()\n\n\n\n\n\nPoisson interval \\(N \\pm \\sqrt{N}\\) only has proper coverage for \\(N \\gtrsim 45\\)\n\nReason: if \\(N\\) fluctuates downwards, interval shrinks\n\nIntervals can be empty in some experiments; this is not ideal, but not a defect\nIf you have an estimate \\(\\mu\\) for the true value of \\(N_i\\), this is better: \\(N_i \\pm \\sqrt{\\mu}\\)\n\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.stats import poisson\n\nN = [0.5, 10, 20]\nK = 100000\n\ndata = []\nfor Ni in N:\n n = poisson(Ni).rvs(size=K, random_state=int(Ni))\n mu = np.mean(n)\n sigma_est = np.sqrt(mu)\n inside = np.abs(Ni - n) < sigma_est\n cov_prob = np.mean(inside)\n data.append((Ni, n, sigma_est * np.ones(len(n)), inside, cov_prob))\n\n\nplot()\n\n\n\n\n\n\nMaximum-likelihood estimator (MLE)\n\nMLE has attractive properties\n\nFor \\(N \\to \\infty\\), as close to true value as possible (proven)\n\nUnbiased and minimum variance\n\nIn finite samples\n\nBiased in general, but bias zero or small for many important distributions\nProbably as close to true value as possible (conjecture)\n\nEasy to compute MLE and its variance estimate (with suitable software)\nBut: Not robust against outliers or model misspecification\n\nMaximum-likelihood estimator for parameter \\(a\\) \\[\n\\begin{aligned}\nh_\\text{MLE}[x] &= \\text{argmax}_{a}\\, \\ln \\mathcal L(a) \\\\\n\\ln \\mathcal L(a) &:= \\ln \\prod_i P(x_i; a) = \\sum_i \\ln f(x_i; a) + \\text{const}\n\\end{aligned}\n\\]\nMLE example: arithmetic mean is MLE for parameter \\(\\mu\\) of normal distribution \\[\n\\begin{aligned}\n\\frac\\partial{\\partial \\mu} \\sum_i \\ln\\left( \\frac1{\\sqrt{2\\pi} \\sigma} \\exp\\left(-\\frac{(x_i - \\mu)^2}{2\\sigma^2} \\right)\\right) &\\overset{!}{=} 0 \\\\\n\\underbrace{\\frac\\partial{\\partial \\mu} \\sum_i \\ln\\left( \\frac1{\\sqrt{2\\pi} \\sigma} \\right)}_{0} - \\frac\\partial{\\partial \\mu} \\sum_i \\frac{(x_i - \\mu)^2}{2\\sigma^2} &= 0\\\\\n\\sum_i \\frac{x_i - \\mu}{\\sigma^2} &= 0 \\\\\n\\sum_i x_i - \\sum_i\\mu &= 0 \\\\\n\\Rightarrow \\hat\\mu = \\frac 1 N \\sum_i x_i &\n\\end{aligned}\n\\]\nIn general: Use numerical optimiser (e.g. MINUIT) to find MLE for any distribution\n\n\n\nUncertainty of maximum likelihood estimate\n\nTwo ways to compute uncertainty estimates for MLE based on theorems by Fisher and Wilks\nRecommended: lecture on MLE by Geyer (2013) https://www.stat.umn.edu/geyer/5601/notes/sand.pdf\n\n\nVariance estimate\n\nVariance estimate for MLE estimator \\(\\hat a\\) of parameter \\(a\\) \\[\n\\hat V[\\hat a] = -\\left(\\frac{\\partial^2 \\ln \\mathcal L(a)}{\\partial a^2}\\Bigg|_{a=\\hat a}\\right)^{-1}\n\\]\nCovariance estimate for multivariate MLE of parameter \\(\\vec a\\) \\[\n\\hat C_{ik}[\\hat{\\vec a}] = -\\left(\\frac{\\partial^2 \\ln \\mathcal L(\\vec a)}{\\partial a_i \\partial a_k}\\Bigg|_{\\vec a=\\hat {\\vec a}}\\right)^{-1}\n\\]\nMINUIT computes this numerically (HESSE method)\nExample: variance estimate for \\(\\hat\\mu\\) for normally distributed data with known width \\(\\sigma\\) \\[\n-\\left(\\frac{\\partial^2 \\ln \\mathcal L(\\mu)}{\\partial \\mu^2}\\right)^{-1} = \\dots = \\left(\\frac{\\partial^2}{\\partial \\mu^2} \\sum_i \\frac{(x_i - \\mu)^2}{2\\sigma^2}\\right)^{-1}\n\\] \\[ = -\\left(\\frac{\\partial}{\\partial \\mu} \\sum_i \\frac{(x_i - \\mu)}{\\sigma^2}\\right)^{-1} = \\left(\\sum_i \\frac1{\\sigma^2}\\right)^{-1} = \\frac{\\sigma^2}{N}\n\\]\nWell-known result for error on the mean\n\n\n\nProfile likelihood scan\n\nProduces asymmetric uncertainty intervals in general\nFor desired confidence level \\(CL\\) (a probability) solve \\(\\chi^2(\\lambda;M) = CL\\) for \\(\\lambda\\), where \\(M\\) is number of parameters of interest (subset of all parameters)\nSolve for contour in \\(\\vec a\\)-space (parameters of interest) $$\n2 = $$ where \\(\\hat b(\\vec a)\\) are maximum-likelihood estimates of nuisance parameters \\(\\vec b\\) for fixed values of \\(\\vec a\\)\nToo complicated? No problem, MINUIT computes all this numerically for you (MINOS method)\nCommonly used to draw uncertainty contours\nSometimes advertised as superior to variance estimate (HESSE method), but isn’t\n\nDoes not produce intervals with better coverage in general\nCannot combine asymmetric intervals/confidence regions from different experiments, see Roger’s lecture (references at the end)\n\nIf you really need this, publish full likelihood function with tools like pyhf or HistFactory\n\n\n# Example: fit parameters of shifted Student's t distribution\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom iminuit import Minuit\nfrom iminuit.cost import UnbinnedNLL\nfrom scipy.stats import t\n\nnd = 3\nmu = 1.0\nsample = t(nd, mu).rvs(size=50, random_state=1)\n\n\ndef model(x, nd, mu):\n return t(nd, mu).pdf(x)\n\ncost = UnbinnedNLL(sample, model)\n\nm = Minuit(cost, nd=2, mu=0)\nm.limits[\"nd\"] = (1, None)\nm.migrad() # also runs equivalent of m.hesse() by default\nm.minos()\n\n\n\n\nMigrad\n\n\nFCN = 165\nNfcn = 131\n\n\nEDM = 2.85e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nnd\n3.6\n1.5\n-1.2\n2.1\n1\n\n\n\n\n1\nmu\n0.97\n0.16\n-0.16\n0.16\n\n\n\n\n\n\n\n\n\n\nnd\nmu\n\n\nError\n-1.2\n2.1\n-0.16\n0.16\n\n\nValid\nTrue\nTrue\nTrue\nTrue\n\n\nAt Limit\nFalse\nFalse\nFalse\nFalse\n\n\nMax FCN\nFalse\nFalse\nFalse\nFalse\n\n\nNew Min\nFalse\nFalse\nFalse\nFalse\n\n\n\n\n\n\n\nnd\nmu\n\n\nnd\n2.36\n0.00743 (0.030)\n\n\nmu\n0.00743 (0.030)\n0.0264\n\n\n\n\n\n\nxm = np.linspace(-10, 10, 1000)\nplt.hist(sample, bins=20, density=True, label=\"sample\")\nplt.plot(xm, t(*m.values).pdf(xm), label=\"fit\")\nplt.plot(xm, t(nd, mu).pdf(xm), label=\"truth\")\nplt.legend();\n\n\n\n\n\nfig, ax = plt.subplots(1, 2, figsize=(14, 4))\nfor axi, par in zip(ax, (\"nd\", \"mu\")):\n plt.sca(axi)\n m.draw_mnprofile(par)\n\n\n\n\n\n\n\nMisspecified models\n\nMaximum-likelihood method assumes that model is correctly specified, all proven properties rely on that\nExamples of misspecified models\n\nAssumed: signal only, actually: signal + background\nAssumed: normal distribution, actually: distorted normal (wider tails, asymmetric, etc)\n\nIf model is misspecified\n\nMLE is biased in general, but may not be\nMLE uncertainties (from either method) no longer have coverage\n\nFundamentalist: models are essentially always misspecified, since they are always approximations\nPragmatist: Small deviations have negligible effect\nWorse at HL-LHC: higher statistics, so more sensitive to small deviations\n\nNeed goodness-of-fit criterion (not discussed here) to check whether model is reliable\n\nHow to fix this: make model more general (by adding parameters) so it is no longer misspecified\n\nUse Student’s t (or double-sided Crystal Ball) instead of normal distribution to fit peak\nUse polynomial of higher order for smooth background\n…" + }, + { + "objectID": "posts/UncertaintiesLecture.html#systematic-uncertainties-known-unknowns", + "href": "posts/UncertaintiesLecture.html#systematic-uncertainties-known-unknowns", + "title": "LHCb Startertalk on Uncertainties", + "section": "Systematic uncertainties: Known unknowns", + "text": "Systematic uncertainties: Known unknowns\n * Donald Rumsfeld: “There are known knowns, things we know that we know; and there are known unknowns, things that we know we don’t know. But there are also unknown unknowns, things we do not know we don’t know.”\n\nWhy data analyses take so long: Study data, detector, and methods to turn unknown unknowns into known knowns or known unknowns\nNo formal way, but generally…\n\nApply checks\n\nSplit data by magnet polarity, fill, etc., analyse splits separately, check for agreement\n\nWorry\n\nWhat could go wrong?\nWhat are my assumptions?\nExample: You take an efficiency or correction factor from simulation, but simulation differs from real experiment; perform control measurement or use standard LHCb control measurements\n\nMethods\n\nIf feasible, start from first principles or use methods that are known to be optimal/robust\nExamples\n\nUse Maximum-likelihood instead of least-squares fit\nUse bootstrap estimate of error\n\n\n\nGuidelines for handling systematic errors\n\nRoger Barlow: Systematic Errors: Facts and Fictions (2002)\nRoger Barlow, “Practical Statistics for Particle Physics” (2019), Section 6.5\n\n\n\n\n\nimage.png" + }, + { + "objectID": "posts/UncertaintiesLecture.html#error-propagation", + "href": "posts/UncertaintiesLecture.html#error-propagation", + "title": "LHCb Startertalk on Uncertainties", + "section": "Error propagation", + "text": "Error propagation\n\nYou have vector of values \\(\\vec a\\) with covariance matrix \\(C_a\\) and a smooth function \\(\\vec b = f(\\vec a)\\), you want to know \\(C_b\\)\n\\(\\vec b\\) can have different dimensionality from \\(\\vec a\\)\n\n\nOption 1: Standard error propagation\n\nBased on first order Taylor expansion \\[\nC_b = J \\, C_a \\, J^T\n\\] with Jacobi matrix \\(J\\) \\[\nJ_{ik} = \\frac{\\partial f_i}{\\partial a_k}\n\\]\nFor derivation, see F. James book (references at end)\nCan be used for statistical and systematic uncertainties!\nUseful special cases\n\nSum of scaled independent random values \\(c = \\alpha a + \\beta b\\) (\\(\\alpha, \\beta\\) are constant) \\[\n \\sigma^2_c = \\alpha^2 \\sigma^2_a + \\beta^2 \\sigma^2_b\n \\]\nProduct of powers of independent random values \\(c = a^\\alpha \\, b^\\beta\\) \\[\n \\left(\\frac{\\sigma_c}{c}\\right)^2 = \\alpha^2 \\left(\\frac{\\sigma_a}{a}\\right)^2 + \\beta^2 \\left(\\frac{\\sigma_b}{b}\\right)^2\n \\]\nDominated by component with largest (relative) uncertainty\nPut your effort in estimating largest component and be more lax on small components\n\nExample\n\nArithmetic mean \\[\n \\hat \\mu = \\frac 1N\\sum_i x_i\n \\]\nVariance of x (unbiased formula for unkown mean) \\[\n \\hat V[x] = \\frac 1{N-1} \\sum_i (x_i - \\hat\\mu)^2\n \\]\nVariance of the mean \\[\n \\hat{V}[\\hat \\mu] = \\frac1{N^2} \\sum_i \\hat{V}[x_i] = \\frac 1 {N(N - 1)}\\sum_i (x_i - \\hat \\mu)^2\n \\]\n\nError-prone to do by hand, use software to compute numerical derivative\n\n\n\nOption 2: Monte-Carlo propagation\n\nTo apply, you need to know distribution of \\(\\vec a\\)\nRandomly sample \\(K\\) times\nCompute sample \\(\\{\\vec b_k \\} = \\{ f(\\vec a_k) \\}\\)\nCompute sample covariance of \\(\\{\\vec b_k \\}\\)\n\n\nimport numpy as np\nfrom jacobi import propagate # uses Option 1\nfrom scipy.stats import multivariate_normal\n\na0, sigma_a0 = 2, 0.2\na1, sigma_a1 = 3, 0.5\n\na = [a0, a1]\ncov_a = np.zeros((2, 2))\ncov_a[0, 0] = sigma_a0 ** 2\ncov_a[1, 1] = sigma_a1 ** 2\n\n# option 1\nb, cov_b = propagate(np.mean, a, cov_a)\nprint(f\"option 1: b = {b:.2f} +/- {np.sqrt(cov_b):.2f}\")\n\n# option 2\na_sample = multivariate_normal(a, cov_a).rvs(size=10000)\nb_sample = np.mean(a_sample, axis=1) # sum over second axis\ncov_b = np.cov(b_sample, ddof=1)\nprint(f\"option 2: b = {b:.2f} +/- {np.sqrt(cov_b):.2f}\")\n\noption 1: b = 2.50 +/- 0.27\noption 2: b = 2.50 +/- 0.27\n\n\n\nResults identical here because function was linear; they differ in general\nWhich is better? Hard to say, depends on particular case\n\nOption 2 is conceptually better, but you may need very large samples (>10000) to beat option 1 in accuracy" + }, + { + "objectID": "posts/UncertaintiesLecture.html#lhcb-statistics-guidelines", + "href": "posts/UncertaintiesLecture.html#lhcb-statistics-guidelines", + "title": "LHCb Startertalk on Uncertainties", + "section": "LHCb Statistics Guidelines", + "text": "LHCb Statistics Guidelines\n\nLiving document on GitLab to collect information about best practices\n\nEveryone can edit this and propose new items!\nLHCb StatML conveners are editors, liaisons check new content\n\nIndex instead of a book\n\nBrief items with advice\nLinks to literature\n\nIn conversation, you can directly link to items, point to title and copy link (right-click menu)" + }, + { + "objectID": "posts/UncertaintiesLecture.html#introductory-statistics-books", + "href": "posts/UncertaintiesLecture.html#introductory-statistics-books", + "title": "LHCb Startertalk on Uncertainties", + "section": "Introductory statistics books", + "text": "Introductory statistics books\n\nList of books on TWiki pages of Statistis and Machine Learning WG\nBooks I personally learned from\n\nGlen Cowan, “Statistical Data Analysis”, Oxford University Press, 1998\nF. James, “Statistical Methods in Experimental Physics”, 2nd edition, World Scientific, 2006\nB. Efron, R.J. Tibshirani, “An Introduction to the Bootstrap”, CRC Press, 1994\n\nRoger Barlow\n\nIntroductory lecture (2019)\nSystematic Errors: Facts and Fictions (2002)\n“Practical Statistics for Particle Physics” (2019), Section 6.5\n\nMore Jupyter notebooks\n\nMatt Kenzie, PDG averaging\nMatt Kenzie, MLE uncertainties\niminuit tutorials on various fitting topics\n\nFree books\n\n“Introduction to Statistics and Data Analysis for Physicists”, Gerhard Bohm and Günter Zech, Verlag-DESY, 2010\n“Statistische und numerische Methoden der Datenanalyse”, V Blobel and E Lohrmann, Open Access ebook CC-BY 3.0, 2012\n\n\n\nThe END" + }, + { + "objectID": "posts/New iminuit displays.html", + "href": "posts/New iminuit displays.html", + "title": "New Fit Result Display", + "section": "", + "text": "The next release of iminuit will feature a new Fit Result display.\nYou can see old vs. new below. Please commment!" + }, + { + "objectID": "posts/New iminuit displays.html#old-displays", + "href": "posts/New iminuit displays.html#old-displays", + "title": "New Fit Result Display", + "section": "Old displays", + "text": "Old displays\n\n!pip install iminuit==1.4.8\nimport iminuit\n\nCollecting iminuit==1.4.8\n Using cached iminuit-1.4.8-cp38-cp38-macosx_10_9_x86_64.whl (390 kB)\nRequirement already satisfied: numpy>=1.11.3 in /usr/local/lib/python3.8/site-packages (from iminuit==1.4.8) (1.19.0)\nInstalling collected packages: iminuit\n Attempting uninstall: iminuit\n Found existing installation: iminuit 1.4.9\n Uninstalling iminuit-1.4.9:\n Successfully uninstalled iminuit-1.4.9\nSuccessfully installed iminuit-1.4.8\n\n\n\nGood fit\n\nm = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)\nm.migrad()\n\n\n\n\nFCN = 1\nNcalls = 19 (19 total)\n\n\nEDM = 0 (Goal: 0.0002)\nup = 1.0\n\n\nValid Min.\nValid Param.\nAbove EDM\nReached call limit\n\n\nTrue\nTrue\nFalse\nFalse\n\n\nHesse failed\nHas cov.\nAccurate\nPos. def.\nForced\n\n\nFalse\nTrue\nTrue\nTrue\nFalse\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n0\n1\n\n\n\n\n\n\n\n1\nb\n0\n1\n\n\n\n\n\n\n\n\n\n\n\nm = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)\nprint(m.migrad())\n\n------------------------------------------------------------------\n| FCN = 1 | Ncalls=19 (19 total) |\n| EDM = 0 (Goal: 0.0002) | up = 1.0 |\n------------------------------------------------------------------\n| Valid Min. | Valid Param. | Above EDM | Reached call limit |\n------------------------------------------------------------------\n| True | True | False | False |\n------------------------------------------------------------------\n| Hesse failed | Has cov. | Accurate | Pos. def. | Forced |\n------------------------------------------------------------------\n| False | True | True | True | False |\n------------------------------------------------------------------\n------------------------------------------------------------------------------------------\n| | Name | Value | Hesse Err | Minos Err- | Minos Err+ | Limit- | Limit+ | Fixed |\n------------------------------------------------------------------------------------------\n| 0 | a | 0 | 1 | | | | | |\n| 1 | b | 0 | 1 | | | | | |\n------------------------------------------------------------------------------------------\n\n\n\n\nBad fits\n\nm = iminuit.Minuit(lambda x, y : 0, pedantic=False)\nm.migrad()\n\n\n\n\nFCN = 0\nNcalls = 20 (20 total)\n\n\nEDM = 0 (Goal: 0.0002)\nup = 1.0\n\n\nValid Min.\nValid Param.\nAbove EDM\nReached call limit\n\n\nFalse\nTrue\nFalse\nFalse\n\n\nHesse failed\nHas cov.\nAccurate\nPos. def.\nForced\n\n\nTrue\nTrue\nFalse\nFalse\nFalse\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx\n0\n0\n\n\n\n\n\n\n\n1\ny\n0\n0\n\n\n\n\n\n\n\n\n\n\n\nm = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)\nm.migrad()\n\n\n\n\nFCN = 8.015e-09\nNcalls = 31 (31 total)\n\n\nEDM = 8.01e-09 (Goal: 0.0002)\nup = 1.0\n\n\nValid Min.\nValid Param.\nAbove EDM\nReached call limit\n\n\nTrue\nTrue\nFalse\nFalse\n\n\nHesse failed\nHas cov.\nAccurate\nPos. def.\nForced\n\n\nFalse\nTrue\nFalse\nFalse\nTrue\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx\n0.3\n0.8\n\n\n0\n1\n\n\n\n1\ny\n0\n16\n\n\n\n\n\n\n\n\n\n\n\nm = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)\nprint(m.migrad())\n\n------------------------------------------------------------------\n| FCN = 8.015e-09 | Ncalls=31 (31 total) |\n| EDM = 8.01e-09 (Goal: 0.0002) | up = 1.0 |\n------------------------------------------------------------------\n| Valid Min. | Valid Param. | Above EDM | Reached call limit |\n------------------------------------------------------------------\n| True | True | False | False |\n------------------------------------------------------------------\n| Hesse failed | Has cov. | Accurate | Pos. def. | Forced |\n------------------------------------------------------------------\n| False | True | False | False | True |\n------------------------------------------------------------------\n------------------------------------------------------------------------------------------\n| | Name | Value | Hesse Err | Minos Err- | Minos Err+ | Limit- | Limit+ | Fixed |\n------------------------------------------------------------------------------------------\n| 0 | x | 0.3 | 0.8 | | | 0 | 1 | |\n| 1 | y | 0 | 16 | | | | | |\n------------------------------------------------------------------------------------------" + }, + { + "objectID": "posts/New iminuit displays.html#new-displays", + "href": "posts/New iminuit displays.html#new-displays", + "title": "New Fit Result Display", + "section": "New displays", + "text": "New displays\nThis part requires a restart of the Jupyter kernel.\n\n!pip install --upgrade ../iminuit\nimport iminuit\n\nProcessing /Users/hdembinski/Extern/iminuit\nRequirement already satisfied, skipping upgrade: numpy>=1.11.3 in /usr/local/lib/python3.8/site-packages (from iminuit==1.4.9) (1.19.0)\nBuilding wheels for collected packages: iminuit\n Building wheel for iminuit (setup.py) ... done\n Created wheel for iminuit: filename=iminuit-1.4.9-cp38-cp38-macosx_10_15_x86_64.whl size=384673 sha256=9ba56dcf93e21dc8844737e5caf597fa22c78cc05bb431ac1b8bc3ddeca07510\n Stored in directory: /private/var/folders/tl/pv6mt7z17tz0stm1fjfg01cc0000gn/T/pip-ephem-wheel-cache-2v6nwkbn/wheels/e7/61/b4/0dd6b8fc20e6d6b170f3e0bd185533497f2fe08c4ae74cc799\nSuccessfully built iminuit\nInstalling collected packages: iminuit\n Attempting uninstall: iminuit\n Found existing installation: iminuit 1.4.9\n Uninstalling iminuit-1.4.9:\n Successfully uninstalled iminuit-1.4.9\nSuccessfully installed iminuit-1.4.9\n\n\n\nGood fit\n\nimport iminuit\nm = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)\nm.migrad()\n\n\n\n\nFCN = 1\nNcalls = 19 (19 total)\n\n\nEDM = 0 (Goal: 0.0002)\nup = 1.0\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM goal\nBelow call limit\n\n\nHesse ok\nHas Covariance\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n0\n1\n\n\n\n\n\n\n\n1\nb\n0\n1\n\n\n\n\n\n\n\n\n\n\n\nm = iminuit.Minuit(lambda a, b: a ** 2 + b ** 2 + 1, pedantic=False)\nprint(m.migrad())\n\n┌──────────────────────────────────┬──────────────────────────────────────┐\n│ FCN = 1 │ Ncalls = 19 (19 total) │\n│ EDM = 0 (Goal: 0.0002) │ up = 1.0 │\n├───────────────┬──────────────────┼──────────────────────────────────────┤\n│ Valid Minimum │ Valid Parameters │ No Parameters at limit │\n├───────────────┴──────────────────┼──────────────────────────────────────┤\n│ Below EDM goal │ Below call limit │\n├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤\n│ Hesse ok │ Has Covariance │ Accurate │ Pos. def. │ Not forced │\n└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘\n┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n│ 0 │ a │ 0 │ 1 │ │ │ │ │ │\n│ 1 │ b │ 0 │ 1 │ │ │ │ │ │\n└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n\n\n\n\nBad fits\n\nm = iminuit.Minuit(lambda x, y : 0, pedantic=False)\nm.migrad()\n\n\n\n\nFCN = 0\nNcalls = 20 (20 total)\n\n\nEDM = 0 (Goal: 0.0002)\nup = 1.0\n\n\nINVALID Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM goal\nBelow call limit\n\n\nHesse FAILED\nHas Covariance\nAPPROXIMATE\nNOT pos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx\n0\n0\n\n\n\n\n\n\n\n1\ny\n0\n0\n\n\n\n\n\n\n\n\n\n\n\nm = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)\nm.migrad()\n\n\n\n\nFCN = 8.015e-09\nNcalls = 31 (31 total)\n\n\nEDM = 8.01e-09 (Goal: 0.0002)\nup = 1.0\n\n\nValid Minimum\nValid Parameters\nSOME Parameters at limit\n\n\nBelow EDM goal\nBelow call limit\n\n\nHesse ok\nHas Covariance\nAPPROXIMATE\nNOT pos. def.\nFORCED\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx\n0.3\n0.8\n\n\n0\n1\n\n\n\n1\ny\n0\n16\n\n\n\n\n\n\n\n\n\n\n\nm = iminuit.Minuit(lambda x, y : (x-y) ** 2, limit_x=(0, 1), pedantic=False)\nprint(m.migrad())\n\n┌──────────────────────────────────┬──────────────────────────────────────┐\n│ FCN = 8.015e-09 │ Ncalls = 31 (31 total) │\n│ EDM = 8.01e-09 (Goal: 0.0002) │ up = 1.0 │\n├───────────────┬──────────────────┼──────────────────────────────────────┤\n│ Valid Minimum │ Valid Parameters │ SOME Parameters at limit │\n├───────────────┴──────────────────┼──────────────────────────────────────┤\n│ Below EDM goal │ Below call limit │\n├───────────────┬──────────────────┼───────────┬─────────────┬────────────┤\n│ Hesse ok │ Has Covariance │APPROXIMATE│NOT pos. def.│ FORCED │\n└───────────────┴──────────────────┴───────────┴─────────────┴────────────┘\n┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n│ 0 │ x │ 0.3 │ 0.8 │ │ │ 0 │ 1 │ │\n│ 1 │ y │ 0 │ 16 │ │ │ │ │ │\n└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘" + }, + { + "objectID": "posts/ratio bias.html", + "href": "posts/ratio bias.html", + "title": "Ratio bias", + "section": "", + "text": "When you compute a ratio of two random numbers, the ratio is biased in general, unless the relative error on the denominator is very small. This is a general consequence of the fact that any non-linear transformation of a random variable introduces a bias in general.\nThe good news is that we can correct for this bias to first order with a simple formula, and thus get an approximately unbiased result.\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n\nrng = np.random.default_rng(1)\n\na_true = 100.\nb_factor = 20.\n\na = rng.poisson(a_true, size=10000)\nb = rng.poisson(a_true / b_factor, size=len(a)) * b_factor\nb[b==0] = np.nan\n\n\nplt.hist(a / b, bins=50, label=\"toys\")\nplt.axvline(1, ls=\"--\", color=\"k\", label=\"true\")\nplt.axvline(np.nanmean(a/b), ls=\"-\", color=\"r\", label=\"avg\")\nplt.legend();\n\n\n\n\n\n@np.vectorize\ndef ratio_bias(a, a_var, b, b_var):\n if np.isnan(a) or np.isnan(b) or a == 0 or b == 0:\n return np.nan\n\n scale_a = a_var / a\n n_eff_a = a / scale_a\n scale_b = b_var / b\n n_eff_b = b / scale_b\n\n rng = np.random.default_rng(1)\n a_toy = rng.poisson(n_eff_a, size=10000) * scale_a\n b_toy = rng.poisson(n_eff_b, size=len(a_toy)) * scale_b\n b_toy[b_toy == 0] = np.nan\n r_toy = np.nanmean(a_toy / b_toy)\n return r_toy / (a / b)\n\n\na = 10.\nb = 10.\n\nfor a_rel_err in (0.01, 0.5):\n b_rel_err = np.geomspace(0.01, 0.5, 10)\n bias = ratio_bias(a, (a * a_rel_err) ** 2, b, (b * b_rel_err) ** 2)\n plt.plot(b_rel_err, bias, label=f\"$\\\\sigma(a)/a = {a_rel_err:.2f}$\")\n\n# analytical formula for bias\nbias = (1+b_rel_err**2)\nplt.plot(b_rel_err, bias, label=f\"analytical\")\n\nplt.ylabel(r\"$\\langle a/b \\rangle$ / truth\")\nplt.xlabel(r\"$\\sigma(b)/b$\")\nplt.legend();\n\n\n\n\nThe analytical formula is a first order correction, therefore it fails to describe the bias for large relative uncertainties." + }, + { + "objectID": "posts/Fit weighted histograms with SPD method.html", + "href": "posts/Fit weighted histograms with SPD method.html", + "title": "Fitting weighted histograms with SPD method", + "section": "", + "text": "We test different fit methods of weighted binned data. The tested weight distributions are normal, exponential, and uniform. The toy case is a common HEP fit of a gaussian peak above a smooth background (here exponential).\nFits of weighted data are discussed in F. James, “Statistical Methods in Experimental Physics” (2nd edition), World Scientific, in section 8.5. Two approximate least-squares methods for binned data are presented in section 8.5.4. Here we test the method \\(Q_1'\\).\nBohm and Zech (2014) derived an approximate maximum-likelihood method for weighted histograms with on average positive weights. The authors show that their approximation yields less biased results than the aforementioned least-squares method.\nStill, a notable bias is present even when the SPD method is used. The largest bias is found for exponentially distributed weights with deviations of 4-5% in the fitted parameters.\nThe remaining bias of the SPD method originates from the fact that the true weight variance in each bin is unknown and is replaced by the sample variance. This is inaccurate in bins with small number of samples and fails for empty bins, which must be skipped. The same is true for the alternative least-squares fit, however, and the biases are even larger there.\nThe study suggest that biases are larger if the weight distribution deviates more from the normal distribution.\n\n#!pip install boost-histogram iminuit scipy matplotlib numpy numba progressbar2 pyik\nimport boost_histogram as bh\nfrom iminuit import Minuit\nfrom scipy.stats import norm, expon, uniform, poisson\nimport multiprocessing\nmultiprocessing.set_start_method('fork') # needed on OSX, see https://github.com/ipython/ipython/issues/12396\nfrom multiprocessing.pool import Pool\nfrom matplotlib import pyplot as plt\nfrom pyik.mplext import plot_stacked\nimport numpy as np\nimport numba as nb\nfrom progressbar import progressbar\nimport math\n\n\nkwd = {\"fastmath\": True}\n\n\n@nb.njit(**kwd)\ndef sum_log(fs, spdf, fb, bpdf):\n return np.sum(np.log(fs * spdf + fb * bpdf))\n\n\n@nb.njit(**kwd)\ndef norm_pdf(x, mu, sigma):\n invs = 1.0 / sigma\n z = (x - mu) * invs\n invnorm = 1 / np.sqrt(2 * np.pi) * invs\n return np.exp(-0.5 * z**2) * invnorm\n\n\n@nb.njit(**kwd)\ndef nb_erf(x):\n y = np.empty_like(x)\n for i in nb.prange(len(x)):\n y[i] = math.erf(x[i])\n return y\n\n\n@nb.njit(**kwd)\ndef norm_cdf(x, mu, sigma):\n invs = 1.0 / (sigma * np.sqrt(2))\n z = (x - mu) * invs\n return 0.5 * (1 + nb_erf(z))\n\n\n@nb.njit(**kwd)\ndef expon_pdf(x, lambd):\n inv_lambd = 1.0 / lambd\n return inv_lambd * np.exp(-inv_lambd * x)\n\n\n@nb.njit(**kwd)\ndef expon_cdf(x, lambd):\n inv_lambd = 1.0 / lambd\n return 1.0 - np.exp(-inv_lambd * x)\n\n\n@nb.njit\ndef cost_lsq(n, mu):\n r = 0.0\n for i in range(len(mu)):\n if n[i] == 0:\n continue\n r += (n[i] - mu[i]) ** 2 / n[i]\n return r\n\n\n@nb.njit\ndef _cost_lsq_w(w, wvar, mu):\n r = 0.0\n n = len(mu)\n for i in range(n):\n if wvar[i] == 0:\n continue\n r += (w[i] - mu[i]) ** 2 / wvar[i]\n return r\n\n\ndef cost_lsq_w(w, mu):\n return _cost_lsq_w(w.value, w.variance, mu)\n\n\n@nb.njit\ndef cost_ml(n, mu):\n r = 0.0\n for i in range(len(mu)):\n r += mu[i] - n[i] * np.log(mu[i])\n return 2 * r\n\n\n@nb.njit\ndef _cost_ml_w(w, wvar, mu):\n # Bohm and Zech, NIMA 748 (2014) 1-6\n r = 0.0\n for i in range(len(mu)):\n if wvar[i] > 0.0:\n if w[i] > 0.0:\n s_inv = w[i] / wvar[i]\n mu_prime = mu[i] * s_inv\n n_prime = w[i] * s_inv\n r += mu_prime - n_prime * np.log(mu_prime)\n else: # extension\n r += 0.5 * (w[i] - mu[i]) ** 2 / wvar[i]\n return 2 * r\n\n\ndef cost_ml_w(w, mu):\n return _cost_ml_w(w.value, w.variance, mu)\n\n\n# these distributions are later shifted and scaled to give E[w] = 1 and Var(w) = 1\nweight_distributions = [None, np.ones, norm().rvs, uniform().rvs, expon().rvs]\nwd_labels = (\"no weights\", \"w=1\", \"norm\", \"uniform\", \"expon\")\n\nmu_npts = 1000\nsignal_fraction = 0.2\n\npar_labels = (\"bkg: fraction\", \"sig: fraction\",\n \"bkg: lambda\", \"sig: mu\", \"sig: sigma\")\ntruth = ((1.0 - signal_fraction), signal_fraction, 1.0, 2.0, 0.2)\n\n\ndef run(wd, wd_label):\n sub_pars = []\n for imc in progressbar(range(nmc), redirect_stdout=True):\n rng = np.random.default_rng(imc)\n all_valid = False\n while not all_valid:\n # sample total number of points from poisson distribution\n npts = rng.poisson(mu_npts)\n\n # generate background\n b = rng.exponential(truth[2], size=int(truth[0] * npts))\n\n # generate signal\n s = rng.normal(truth[3], truth[4], size=int(truth[1] * npts))\n x = np.append(b, s)\n\n if wd is None:\n h = bh.Histogram(bh.axis.Regular(20, 0, 3))\n h.fill(x)\n xe = h.axes[0].edges\n w = h.view()\n\n else:\n # generate weights\n wi = wd(len(x))\n if np.std(wi) > 0:\n # wi have mean(wi) = 1 and std(wi) = 1\n wi = (wi - np.mean(wi)) / np.std(wi) + 1.0\n h = bh.Histogram(bh.axis.Regular(20, 0, 3),\n storage=bh.storage.Weight())\n h.fill(x, weight=wi)\n xe = h.axes[0].edges\n w = h.view()\n\n \n sub_sub_pars = []\n all_valid = True\n for fit_mode in range(2):\n if wd is None:\n if fit_mode == 0:\n cost = cost_lsq\n else:\n cost = cost_ml\n else:\n if fit_mode == 0:\n cost = cost_lsq_w\n else:\n cost = cost_ml_w\n\n def cost_model(par):\n bkg = expon_cdf(xe, par[2])\n sig = norm_cdf(xe, par[3], par[4])\n mu_bkg = par[0] * npts * np.diff(bkg)\n mu_sig = par[1] * npts * np.diff(sig)\n res = cost(w, mu_bkg + mu_sig)\n return res\n\n m = Minuit.from_array_func(\n cost_model,\n truth,\n limit=((0, None), (0, None),\n (0, None), None, (0, None)),\n pedantic=False,\n )\n\n try:\n m.migrad()\n all_valid &= m.valid\n sub_sub_pars.append(m.np_values())\n except RuntimeError:\n all_valid = False\n\n if nmc == 1 and wd is not None:\n plt.figure()\n h_bkg = bh.Histogram(h.axes[0], storage=bh.storage.Weight())\n h_bkg.fill(x[:len(b)], weight=wi[:len(b)])\n h_sig = bh.Histogram(h.axes[0], storage=bh.storage.Weight())\n h_sig.fill(x[len(b):], weight=wi[len(b):])\n plot_stacked(xe, np.array((h_bkg.view().value, h_sig.view().value)),\n colors=(\"0.8\", \"r\"), labels=(\"bkg\", \"sig\"))\n plt.errorbar(h.axes[0].centers, w.value,\n w.variance ** 0.5, fmt=\"ok\")\n plt.title(wd_label)\n plt.ylim(0, None)\n plt.xlim(xe[0], xe[-1])\n\n sub_pars.append(sub_sub_pars)\n return sub_pars\n\n\n# for drawing examples\nnmc = 1\nfor wd, wd_label in zip(weight_distributions, wd_labels):\n run(wd, wd_label)\n\n100% (1 of 1) |##########################| Elapsed Time: 0:00:00 Time: 0:00:00\n100% (1 of 1) |##########################| Elapsed Time: 0:00:00 Time: 0:00:00\n100% (1 of 1) |##########################| Elapsed Time: 0:00:00 Time: 0:00:00\n100% (1 of 1) |##########################| Elapsed Time: 0:00:00 Time: 0:00:00\n100% (1 of 1) |##########################| Elapsed Time: 0:00:00 Time: 0:00:00\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nnmc = 10000\nwith Pool() as p:\n pars = p.starmap(run, zip(weight_distributions, wd_labels))\npars = np.array(pars)\n\n100% (10000 of 10000) |##################| Elapsed Time: 0:01:15 Time: 0:01:15\n100% (10000 of 10000) |##################| Elapsed Time: 0:01:31 Time: 0:01:31\n100% (10000 of 10000) |##################| Elapsed Time: 0:01:35 Time: 0:01:35\n100% (10000 of 10000) |##################| Elapsed Time: 0:01:35 Time: 0:01:35\n100% (10000 of 10000) |##################| Elapsed Time: 0:01:36 Time: 0:01:36\n\n\n\nfit_labels = (\"LSQ\", \"ML\")\npars_range = ((0.4, 1.0), (0.0, 0.4), (0.2, 1.6), (1.75, 2.15), (0.0, 0.4))\nfor ipar, (par_label, t) in enumerate(zip(par_labels, truth)):\n fig, ax = plt.subplots(1, 2, figsize=(14, 4), sharex=True, sharey=True)\n plt.suptitle(par_label)\n for iwd, wd_label in enumerate(wd_labels):\n for ifit, fit_label in enumerate(fit_labels):\n plt.sca(ax[ifit])\n p = pars[iwd, :, ifit, ipar]\n plt.hist(p, alpha=0.5, label=wd_label,\n bins=100, range=pars_range[ipar])\n for label, axi in zip(fit_labels, ax):\n axi.axvline(t, color=\"r\")\n axi.legend(loc=\"upper left\", title=label)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nfor ipar, (par_label, t) in enumerate(zip(par_labels, truth)):\n fig, ax = plt.subplots(1, 2, figsize=(14, 3), sharex=True, sharey=True)\n plt.suptitle(f\"{par_label}\", y=1.05)\n for iwd, wd_label in enumerate(wd_labels):\n for ifit in range(2):\n plt.sca(ax[ifit])\n p = pars[iwd, :, ifit, ipar]\n mean = np.mean(p)\n sigma = np.std(p) / len(p) ** 0.5\n plt.errorbar(mean, iwd, 0, sigma, fmt=\"o\", ms=10)\n for fit_label, axi in zip(fit_labels, ax):\n axi.set_title(fit_label, y=1.05)\n axi.axvline(t, color=\"r\")\n for percent in (1, 2, 3, 4, 5):\n dt = t * 1e-2 * percent\n axi.axvline(t - dt, color=\"0.5\", ls=\":\")\n axi.axvline(t + dt, color=\"0.5\", ls=\":\")\n axi.text(t + dt, 1.01, f\"+{percent}%\", transform=axi.get_xaxis_transform(),\n ha=\"center\", fontsize=\"x-small\")\n axi.text(t - dt, 1.01, f\"-{percent}%\", transform=axi.get_xaxis_transform(),\n ha=\"center\", fontsize=\"x-small\")\n plt.sca(ax[0])\n plt.yticks(np.arange(len(wd_labels)), wd_labels)" + }, + { + "objectID": "posts/visual_cross_section.html", + "href": "posts/visual_cross_section.html", + "title": "Visual cross-section for particle production", + "section": "", + "text": "When you look for a somewhat rare decay, you want to know how much luminosity is required to see N decays in your detector. For this one needs to compute the visual cross-section, this is the part of the production cross-section for the particle for which the decay products all fall into the acceptance of the detector.\nWe compute the visual cross-section here for the decay of Omega and Xi with chromo. We then go on and compute the expected rate of interactions for running LHCb in fixed-target mode." + }, + { + "objectID": "posts/visual_cross_section.html#visual-cross-section", + "href": "posts/visual_cross_section.html#visual-cross-section", + "title": "Visual cross-section for particle production", + "section": "Visual cross-section", + "text": "Visual cross-section\nWe compute the inelastic cross-section for p-He collisions in fixed-target configuration where the proton has 13 TeV with chromo.\n\nfrom chromo.models import Pythia8, EposLHC\nfrom chromo.kinematics import FixedTarget, TeV, GeV\nimport numpy as np\nfrom particle import literals as lp\nfrom chromo.util import pdg2name\nfrom rich.progress import track\n\n\nkin = FixedTarget(13*TeV, \"p\", \"p\")\n\n# epos = EposLHC(kin)\ngen = Pythia8(kin, seed=1)\ncs = gen.cross_section()\n\n\n *------------------------------------------------------------------------------------* \n | | \n | *------------------------------------------------------------------------------* | \n | | | | \n | | | | \n | | PPP Y Y TTTTT H H III A Welcome to the Lund Monte Carlo! | | \n | | P P Y Y T H H I A A This is PYTHIA version 8.308 | | \n | | PPP Y T HHHHH I AAAAA Last date of change: 16 Nov 2022 | | \n | | P Y T H H I A A | | \n | | P Y T H H III A A Now is 31 Mar 2023 at 15:19:42 | | \n | | | | \n | | Program documentation and an archive of historic versions is found on: | | \n | | | | \n | | https://pythia.org/ | | \n | | | | \n | | PYTHIA is authored by a collaboration consisting of: | | \n | | | | \n | | Christian Bierlich, Nishita Desai, Leif Gellersen, Ilkka Helenius, Philip | | \n | | Ilten, Leif Lonnblad, Stephen Mrenna, Stefan Prestel, Christian Preuss, | | \n | | Torbjorn Sjostrand, Peter Skands, Marius Utheim and Rob Verheyen. | | \n | | | | \n | | The complete list of authors, including contact information and | | \n | | affiliations, can be found on https://pythia.org/. | | \n | | Problems or bugs should be reported on email at authors@pythia.org. | | \n | | | | \n | | The main program reference is C. Bierlich et al, | | \n | | 'A comprehensive guide to the physics and usage of Pythia 8.3', | | \n | | SciPost Phys. Codebases 8-r8.3 (2022) [arXiv:2203.11601 [hep-ph]] | | \n | | | | \n | | PYTHIA is released under the GNU General Public Licence version 2 or later.| | \n | | Please respect the MCnet Guidelines for Event Generator Authors and Users. | | \n | | | | \n | | Disclaimer: this program comes without any guarantees. | | \n | | Beware of errors and use common sense when interpreting results. | | \n | | | | \n | | Copyright (C) 2022 Torbjorn Sjostrand | | \n | | | | \n | | | | \n | *------------------------------------------------------------------------------* | \n | | \n *------------------------------------------------------------------------------------* \n\n\n\n\ncs.inelastic\n\n40.543016683445494\n\n\nNow we compute which fraction of inelastic events produce the following decays so that all charged final state decay products are in the LHCb acceptance, given by \\(p > 2\\) GeV/c and \\(2 < \\eta < 5\\). \\[\n\\Xi^- \\to \\Lambda \\pi^- \\; (99.887\\,\\%)\n\\] \\[\n\\Omega^- \\to \\Lambda K^- \\; (67.8\\,\\%)\n\\] We don’t need to take special care of the branching fractions. We get these reduction factors automatically if we select the specific decay we are interested in from the generator output.\n\napid_xi = abs(lp.Xi_minus.pdgid)\napid_omega = abs(lp.Omega_minus.pdgid)\napid_lambda = abs(lp.Lambda.pdgid)\napid_pi = abs(lp.pi_plus.pdgid)\napid_proton = abs(lp.proton.pdgid)\napid_K = abs(lp.K_plus.pdgid)\n\ngen.set_stable(apid_xi, False)\ngen.set_stable(-apid_xi, False)\ngen.set_stable(apid_omega, False)\ngen.set_stable(-apid_omega, False)\ngen.set_stable(apid_lambda, False)\ngen.set_stable(-apid_lambda, False)\n\n\ndef is_visible_decay(ptot, eta, apid, children, par, apid1, apid2):\n ch = children[par]\n if len(ch) != 2:\n return False\n for i in ch:\n if apid[i] != apid1 and apid[i] != apid2:\n return False\n if apid[i] == apid_lambda:\n if not is_visible_decay(ptot, eta, apid, children, i, apid_proton, apid_pi):\n return False\n if ptot[i] < 2*GeV or eta[i] < 2 or eta[i] > 5:\n return False\n return True\n\nn_inel = 100000\nn_decays = {\"Xi\": 0, \"Omega\": 0}\nfor event in track(gen(n_inel), total=n_inel):\n apid = np.abs(event.pid)\n mask_xi = apid == apid_xi\n mask_omega = apid == apid_omega\n\n if np.any(mask_xi | mask_omega):\n # collect children of each parent\n children = {}\n for i, par in enumerate(event.parents):\n if par[0] == 0:\n continue\n children.setdefault(par[0] - 1, []).append(i)\n\n ptot = event.p_tot\n eta = event.eta\n\n # investigate Xi decays\n for par in np.arange(len(mask_xi))[mask_xi]:\n if is_visible_decay(ptot, eta, apid, children, par, apid_lambda, apid_pi):\n n_decays[\"Xi\"] += 1\n\n # investigate Omega decays\n for par in np.arange(len(mask_omega))[mask_omega]:\n if is_visible_decay(ptot, eta, apid, children, par, apid_lambda, apid_K):\n n_decays[\"Omega\"] += 1\n\nprint(n_decays, \"out of\", n_inel)\n\n\n\n\n\n\n\n\n\n\n{'Xi': 122, 'Omega': 3} out of 100000\n\n\n\nsigma_vis = {k: v / n_inel * cs.inelastic for k, v in n_decays.items()}\n\nprint(\"visible cross-section\")\nfor k, s in sigma_vis.items():\n print(f\"{k:10} {s:.4f} mb\")\n\nvisible cross-section\nXi 0.0495 mb\nOmega 0.0012 mb" + }, + { + "objectID": "posts/visual_cross_section.html#lhcb-in-fixed-target-mode", + "href": "posts/visual_cross_section.html#lhcb-in-fixed-target-mode", + "title": "Visual cross-section for particle production", + "section": "LHCb in fixed-target mode", + "text": "LHCb in fixed-target mode\nLHCb in fixed-target mode uses a gas target. We compute the event rate for the SMOG 1 configuration where the gas is injected directly into the LHCb Vertex Locator (VELO).\nThe average number of interactions per bunch crossing is \\[\n\\mu = \\sigma L N p \\frac{N_A}{22400}\n\\] where \\(L = 60\\) cm is the length of the detection region in the VELO, \\(N=10^{11}\\) is the proton bunch population in Run 2, \\(p\\) is gas pressure in bar, and \\(N_A\\) is Avogadro’s constant.\n\nm = 1\nb = 1e-28 * m ** 2\nmb = 1e-3 * b\ncm = 1e-2 * m\nPa = 1\nbar = 1e5 * Pa\nK = 1\nkb = 1.380649e-23 # Boltzmann constant in J/K\n\n\n# cylinder volume in which a single proton would interact on average\ncross_section = 200 * mb\ndetection_region_length = 60 * cm\nvolume = cross_section * detection_region_length\n\n# number of collisions given by number of gas molecules per volume: \n# pV = N k T -> N = p V / k T\np = 1e-10 * bar\nT = 273 * K\nn_collisions_per_proton = volume * p / (kb * T)\n\nn_protons_per_bunch = 1e11\nn_collisions = n_protons_per_bunch * n_collisions_per_proton\nn_collisions\n\n0.0031837233037538114\n\n\n\n# Helium injection, 24/11/2022, target pressure on the VELO 1.2E-8mbar\np = 1.2e-8 * 1e-3" + }, + { + "objectID": "posts/naive_calibration_bias.html", + "href": "posts/naive_calibration_bias.html", + "title": "Statistical issues in naive calibration analyses", + "section": "", + "text": "The statistical problem of calibrating a sample with observable pairs \\(\\hat a\\) and \\(\\hat b\\) is challenging if random stochastic fluctuations affect both numbers of the pair. Even if the unknown true values \\((a, b)\\) have a perfectly linear relationship and if the stochastic fluctuations around \\((a, b)\\) are symmetric, a binned profile of the \\(\\hat b\\) values as a function of the \\(\\hat a\\) values may indicate a departure from a linear dependence. This happens if the stochastic fluctuations of \\(\\hat a\\) around \\(a\\) are not negligible compared to the bin size.\nWe demonstrate this in the following with toy simulation. We assume fluctuations that are relative to the true values and that the true pairs \\((a,b)\\) are not uniformly distributed, but follow an exponential distribution. This is commonly the case in particle physics.\nimport numpy as np\nfrom matplotlib import pyplot as plt\nimport boost_histogram as bh\nfrom iminuit import Minuit\nfrom iminuit.cost import LeastSquares\n\nrng = np.random.default_rng(1)\n\nx_true = rng.exponential(1, size=1000)\n# x_true = rng.uniform(0, 10, size=1000)\ny_true = x_true.copy()\n\ndef sigma_x(x_true):\n return x_true * 0.1\n\ndef sigma_y(y_true):\n return y_true * 0.5\n\nx0 = rng.normal(x_true, sigma_x(x_true))\ny0 = rng.normal(y_true, sigma_y(y_true))\n\nx = x0\ny = y0\n\nh = bh.Histogram(bh.axis.Variable([0, 0.1, 0.3, 0.5, 0.7, 1, 1.5, 1.75, 2, 3, 4, 6, 10]), storage=bh.storage.Mean())\nh.fill(x, sample=y)\n\ndef line(x, a, b):\n return a + x * b\n\nm = Minuit(LeastSquares(h.axes[0].centers, h.values(), h.variances() ** 0.5, line), 0, 0)\nm.migrad()\n\nplt.scatter(x, y, marker=\".\", color=\"C1\")\nm.visualize()\nprint(m)\n\n┌─────────────────────────────────────────────────────────────────────────┐\n│ Migrad │\n├──────────────────────────────────┬──────────────────────────────────────┤\n│ FCN = 5.194 (χ²/ndof = 0.5) │ Nfcn = 31 │\n│ EDM = 1.42e-21 (Goal: 0.0002) │ │\n├──────────────────────────────────┼──────────────────────────────────────┤\n│ Valid Minimum │ Below EDM threshold (goal x 10) │\n├──────────────────────────────────┼──────────────────────────────────────┤\n│ No parameters at limit │ Below call limit │\n├──────────────────────────────────┼──────────────────────────────────────┤\n│ Hesse ok │ Covariance accurate │\n└──────────────────────────────────┴──────────────────────────────────────┘\n┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n│ 0 │ a │ 0.005 │ 0.005 │ │ │ │ │ │\n│ 1 │ b │ 0.997 │ 0.020 │ │ │ │ │ │\n└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n┌───┬─────────────────────┐\n│ │ a b │\n├───┼─────────────────────┤\n│ a │ 2.18e-05 -0.050e-3 │\n│ b │ -0.050e-3 0.000401 │\n└───┴─────────────────────┘\nWe find excellent linearity, the fitted line is compatible with zero offset and a slope of one.\nNow we swap the roles of x and y and perform the exact same analysis again. An analysis based on a profile is not treating x and y symmetrically, therefore it is possible that we get another answer, although fundamentally it does not matter whether we study pairs (x, y) or (y, x).\nx = y0\ny = x0\n\nh = bh.Histogram(bh.axis.Variable([0, 0.1, 0.3, 0.5, 0.7, 1, 1.5, 1.75, 2, 3, 4, 6, 10]), storage=bh.storage.Mean())\nh.fill(x, sample=y)\n\nm = Minuit(LeastSquares(h.axes[0].centers, h.values(), h.variances(), line), 0, 0)\nm.migrad()\n\nplt.scatter(x, y, marker=\".\", color=\"C1\")\nm.visualize()\nprint(m)\n\n┌─────────────────────────────────────────────────────────────────────────┐\n│ Migrad │\n├──────────────────────────────────┬──────────────────────────────────────┤\n│ FCN = 8884 (χ²/ndof = 888.4) │ Nfcn = 35 │\n│ EDM = 1.01e-20 (Goal: 0.0002) │ │\n├──────────────────────────────────┼──────────────────────────────────────┤\n│ Valid Minimum │ Below EDM threshold (goal x 10) │\n├──────────────────────────────────┼──────────────────────────────────────┤\n│ No parameters at limit │ Below call limit │\n├──────────────────────────────────┼──────────────────────────────────────┤\n│ Hesse ok │ Covariance accurate │\n└──────────────────────────────────┴──────────────────────────────────────┘\n┌───┬──────┬───────────┬───────────┬────────────┬────────────┬─────────┬─────────┬───────┐\n│ │ Name │ Value │ Hesse Err │ Minos Err- │ Minos Err+ │ Limit- │ Limit+ │ Fixed │\n├───┼──────┼───────────┼───────────┼────────────┼────────────┼─────────┼─────────┼───────┤\n│ 0 │ a │ 163.2e-3 │ 0.6e-3 │ │ │ │ │ │\n│ 1 │ b │ 0.7905 │ 0.0014 │ │ │ │ │ │\n└───┴──────┴───────────┴───────────┴────────────┴────────────┴─────────┴─────────┴───────┘\n┌───┬───────────────────┐\n│ │ a b │\n├───┼───────────────────┤\n│ a │ 3.05e-07 -0.50e-6 │\n│ b │ -0.50e-6 1.93e-06 │\n└───┴───────────────────┘\nBoth eye inspection and the fit indicate that the data is not linear, although it actually is. This surprising result is obtained even though the fluctuations in both x and y are symmetric around the true value. It originates from bin migration effects, which are not taken into account in this simple study.\nIt is possible to avoid the bias with a more sophisticated analysis that properly treats this problem with the maximum likelihood method. Such an analysis was developed to calibrate the Pierre Auger Observatory, which can be transferred to other calibration problems.\nH. Dembinski, B. Kégl, I.C. Maris, M. Roth, D. Veberič, Astropart.Phys. 73 (2016) 44-51\nIf the proper likelihood analysis of the problem cannot be performed, one should always bin in the variable which depicts smaller fluctuations and try to make the bins large compared to these fluctuations to minimize bin migration. However, this can also be counter-productive as making bins larger increases another bias from mixing samples that have locally different expectation values and densities.\nEven then, however, it is not guaranteed that biases are absent. This needs to confirmed with toy simulations.\nIf the extra effort needed to patch up a naive analysis is included, the amount of work is as great or larger to using the proper likelihood analysis described in the paper, which makes the likelihood analysis the economic choice." + }, + { + "objectID": "posts/naive_calibration_bias.html#partial-solution-modified-least-squares-method", + "href": "posts/naive_calibration_bias.html#partial-solution-modified-least-squares-method", + "title": "Statistical issues in naive calibration analyses", + "section": "Partial solution: modified least-squares method", + "text": "Partial solution: modified least-squares method\nA partial solution that may produce good results on some data is the modified least-squares method. It requires point estimates for the uncertainties of the x and y values. The standard least-squares method assumes that the x-values are known and only the y-values carry Gaussian uncertainties. Gaussian uncertainties on x-values can be approximately taken into account via error-propagation inside the cost function, using the current estimate of the fitting function.\nThis method works well, if the point estimates of the errors are fairly accurate and if the fitted function is not far from linear on the scale of the x-errors.\nIn our toy example, we get good results if we use the true uncertainties per point, but the results are poor if we try to estimate the uncertainties from the observed values.\n\n# with small error on x\nsx_true = sigma_x(x_true)\nsy_true = sigma_y(y_true)\n\nx, y = x0, y0\nsx, sy = sx_true, sy_true\n\ndef chi2(a, b):\n ym = line(x, a, b)\n dy_dx = b\n var = (dy_dx * sx) ** 2 + sy ** 2\n res = (y - ym) ** 2 / var\n return np.sum(res)\n\nchi2.ndata = len(x)\n\n\ndef visualize(args):\n a, b = args\n ym = line(x, a, b)\n plt.plot(x, ym, zorder=3)\n plt.errorbar(x, y, np.abs(sigma_y(y)), np.abs(sigma_x(x)), fmt=\".\", alpha=0.5)\n\nchi2.visualize = visualize\n\nm = Minuit(chi2, 0, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 953.6 (χ²/ndof = 1.0)\nNfcn = 31\n\n\nEDM = 1.57e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n-1.2e-3\n0.8e-3\n\n\n\n\n\n\n\n1\nb\n1.013\n0.017\n\n\n\n\n\n\n\n\n\n\n\n\na\nb\n\n\na\n6.42e-07\n-3.5e-6 (-0.260)\n\n\nb\n-3.5e-6 (-0.260)\n0.000277\n\n\n\n\n\n\n \n \n \n \n 2023-07-10T11:27:45.162470\n image/svg+xml\n \n \n Matplotlib v3.6.3, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 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\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n# with large error on x\nsx_true = sigma_x(x_true)\nsy_true = sigma_y(y_true)\n\ny, x = x0, y0\nsy, sx = sx_true, sy_true\n\nm = Minuit(chi2, 0, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 953.6 (χ²/ndof = 1.0)\nNfcn = 38\n\n\nEDM = 1.13e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n1.2e-3\n0.8e-3\n\n\n\n\n\n\n\n1\nb\n0.987\n0.016\n\n\n\n\n\n\n\n\n\n\n\n\na\nb\n\n\na\n6.17e-07\n-3.0e-6 (-0.236)\n\n\nb\n-3.0e-6 (-0.236)\n0.000263\n\n\n\n\n\n\n \n \n \n \n 2023-07-10T11:27:45.502446\n image/svg+xml\n \n \n Matplotlib v3.6.3, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nThis fit is already slightly biased due to the points at negative x-values.\nThe fit gets badly biased if we don’t know the true uncertainties per point and have to estimate them based on the observed x- and y-values. For the y-values, we can do a bit better by using the predicted y-value from the model to compute the uncertainty estimate, but this trick does not work for the x-values.\n\n# with small error on x, but errors estimated from observed x,y\nx, y = x0, y0\nsx = np.abs(sigma_x(x))\n\ndef chi2(a, b):\n ym = line(x, a, b)\n dy_dx = b\n # estimate y-error using predicted y-values\n sy = sigma_y(ym)\n var = (dy_dx * sx) ** 2 + sy ** 2\n res = (y - ym) ** 2 / var\n return np.sum(res)\n\nchi2.ndata = len(x)\nchi2.visualize = visualize\n\nm = Minuit(chi2, 0, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 753.9 (χ²/ndof = 0.8)\nNfcn = 71\n\n\nEDM = 1.76e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n-1.7e-3\n0.8e-3\n\n\n\n\n\n\n\n1\nb\n1.280\n0.023\n\n\n\n\n\n\n\n\n\n\n\n\na\nb\n\n\na\n6.36e-07\n-4.6e-6 (-0.245)\n\n\nb\n-4.6e-6 (-0.245)\n0.000554\n\n\n\n\n\n\n \n \n \n \n 2023-07-10T11:27:45.851819\n image/svg+xml\n \n \n Matplotlib v3.6.3, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nWe are not recovering the expected (0, 1) parameters." + }, + { + "objectID": "posts/naive_calibration_bias.html#general-solution-maximum-likelihood-method-with-hierarchical-statistical-model", + "href": "posts/naive_calibration_bias.html#general-solution-maximum-likelihood-method-with-hierarchical-statistical-model", + "title": "Statistical issues in naive calibration analyses", + "section": "General solution: maximum-likelihood method with hierarchical statistical model", + "text": "General solution: maximum-likelihood method with hierarchical statistical model\nThe correct solution from first principles is to write down a statistical model of the calibration process. This model is hierarchical. The first step is to draw a sample of unknown true values \\(x, y\\) from the underlying frequency distribution (here an exponential). This needs to be folded with normal distributions to model the observed distribution.\nFollowing section A of H. Dembinski, B. Kégl, I.C. Maris, M. Roth, D. Veberič, Astropart.Phys. 73 (2016) 44-51, we get for this toy example\n\\[\nf(x, y; \\vec p) = \\int \\text{d}x_\\text{true} \\, g(x; x_\\text{true}, \\sigma_x) \\, g(y; f(x_\\text{true}, \\vec{p}), \\sigma_y) \\, h(x_\\text{true})\n\\]\nwhere \\(g\\) is a normal distribution and \\(h\\) is an exponential. This solution treats x- and y-values on equal footing and is derived from first principles.\nAs explained in the paper, solving this equation numerically is slow. However, a fast approximation is given with approximation B (Eq. 26) which replaces \\(h(x_\\text{true})\\) with its empirical density (an idea borrowed from bootstrap theory). With this we get\n\\[\nf(x, y; \\vec p) = \\frac 1 N \\sum_i \\, g(x; x_i, \\sigma_{x,i}) \\, g(y; f(x_i), \\sigma_{y,i})\n\\]\nwhere \\((x_i, y_i)\\) are the observed values and the sum \\(\\sum_i\\) runs over the observed sample. The log-likelihood to maximize is then\n\\[\n\\ln L(\\vec p) = \\sum_k \\ln f(x_k, y_k; \\vec p)\n\\]\nThe approximate solution for this problem contains a curious double sum over the event sample. This is not an error, but the consequence of using the empirical distribution to replace \\(h(x_\\text{true})\\).\n\nimport numba as nb\n\nx, y = x0, y0\nsx, sy = np.abs(sigma_x(x)), np.abs(sigma_y(y))\n\n\n@nb.njit\ndef norm_pdf(x, m, s):\n z = (x-m)/s\n return np.exp(-0.5 * z ** 2) / np.sqrt(2 * np.pi) / s\n\n\n\n@nb.njit\ndef pdf(xi, yi, a, b):\n p = 0\n for xk, sxk, syk in zip(x, sx, sy):\n ym = a + b * xk\n p += norm_pdf(xi, xk, sxk) * norm_pdf(yi, ym, syk)\n return p\n\n\n@nb.njit\ndef chi2(a, b):\n result = 0\n for (xi, yi) in zip(x, y):\n result += np.log(pdf(xi, yi, a, b))\n return -2 * result\n\n\nchi2.visualize = visualize\n\nm = Minuit(chi2, 0, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -1.14e+04\nNfcn = 40\n\n\nEDM = 7.52e-05 (Goal: 0.0002)\ntime = 1.1 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n-1.7e-3\n0.5e-3\n\n\n\n\n\n\n\n1\nb\n1.018\n0.008\n\n\n\n\n\n\n\n\n\n\n\n\na\nb\n\n\na\n2.55e-07\n-0.52e-6 (-0.136)\n\n\nb\n-0.52e-6 (-0.136)\n5.81e-05\n\n\n\n\n\n\n \n \n \n \n 2023-07-10T11:27:47.294393\n image/svg+xml\n \n \n Matplotlib v3.6.3, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nThe result is fairly good despite the fact that we used a poor estimate of \\(\\sigma_y\\), since it is based on the observed value which can deviate quite a bit from the true value. We can improve this approximation by computing the estimate from the predicted value \\(f(x)\\) instead.\n\n@nb.njit\ndef pdf(xi, yi, a, b):\n p = 0\n for xk, sxk in zip(x, sx):\n ym = a + b * xk\n syk = 0.5 * np.abs(ym)\n p += norm_pdf(xi, xk, sxk) * norm_pdf(yi, ym, syk)\n return p\n\n\ndef visualize(args):\n a, b = args\n ym = line(x, a, b)\n plt.plot(x, ym, zorder=3)\n plt.errorbar(x, y, np.abs(sigma_y(ym)), np.abs(sigma_x(x)), fmt=\".\", alpha=0.5)\n\n\nchi2.visualize = visualize\n\nm = Minuit(chi2, 0, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -1.14e+04\nNfcn = 40\n\n\nEDM = 7.52e-05 (Goal: 0.0002)\ntime = 0.8 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n-1.7e-3\n0.5e-3\n\n\n\n\n\n\n\n1\nb\n1.018\n0.008\n\n\n\n\n\n\n\n\n\n\n\n\na\nb\n\n\na\n2.55e-07\n-0.52e-6 (-0.136)\n\n\nb\n-0.52e-6 (-0.136)\n5.81e-05\n\n\n\n\n\n\n \n \n \n \n 2023-07-10T11:27:48.392241\n image/svg+xml\n \n \n Matplotlib v3.6.3, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nStrictly speaking, this visualization is not correct, since we cannot assign error bars to the data points in the likelihood approach. A proper visualization is showing the predicted 2D PDF and the points overlaid.\n\na, b = m.values\nym = line(x, a, b)\n\n# Simple Monte-Carlo integration of pdf over bins in x,y\nxg = rng.uniform(0, 10, 100000)\nyg = rng.uniform(np.min(y), np.max(y), 100000)\np = np.vectorize(pdf)(xg, yg, a, b)\nw, xe, ye = np.histogram2d(xg, yg, bins=20, range=((0, 10), None), weights=p)\n\n# normalize along y-axis to enhance pdf visualization at large x\nfor i in range(w.shape[0]):\n w[i] /= np.sum(w[i])\n\nplt.plot(x, ym, zorder=3)\nplt.pcolormesh(xe, ye, w.T, cmap=\"Blues\")\nplt.plot(x, y, \".C1\", ms=1, alpha=0.5)\nplt.colorbar();\n\n\n\n\nThe agreement of the PDF with the data can be judged by looking at slices along the y-axis.\n\nwd = np.histogram2d(x, y, bins=(xe, ye))[0]\ncy = 0.5 * (ye[1:] + ye[:-1])\nfor i in range(w.shape[0]):\n wsum = np.sum(wd[i])\n if wsum > 0:\n plt.figure()\n plt.title(f\"x = [{xe[i]:.1f}, {xe[i+1]:.1f})\")\n plt.stairs(w[i], ye)\n plt.errorbar(cy, wd[i] / wsum, wd[i] ** 0.5 / wsum, fmt=\"o\")" + }, + { + "objectID": "posts/render_latex_to_svg.html", + "href": "posts/render_latex_to_svg.html", + "title": "Render LaTeX to SVG with matplotlib backend", + "section": "", + "text": "We use matplotlib’s mathtext system and SVG backend to generate an SVG rendering of LaTeX code.\n\nfrom matplotlib.backends.backend_svg import RendererSVG\nfrom matplotlib.font_manager import FontProperties\nfrom io import StringIO\nfrom IPython.display import display_svg, display_html\n\n\nfp = FontProperties()\n\ns = r\"$D^{\\ast\\ast}\\ell\\nu$\"\n\n# get bounding box of rendered text\nr = RendererSVG(0, 0, StringIO())\nw, h, _ = r.get_text_width_height_descent(s, fp, ismath=True)\n\nwith StringIO() as f:\n # render LaTeX as path\n r = RendererSVG(w, h, f)\n r.draw_text(r.new_gc(), 0, h, s, fp, angle=0, ismath=True)\n f.seek(0)\n svg_code = f.read()\n\ndisplay_svg(svg_code, raw=True)\n\n\n\n\nWe can insert the SVG image into an HTML table.\n\ndisplay_html(f\"\"\"\n<html>\n<body>\n<table>\n<tr>\n <td style=\"fill: #f0f0f0\"> {svg_code} </td> <td> foo </td>\n</tr>\n<tr>\n <td> bar </td> <td> baz </td>\n</tr>\n</table>\n</body>\n</html>\n\"\"\", raw=True)\n\n\n\n\n\n\n\n \n\n\n \n \n \n \n 2024-05-12T19:17:48.899755\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n foo \n\n\n bar baz" + }, + { + "objectID": "posts/error_propagation_with_sympy.html", + "href": "posts/error_propagation_with_sympy.html", + "title": "Error propagation with Sympy", + "section": "", + "text": "Sympy is a Python module for symbolic computation (like Mathematica and Matlab) with an elegant Python design.\nIt can easily do mechanical tasks such as computing first derivatives for error propagation. We use this here to write a generic error propagation function in only 20+ lines.\nimport sympy\nprint(f\"Sympy version {sympy.__version__}\")\n\nSympy version 1.8" + }, + { + "objectID": "posts/error_propagation_with_sympy.html#independent-variables", + "href": "posts/error_propagation_with_sympy.html#independent-variables", + "title": "Error propagation with Sympy", + "section": "Independent variables", + "text": "Independent variables\n\ndef value_and_covariance_gen(expr, variables):\n expr = sympy.parse_expr(expr)\n\n symbols = sympy.symbols(variables)\n cov_symbols = sympy.symbols(tuple(\"C_\" + k for k in variables))\n expr2 = sum(expr.diff(s) ** 2 * c for s, c in zip(symbols, cov_symbols))\n expr2 = expr2.simplify() # recommended for speed and accuracy\n\n fval = sympy.lambdify(symbols, expr)\n fcov = sympy.lambdify(symbols + cov_symbols, expr2)\n\n def fn(**kwargs):\n x = tuple(v[0] for v in kwargs.values())\n c = tuple(v[1] for v in kwargs.values())\n return fval(*x), fcov(*x, *c)\n\n return fn\n\n\ndef value_and_covariance(expr, **kwargs):\n return value_and_covariance_gen(expr, tuple(kwargs))(**kwargs)\n\nThat’s all, folks!\nvalue_and_covariance_gen generates a Python function that computes the value of the expression and the propagated covariance.\nvalue_and_covariance is just a shortcut to generate and immediately call the function. If the generated function is called with different values several times it is more efficient to generate it once with the first command and then call it several times.\nGenerating the propagating function takes about 6ms on my computer. The generated Python code is as fast to evaluate as any other function that calls numpy functions.\nLimitations of this implemention - Only independent (uncorrelated) inputs are supported here - Expression must consist of basic math (must be parsable by Sympy)\n\nExample usage\n\nvalue_and_covariance(\"a + b\", a=(1, 0.1), b=(2, 0.2)\n\n(3, 0.30000000000000004)\n\n\n\nvalue_and_covariance(\"s / (s + b)\", s=(5, 0.5), b=(10, 0.1))\n\n(0.3333333333333333, 0.001037037037037037)" + }, + { + "objectID": "posts/From fixed target to sqrt_s_nn and back.html", + "href": "posts/From fixed target to sqrt_s_nn and back.html", + "title": "From a fixed-target to center-of-mass frame and back", + "section": "", + "text": "I compute with Sympy how to transform a 4-vector from a fixed-target frame (aka laboratory frame) to the center-of-mass frame, where the colliding particles have equal but opposite momenta. It is fun to do this with Sympy, because it can compute the Taylor series expansion for the common case where the momenta are large compared to the particle masses.\n\nfrom sympy import *\n\n\ne1, p1, e2, p2, m = symbols(\"E_1 p_1 E_2 p_2 m\",real=True, positive=True)\n\nWe work in 1-dimensional coordinates with \\(p = p_z\\). We generally use \\(m = m_p = m_n\\), in other words we neglect the mass difference between protons and neutrons. We consider a proton projectile and a nuclear target with \\(A\\) nucleons.\nWe first compute the cms energy \\(\\sqrt{s_{nn}}\\) in the nucleon-nucleon system from the fixed target system.\nIn this system, we have these 4-vectors: * projectile \\((E, p)\\) * target \\((m, 0)\\)\n\ns2 = (e1 + m)**2 - (p1 + 0)**2; s2\n\n\\(\\displaystyle - p_{1}^{2} + \\left(E_{1} + m\\right)^{2}\\)\n\n\n\ns2 = s2.subs(e1, sqrt(p1 ** 2 + m ** 2)); s2\n\n\\(\\displaystyle - p_{1}^{2} + \\left(m + \\sqrt{m^{2} + p_{1}^{2}}\\right)^{2}\\)\n\n\n\ns2.series(m, 0, 2)\n\n\\(\\displaystyle 2 m p_{1} + O\\left(m^{2}\\right)\\)\n\n\n\ns = sqrt((e1 + m)**2 - (p1 + 0)**2); s\n\n\\(\\displaystyle \\sqrt{- p_{1}^{2} + \\left(E_{1} + m\\right)^{2}}\\)\n\n\n\ns = s.subs(e1, sqrt(p1 ** 2 + m ** 2)); s\n\n\\(\\displaystyle \\sqrt{- p_{1}^{2} + \\left(m + \\sqrt{m^{2} + p_{1}^{2}}\\right)^{2}}\\)\n\n\n\ns.series(m, 0, 2)\n\n\\(\\displaystyle \\sqrt{2} \\sqrt{m} \\sqrt{p_{1}} + \\frac{\\sqrt{2} m^{\\frac{3}{2}}}{2 \\sqrt{p_{1}}} + O\\left(m^{2}\\right)\\)\n\n\n\ndef sqrt_snn(energy):\n mass = (938.27 + 939.57) * 0.5e-9 # nucleon mass in PeV\n return sqrt(2 * mass * energy) * 1e3 # sqrt(sNN) in TeV\n\n\n0.5 * (sqrt_snn(320000+90000) - sqrt_snn(320000-90000))\n\n\\(\\displaystyle 110.127117815583\\)\n\n\n\ndef beam_energy(sqrt_snn):\n mass = (938.27 + 939.57) * 0.5e-3 # nucleon mass in GeV\n return (sqrt_snn * 1e3) ** 2 / (2 * mass) / 1e6 # beam in PeV\n\n\nbeam_energy(14)\n\n104.37523963702976" + }, + { + "objectID": "posts/regression.html", + "href": "posts/regression.html", + "title": "Regression example with a neural network", + "section": "", + "text": "import pickle\nimport gzip\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sklearn.preprocessing import StandardScaler\n\n\nwith gzip.open(\"rec_hist_pt_25.pkl.gz\") as f:\n rec = pickle.load(f)\n\nwith gzip.open(\"gen_hist_pt_25.pkl.gz\") as f:\n gen = pickle.load(f)\n\n# fix for divide by zero\ngen.values()[gen.values() == 0] = 1\n\nn_gen = gen.values()\nn_rec = rec.values()\n# fix for rec > n\nn_rec = np.where(n_rec > n_gen, n_gen, n_rec)\n\nX = []\ndata_k = []\ndata_n = []\nfor ieta, eta in enumerate(rec.axes[0].centers):\n for ipt, pt in enumerate(rec.axes[1].centers):\n for iphi, phi in enumerate(rec.axes[2].centers):\n for ich, _ in enumerate(rec.axes[3]):\n for isp, _ in enumerate(rec.axes[4]):\n X.append((eta, np.log(pt), np.cos(phi), np.sin(phi), ich, isp))\n data_k.append(n_rec[ieta, ipt, iphi, ich, isp])\n data_n.append(n_gen[ieta, ipt, iphi, ich, isp])\n\nX0 = np.array(X).astype(np.float32)\ndata_k = np.array(data_k).astype(np.float32)\ndata_n = np.array(data_n).astype(np.float32)\ny = data_k / data_n\n\nscaler = StandardScaler()\nscaler.fit(X0)\n\nX = scaler.transform(X0)\n\nprint(\"data points\", X.shape[0])\n\ndata points 14400\n\n\n\ndef draw(model):\n labels = (\"π\", \"K\", \"p\", \"other\")\n for phii in rec.axes[2].centers[:]:\n if phii < 0:\n continue\n fig, axes = plt.subplots(3, 2, sharex=True, sharey=True, layout=\"compressed\")\n plt.suptitle(rf\"$\\phi = {np.degrees(phii):.0f}$ deg\")\n for etai, axi in zip(rec.axes[0].centers, axes.flat):\n plt.sca(axi)\n for ich, charge in enumerate((-1, 1)):\n for isp, label in enumerate(labels):\n ma = X0[:, 0] == etai\n ma &= X0[:, 2] == np.cos(phii)\n ma &= X0[:, 3] == np.sin(phii)\n ma &= X0[:, 4] == ich\n ma &= X0[:, 5] == isp\n plt.plot(np.exp(X0[ma, 1]), charge * y[ma], \"o\", ms=4, color=f\"C{isp}\", label=f\"{label}\" if ich else None)\n\n mpt = np.geomspace(10, 1e4, 2000)\n Xp = np.empty((len(mpt), 6), dtype=np.float32)\n Xp[:, 0] = etai\n Xp[:, 1] = np.log(mpt)\n Xp[:, 2] = np.cos(phii)\n Xp[:, 3] = np.sin(phii)\n Xp[:, 4] = ich\n Xp[:, 5] = isp\n yp = model(scaler.transform(Xp))\n plt.plot(np.exp(Xp[:, 1]), charge * yp, color=f\"C{isp}\")\n plt.sca(axes[0,0])\n plt.semilogx()\n plt.ylim(-1.1, 1.1)\n plt.legend(fontsize=\"xx-small\", ncol=2, frameon=False);" + }, + { + "objectID": "posts/regression.html#prepare-data", + "href": "posts/regression.html#prepare-data", + "title": "Regression example with a neural network", + "section": "", + "text": "import pickle\nimport gzip\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sklearn.preprocessing import StandardScaler\n\n\nwith gzip.open(\"rec_hist_pt_25.pkl.gz\") as f:\n rec = pickle.load(f)\n\nwith gzip.open(\"gen_hist_pt_25.pkl.gz\") as f:\n gen = pickle.load(f)\n\n# fix for divide by zero\ngen.values()[gen.values() == 0] = 1\n\nn_gen = gen.values()\nn_rec = rec.values()\n# fix for rec > n\nn_rec = np.where(n_rec > n_gen, n_gen, n_rec)\n\nX = []\ndata_k = []\ndata_n = []\nfor ieta, eta in enumerate(rec.axes[0].centers):\n for ipt, pt in enumerate(rec.axes[1].centers):\n for iphi, phi in enumerate(rec.axes[2].centers):\n for ich, _ in enumerate(rec.axes[3]):\n for isp, _ in enumerate(rec.axes[4]):\n X.append((eta, np.log(pt), np.cos(phi), np.sin(phi), ich, isp))\n data_k.append(n_rec[ieta, ipt, iphi, ich, isp])\n data_n.append(n_gen[ieta, ipt, iphi, ich, isp])\n\nX0 = np.array(X).astype(np.float32)\ndata_k = np.array(data_k).astype(np.float32)\ndata_n = np.array(data_n).astype(np.float32)\ny = data_k / data_n\n\nscaler = StandardScaler()\nscaler.fit(X0)\n\nX = scaler.transform(X0)\n\nprint(\"data points\", X.shape[0])\n\ndata points 14400\n\n\n\ndef draw(model):\n labels = (\"π\", \"K\", \"p\", \"other\")\n for phii in rec.axes[2].centers[:]:\n if phii < 0:\n continue\n fig, axes = plt.subplots(3, 2, sharex=True, sharey=True, layout=\"compressed\")\n plt.suptitle(rf\"$\\phi = {np.degrees(phii):.0f}$ deg\")\n for etai, axi in zip(rec.axes[0].centers, axes.flat):\n plt.sca(axi)\n for ich, charge in enumerate((-1, 1)):\n for isp, label in enumerate(labels):\n ma = X0[:, 0] == etai\n ma &= X0[:, 2] == np.cos(phii)\n ma &= X0[:, 3] == np.sin(phii)\n ma &= X0[:, 4] == ich\n ma &= X0[:, 5] == isp\n plt.plot(np.exp(X0[ma, 1]), charge * y[ma], \"o\", ms=4, color=f\"C{isp}\", label=f\"{label}\" if ich else None)\n\n mpt = np.geomspace(10, 1e4, 2000)\n Xp = np.empty((len(mpt), 6), dtype=np.float32)\n Xp[:, 0] = etai\n Xp[:, 1] = np.log(mpt)\n Xp[:, 2] = np.cos(phii)\n Xp[:, 3] = np.sin(phii)\n Xp[:, 4] = ich\n Xp[:, 5] = isp\n yp = model(scaler.transform(Xp))\n plt.plot(np.exp(Xp[:, 1]), charge * yp, color=f\"C{isp}\")\n plt.sca(axes[0,0])\n plt.semilogx()\n plt.ylim(-1.1, 1.1)\n plt.legend(fontsize=\"xx-small\", ncol=2, frameon=False);" + }, + { + "objectID": "posts/regression.html#scikit-learn", + "href": "posts/regression.html#scikit-learn", + "title": "Regression example with a neural network", + "section": "Scikit-Learn", + "text": "Scikit-Learn\nIt turns out that the simple MLPRegressor in Scikit-Learn works very well on small datasets.\n\nfrom sklearn.neural_network import MLPRegressor\nfrom sklearn.metrics import mean_squared_error\n\nclf = MLPRegressor(\n hidden_layer_sizes=(2**6,) * 4,\n batch_size=1000,\n alpha=0,\n tol=1e-6,\n max_iter=1000,\n verbose=1,\n)\n\nclf.fit(X, y)\n\n# coefficients only available after training\nn = 0\nfor c in clf.coefs_:\n n += np.prod(c.shape)\nfor c in clf.intercepts_:\n n += c.shape[0]\nprint(\"number of parameters\", n)\n\nprint(\"MLPRegressor\", mean_squared_error(clf.predict(X), y))\n\nIteration 1, loss = 0.03872877\nIteration 2, loss = 0.01838284\nIteration 3, loss = 0.01322917\nIteration 4, loss = 0.00992924\nIteration 5, loss = 0.00764789\nIteration 6, loss = 0.00596758\nIteration 7, loss = 0.00476454\nIteration 8, loss = 0.00403427\nIteration 9, loss = 0.00354749\nIteration 10, loss = 0.00312269\nIteration 11, loss = 0.00285058\nIteration 12, loss = 0.00266859\nIteration 13, loss = 0.00250857\nIteration 14, loss = 0.00243164\nIteration 15, loss = 0.00227876\nIteration 16, loss = 0.00218516\nIteration 17, loss = 0.00204908\nIteration 18, loss = 0.00195609\nIteration 19, loss = 0.00188667\nIteration 20, loss = 0.00181428\nIteration 21, loss = 0.00181260\nIteration 22, loss = 0.00172098\nIteration 23, loss = 0.00166783\nIteration 24, loss = 0.00162889\nIteration 25, loss = 0.00162617\nIteration 26, loss = 0.00151559\nIteration 27, loss = 0.00148309\nIteration 28, loss = 0.00146949\nIteration 29, loss = 0.00142209\nIteration 30, loss = 0.00137413\nIteration 31, loss = 0.00133488\nIteration 32, loss = 0.00132600\nIteration 33, loss = 0.00133857\nIteration 34, loss = 0.00133807\nIteration 35, loss = 0.00125656\nIteration 36, loss = 0.00125801\nIteration 37, loss = 0.00124734\nIteration 38, loss = 0.00116203\nIteration 39, loss = 0.00112787\nIteration 40, loss = 0.00113262\nIteration 41, loss = 0.00111350\nIteration 42, loss = 0.00109534\nIteration 43, loss = 0.00109662\nIteration 44, loss = 0.00103106\nIteration 45, loss = 0.00099346\nIteration 46, loss = 0.00098687\nIteration 47, loss = 0.00103844\nIteration 48, loss = 0.00099100\nIteration 49, loss = 0.00096235\nIteration 50, loss = 0.00097157\nIteration 51, loss = 0.00092747\nIteration 52, loss = 0.00091701\nIteration 53, loss = 0.00089235\nIteration 54, loss = 0.00087139\nIteration 55, loss = 0.00087560\nIteration 56, loss = 0.00087910\nIteration 57, loss = 0.00090950\nIteration 58, loss = 0.00085914\nIteration 59, loss = 0.00082885\nIteration 60, loss = 0.00081474\nIteration 61, loss = 0.00084764\nIteration 62, loss = 0.00081955\nIteration 63, loss = 0.00083712\nIteration 64, loss = 0.00080643\nIteration 65, loss = 0.00078607\nIteration 66, loss = 0.00077217\nIteration 67, loss = 0.00080849\nIteration 68, loss = 0.00081811\nIteration 69, loss = 0.00078761\nIteration 70, loss = 0.00077427\nIteration 71, loss = 0.00075709\nIteration 72, loss = 0.00075573\nIteration 73, loss = 0.00075636\nIteration 74, loss = 0.00074296\nIteration 75, loss = 0.00075769\nIteration 76, loss = 0.00078198\nIteration 77, loss = 0.00074714\nIteration 78, loss = 0.00076107\nIteration 79, loss = 0.00073871\nIteration 80, loss = 0.00071247\nIteration 81, loss = 0.00073748\nIteration 82, loss = 0.00074470\nIteration 83, loss = 0.00070695\nIteration 84, loss = 0.00069866\nIteration 85, loss = 0.00068365\nIteration 86, loss = 0.00067912\nIteration 87, loss = 0.00068739\nIteration 88, loss = 0.00067988\nIteration 89, loss = 0.00070646\nIteration 90, loss = 0.00067668\nIteration 91, loss = 0.00068584\nIteration 92, loss = 0.00066119\nIteration 93, loss = 0.00065713\nIteration 94, loss = 0.00079503\nIteration 95, loss = 0.00070443\nIteration 96, loss = 0.00069255\nIteration 97, loss = 0.00064721\nIteration 98, loss = 0.00066456\nIteration 99, loss = 0.00066503\nIteration 100, loss = 0.00064768\nIteration 101, loss = 0.00063161\nIteration 102, loss = 0.00065152\nIteration 103, loss = 0.00063627\nIteration 104, loss = 0.00062377\nIteration 105, loss = 0.00063346\nIteration 106, loss = 0.00065331\nIteration 107, loss = 0.00063847\nIteration 108, loss = 0.00061856\nIteration 109, loss = 0.00061806\nIteration 110, loss = 0.00060691\nIteration 111, loss = 0.00061332\nIteration 112, loss = 0.00061673\nIteration 113, loss = 0.00064613\nIteration 114, loss = 0.00060685\nIteration 115, loss = 0.00059055\nIteration 116, loss = 0.00060077\nIteration 117, loss = 0.00062042\nIteration 118, loss = 0.00065500\nIteration 119, loss = 0.00060160\nIteration 120, loss = 0.00060204\nIteration 121, loss = 0.00059747\nIteration 122, loss = 0.00061649\nIteration 123, loss = 0.00060509\nIteration 124, loss = 0.00063298\nIteration 125, loss = 0.00060878\nIteration 126, loss = 0.00061471\nTraining loss did not improve more than tol=0.000001 for 10 consecutive epochs. Stopping.\nnumber of parameters 12993\nMLPRegressor 0.0011639113\n\n\n\ndraw(clf.predict)" + }, + { + "objectID": "posts/regression.html#pytorch", + "href": "posts/regression.html#pytorch", + "title": "Regression example with a neural network", + "section": "PyTorch", + "text": "PyTorch\n\nimport torch\nfrom torch import nn\nfrom torch.optim.lr_scheduler import ReduceLROnPlateau\nfrom torch.optim import Adam\n\ndevice = \"cuda\" if torch.cuda.is_available() else \"cpu\"\ntorch.set_default_device(device)\ntorch.set_default_dtype(torch.float32)\n\nnonlin = nn.ReLU()\nnum = 2**6\ntorch_model = nn.Sequential(\n nn.Linear(6, num),\n nonlin,\n nn.Linear(num, num),\n nonlin,\n nn.Linear(num, num),\n nonlin,\n nn.Linear(num, num),\n nonlin,\n nn.Linear(num, 1),\n # adding a ReLU here makes training unstable\n nn.Flatten(0, 1),\n)\n\nn = 0\nfor par in torch_model.parameters():\n n += np.prod(par.shape)\nprint(\"number of parameters\", n)\n\n# must make copies here\ntorch_X = torch.tensor(X.copy())\ntorch_y = torch.tensor(y.copy())\n\nmax_epoch = 5000\nshrink_patience = 10\nabort_patience = 100\nlearning_rate = 1e-3\nlearning_rate_shrink_factor = 0.8\ntol = 1e-6\n\nloss_fn = nn.MSELoss()\nopt = Adam(torch_model.parameters(), lr=learning_rate)\nscheduler = ReduceLROnPlateau(\n opt, patience=shrink_patience, factor=learning_rate_shrink_factor, verbose=True\n)\n\nlosses = []\nno_improvement_below_tolerance = 0\nprev_loss = np.inf\nfor epoch in range(1, max_epoch + 1):\n y_pred = torch_model(torch_X)\n loss = loss_fn(y_pred, torch_y)\n opt.zero_grad()\n loss.backward()\n opt.step()\n scheduler.step(loss)\n\n loss = loss.item()\n losses.append(loss)\n if epoch == 1 or epoch % 100 == 0:\n print(\"epoch\", epoch, \"loss\", loss)\n\n if loss < prev_loss - tol:\n no_improvement_below_tolerance = 0\n prev_loss = loss\n else:\n no_improvement_below_tolerance += 1\n if no_improvement_below_tolerance > abort_patience:\n break\n\nplt.plot(losses)\nplt.semilogy();\n\nnumber of parameters 12993\nepoch 1 loss 0.25884100794792175\nepoch 100 loss 0.010155524127185345\nepoch 200 loss 0.003915281500667334\nepoch 300 loss 0.002415228867903352\nepoch 400 loss 0.0017484889831393957\nepoch 500 loss 0.0014784386148676276\nEpoch 00508: reducing learning rate of group 0 to 8.0000e-04.\nepoch 600 loss 0.0013304103631526232\nEpoch 00695: reducing learning rate of group 0 to 6.4000e-04.\nepoch 700 loss 0.001243863138370216\nepoch 800 loss 0.0011801073560491204\nepoch 900 loss 0.0011410759761929512\nEpoch 00905: reducing learning rate of group 0 to 5.1200e-04.\nepoch 1000 loss 0.00109128060285002\nepoch 1100 loss 0.0010561983799561858\nEpoch 01152: reducing learning rate of group 0 to 4.0960e-04.\nepoch 1200 loss 0.0010263388976454735\nepoch 1300 loss 0.001002235570922494\nepoch 1400 loss 0.0009775016224011779\nEpoch 01450: reducing learning rate of group 0 to 3.2768e-04.\nepoch 1500 loss 0.0009563906933180988\nepoch 1600 loss 0.0009381748968735337\nepoch 1700 loss 0.0009208598639816046\nepoch 1800 loss 0.0009038595599122345\nEpoch 01887: reducing learning rate of group 0 to 2.6214e-04.\nepoch 1900 loss 0.0008835234912112355\nepoch 2000 loss 0.0008701192564330995\nepoch 2100 loss 0.0008563698502257466\nepoch 2200 loss 0.000845507369376719\nepoch 2300 loss 0.0008326928946189582\nEpoch 02305: reducing learning rate of group 0 to 2.0972e-04.\nepoch 2400 loss 0.0008199398289434612\nepoch 2500 loss 0.0008091804338619113\nepoch 2600 loss 0.000800078792963177\nepoch 2700 loss 0.0007896188762970269\nepoch 2800 loss 0.000779114430770278\nEpoch 02864: reducing learning rate of group 0 to 1.6777e-04.\nepoch 2900 loss 0.0007669864571653306\nepoch 3000 loss 0.0007585080456919968\nepoch 3100 loss 0.0007498746854253113\nepoch 3200 loss 0.0007412403356283903\nepoch 3300 loss 0.0007329632644541562\nepoch 3400 loss 0.0007246120949275792\nEpoch 03408: reducing learning rate of group 0 to 1.3422e-04.\nepoch 3500 loss 0.0007171895122155547\nepoch 3600 loss 0.0007101951632648706\nepoch 3700 loss 0.0007029927219264209\nepoch 3800 loss 0.0006956694996915758\nEpoch 03889: reducing learning rate of group 0 to 1.0737e-04.\nepoch 3900 loss 0.0006882176967337728\nepoch 4000 loss 0.0006816069362685084\nepoch 4100 loss 0.0006751783657819033\nepoch 4200 loss 0.0006688352441415191\nepoch 4300 loss 0.0006628871196880937\nepoch 4400 loss 0.0006569336983375251\nepoch 4500 loss 0.0006518965819850564\nepoch 4600 loss 0.0006464265170507133\nepoch 4700 loss 0.00063846365083009\nepoch 4800 loss 0.0006334380595944822\nepoch 4900 loss 0.0006282057147473097\nepoch 5000 loss 0.0006211302825249732\n\n\n\n\n\n\nprint(\"MLPRegressor\", mean_squared_error(clf.predict(X), y))\nprint(\"PyTorch \", mean_squared_error(torch_model(torch_X).detach().numpy(), y))\n\nMLPRegressor 0.0011639113\nPyTorch 0.00062100013\n\n\n\ndraw(lambda x: torch_model(torch.tensor(x)).detach().numpy())" + }, + { + "objectID": "posts/regression.html#flax-jax", + "href": "posts/regression.html#flax-jax", + "title": "Regression example with a neural network", + "section": "Flax / JAX", + "text": "Flax / JAX\nFlax with JAX is a modern functional programming library which is great for trying out new things. It comes with fewer high-level functionality than PyTorch. Where PyTorch offers ReduceLROnPlateau, we need to implement something analog from scratch.\n\nfrom typing import Callable, Iterable\nfrom types import SimpleNamespace\nimport jax\nfrom jax import numpy as jnp\nfrom flax import linen as nn\nimport optax\n\n\nclass Model(nn.Module):\n sizes: Iterable[int] # Number of hidden neurons\n nonlin: Callable\n\n @nn.compact # Tells Flax to look for defined submodules\n def __call__(self, x):\n for size in self.sizes:\n x = nn.Dense(size)(x)\n x = self.nonlin(x)\n return nn.Dense(1)(x).flatten()\n\n\ndef train(config: SimpleNamespace, x, y):\n x = jax.device_put(x)\n y = jax.device_put(y)\n\n model = Model(config.layers, config.nonlin)\n\n @jax.jit\n def loss_fn(theta, x, y):\n def squared_error(x, y):\n y_pred = model.apply(theta, x)\n return jnp.square(y - y_pred)\n return jnp.mean(jax.vmap(squared_error, in_axes=0)(x, y))\n\n key1, key2 = jax.random.split(jax.random.PRNGKey(0))\n theta = model.init(key2, jax.random.normal(key1, x.shape[1:]))\n\n n = 0\n for layer in theta[\"params\"].values():\n n += np.prod(layer[\"kernel\"].shape) + layer[\"bias\"].shape[0]\n print(\"number of parameters\", n)\n\n tx = optax.inject_hyperparams(lambda learning_rate: optax.chain(\n optax.clip(1), # equivalent to using huber loss\n optax.adabelief(learning_rate),\n ))(learning_rate=config.learning_rate)\n tx_state = tx.init(theta)\n loss_grad_fn = jax.value_and_grad(loss_fn)\n\n losses = []\n prev_loss = np.inf\n epochs_without_improvement = 0\n for epoch in range(1, config.max_epoch + 1):\n\n loss, grads = loss_grad_fn(theta, x, y)\n updates, tx_state = tx.update(grads, tx_state)\n theta = optax.apply_updates(theta, updates)\n\n losses.append(loss)\n\n if epoch == 1 or epoch % config.print_freq == 0:\n print(f\"epoch {epoch} loss={loss}\")\n\n if loss < prev_loss:\n epochs_without_improvement = 0\n prev_loss = loss\n else:\n epochs_without_improvement += 1\n if epochs_without_improvement >= config.shrink_patience:\n tx_state.hyperparams[\"learning_rate\"] *= config.learning_rate_shrink_factor\n print(f\"epoch {epoch } learning rate={tx_state.hyperparams['learning_rate']}\")\n epochs_without_improvement = 0\n continue\n if tx_state.hyperparams[\"learning_rate\"] < config.tolerance:\n break\n\n return model, theta, losses\n\nprint(f\"JAX process: {jax.process_index()} / {jax.process_count()}\")\nprint(f\"JAX local devices: {jax.local_devices()}\")\n\nconfig = SimpleNamespace()\nconfig.layers = (2 ** 6,) * 4\nconfig.nonlin = nn.relu\nconfig.max_epoch = 5000\nconfig.learning_rate = 1e-2\nconfig.shrink_patience = 10\nconfig.learning_rate_shrink_factor = 0.8\nconfig.print_freq = 100\nconfig.tolerance = 1e-6\n\nprev_losses = locals().get(\"prev_losses\", None)\n\nflax_model, flax_theta, losses = train(config, X, y)\n\nbest_losses = locals().get(\"best_losses\", losses)\nif best_losses[-1] < losses[-1]:\n best_losses = losses\n\nplt.plot(losses, label=\"now\")\nif prev_losses is not None:\n plt.plot(prev_losses, label=\"previous\")\nplt.plot(best_losses, label=\"best\")\nplt.legend()\nplt.semilogy();\n\nJAX process: 0 / 1\nJAX local devices: [CpuDevice(id=0)]\nnumber of parameters 12993\nepoch 1 loss=0.1615249365568161\nepoch 100 loss=0.004056446719914675\nepoch 200 loss=0.0022707092575728893\nepoch 237 learning rate=0.00800000037997961\nepoch 247 learning rate=0.006400000303983688\nepoch 300 loss=0.0015574624994769692\nepoch 400 loss=0.0013330463552847505\nepoch 500 loss=0.0012052441015839577\nepoch 600 loss=0.0011462707770988345\nepoch 604 learning rate=0.005120000336319208\nepoch 700 loss=0.0010661932174116373\nepoch 773 learning rate=0.0040960004553198814\nepoch 800 loss=0.0010216122027486563\nepoch 900 loss=0.0009868466295301914\nepoch 982 learning rate=0.0032768005039542913\nepoch 1000 loss=0.0009563774801790714\nepoch 1100 loss=0.0009271150338463485\nepoch 1200 loss=0.0008946977904997766\nepoch 1256 learning rate=0.002621440449729562\nepoch 1300 loss=0.0008701515616849065\nepoch 1400 loss=0.0008422626415267587\nepoch 1500 loss=0.0008212537504732609\nepoch 1600 loss=0.0008094692602753639\nepoch 1602 learning rate=0.002097152406349778\nepoch 1700 loss=0.0007778366561979055\nepoch 1800 loss=0.0007623148267157376\nepoch 1867 learning rate=0.0016777219716459513\nepoch 1900 loss=0.0007488274131901562\nepoch 2000 loss=0.0007354258559644222\nepoch 2100 loss=0.0007237782701849937\nepoch 2200 loss=0.0007126012351363897\nepoch 2236 learning rate=0.0013421776238828897\nepoch 2300 loss=0.0007015920709818602\nepoch 2400 loss=0.0006939161103218794\nepoch 2500 loss=0.000684640312101692\nepoch 2544 learning rate=0.001073742168955505\nepoch 2600 loss=0.0006755936774425209\nepoch 2700 loss=0.0006690087029710412\nepoch 2768 learning rate=0.0008589937351644039\nepoch 2800 loss=0.0006616609753109515\nepoch 2853 learning rate=0.0006871949881315231\nepoch 2900 loss=0.0006564840441569686\nepoch 3000 loss=0.0006521755130961537\nepoch 3100 loss=0.0006475721020251513\nepoch 3200 loss=0.0006428791675716639\nepoch 3300 loss=0.0006390072521753609\nepoch 3344 learning rate=0.0005497559905052185\nepoch 3386 learning rate=0.000439804804045707\nepoch 3400 loss=0.000634271593298763\nepoch 3500 loss=0.0006310778553597629\nepoch 3600 loss=0.0006278276559896767\nepoch 3700 loss=0.0006245431723073125\nepoch 3800 loss=0.000620958860963583\nepoch 3900 loss=0.0006179117481224239\nepoch 4000 loss=0.0006144311628304422\nepoch 4091 learning rate=0.00035184386069886386\nepoch 4100 loss=0.0006107029039412737\nepoch 4200 loss=0.0006076746503822505\nepoch 4300 loss=0.0006047412171028554\nepoch 4400 loss=0.0006017453270033002\nepoch 4500 loss=0.0005980761488899589\nepoch 4510 learning rate=0.00028147510602138937\nepoch 4600 loss=0.0005954525549896061\nepoch 4700 loss=0.0005925073637627065\nepoch 4800 loss=0.0005897046066820621\nepoch 4900 loss=0.0005866923602297902\nepoch 5000 loss=0.0005837904172949493\n\n\n\n\n\n\nprint(\"MLPRegressor\", mean_squared_error(clf.predict(X), y))\nprint(\"PyTorch \", mean_squared_error(torch_model(torch_X).detach().numpy(), y))\nprint(\"Flax \", mean_squared_error(flax_model.apply(flax_theta, X), y))\n\nMLPRegressor 0.0011639113\nPyTorch 0.00062100013\nFlax 0.0005836923\n\n\n\ndraw(lambda x: flax_model.apply(flax_theta, x))\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAnother version of learning with FLAX using the meta-learning technique, where the learning rate of the model optimization is learned by an outer learning step. Meta-learning outperforms our previous approach: the loss after 5000 epochs is smaller, and the run-time is a bit shorter.\n\nfrom typing import Callable, Iterable\nfrom types import SimpleNamespace\nimport jax\nfrom jax import numpy as jnp\nfrom flax import linen as nn\nimport optax\n\n\nclass Model(nn.Module):\n sizes: Iterable[int] # Number of hidden neurons\n nonlin: Callable\n\n @nn.compact # Tells Flax to look for defined submodules\n def __call__(self, x):\n for size in self.sizes:\n x = nn.Dense(size)(x)\n x = self.nonlin(x)\n return nn.Dense(1)(x).flatten()\n\n\ndef train(config, x, y):\n x = jax.device_put(x)\n y = jax.device_put(y)\n model = Model(config.layers, config.nonlin)\n\n root_key = jax.random.PRNGKey(0)\n param_key, perm_key = jax.random.split(root_key)\n theta = model.init(param_key, x)\n\n opt = optax.inject_hyperparams(optax.rmsprop)(learning_rate=config.learning_rate)\n\n meta_opt = optax.adabelief(learning_rate=config.meta_learning_rate)\n\n state = opt.init(theta)\n eta = -np.log(1.0 / config.learning_rate - 1)\n meta_state = meta_opt.init(eta)\n\n @jax.jit\n def loss_fn(theta, x, y):\n def squared_error(x, y):\n y_pred = model.apply(theta, x)\n return jnp.square(y - y_pred)\n return jnp.mean(jax.vmap(squared_error, in_axes=0)(x, y))\n\n @jax.jit\n def step(theta, state, x, y):\n loss, grad = jax.value_and_grad(loss_fn)(theta, x, y)\n updates, state = opt.update(grad, state)\n theta = optax.apply_updates(theta, updates)\n return loss, theta, state\n\n @jax.jit\n def outer_loss(eta, theta, state, x, y):\n state.hyperparams[\"learning_rate\"] = jax.nn.sigmoid(eta)\n loss, theta, state = step(theta, state, x[:-1], y[:-1])\n return loss_fn(theta, x[-1:], y[-1:]), (loss, theta, state)\n\n @jax.jit\n def outer_step(eta, theta, meta_state, state, x, y):\n grad, (loss, theta, state) = jax.grad(outer_loss, has_aux=True)(eta, theta, state, x, y)\n meta_updates, meta_state = meta_opt.update(grad, meta_state)\n eta = optax.apply_updates(eta, meta_updates)\n return eta, theta, meta_state, state, loss\n\n learning_rates = []\n losses = []\n\n for epoch in range(1, config.max_epoch + 1):\n perm_train_key = jax.random.fold_in(perm_key, epoch)\n perm = jax.random.permutation(perm_train_key, len(x))\n eta, theta, meta_state, state, loss = outer_step(eta, theta, meta_state, state, x[perm], y[perm])\n learning_rate = jax.nn.sigmoid(eta)\n\n losses.append(loss)\n learning_rates.append(learning_rate)\n\n if epoch == 1 or epoch % config.print_freq == 0:\n print(f\"epoch {epoch} loss={loss} lr={learning_rate}\")\n\n return model, theta, losses, learning_rates\n\nconfig = SimpleNamespace()\nconfig.layers = (2 ** 6,) * 4\nconfig.nonlin = nn.relu\nconfig.max_epoch = 5000\nconfig.learning_rate = 1e-2\nconfig.meta_learning_rate = 0.03\nconfig.print_freq = 100\n\nflax_model, flax_ml_theta, losses, learning_rates = train(config, X, y)\n\nfig, ax = plt.subplots(2, 1, sharex=True)\nax[0].plot(losses)\nax[0].set(ylabel=\"loss\", yscale=\"log\")\nax[1].plot(learning_rates)\nax[1].set(xlabel=\"epoch\", ylabel=\"lr\", yscale=\"log\");\n\nepoch 1 loss=0.31326305866241455 lr=0.00970732606947422\nepoch 100 loss=0.022382477298378944 lr=0.008342958986759186\nepoch 200 loss=0.009440391324460506 lr=0.008291025646030903\nepoch 300 loss=0.005982980597764254 lr=0.008259016089141369\nepoch 400 loss=0.004722251556813717 lr=0.008235295303165913\nepoch 500 loss=0.0026633746456354856 lr=0.008185734041035175\nepoch 600 loss=0.0032396698370575905 lr=0.008146436884999275\nepoch 700 loss=0.002778511494398117 lr=0.008116255514323711\nepoch 800 loss=0.003227198962122202 lr=0.00809383299201727\nepoch 900 loss=0.0033369455486536026 lr=0.008060149848461151\nepoch 1000 loss=0.0021190992556512356 lr=0.008024145849049091\nepoch 1100 loss=0.002638544887304306 lr=0.007992972619831562\nepoch 1200 loss=0.002345192711800337 lr=0.007957393303513527\nepoch 1300 loss=0.001486414228565991 lr=0.007931879721581936\nepoch 1400 loss=0.0017564770532771945 lr=0.007895255461335182\nepoch 1500 loss=0.0017623923486098647 lr=0.007862220518290997\nepoch 1600 loss=0.0017055724747478962 lr=0.007831801660358906\nepoch 1700 loss=0.0017303635831922293 lr=0.007792978081852198\nepoch 1800 loss=0.0011820418294519186 lr=0.007761101704090834\nepoch 1900 loss=0.0019437490263953805 lr=0.007730470504611731\nepoch 2000 loss=0.0014381277142092586 lr=0.007678112480789423\nepoch 2100 loss=0.001700665452517569 lr=0.007647846359759569\nepoch 2200 loss=0.0012748426524922252 lr=0.007603880017995834\nepoch 2300 loss=0.0014644829789176583 lr=0.007567934691905975\nepoch 2400 loss=0.0017952200723811984 lr=0.007531826850026846\nepoch 2500 loss=0.0014581787399947643 lr=0.0074939788319170475\nepoch 2600 loss=0.0011716835433617234 lr=0.0074319071136415005\nepoch 2700 loss=0.0018534348346292973 lr=0.007394589018076658\nepoch 2800 loss=0.0011852614115923643 lr=0.0073562441393733025\nepoch 2900 loss=0.0014049195451661944 lr=0.007317622657865286\nepoch 3000 loss=0.0016386691713705659 lr=0.007295084185898304\nepoch 3100 loss=0.0010843385243788362 lr=0.007261158432811499\nepoch 3200 loss=0.0009666284313425422 lr=0.007225275970995426\nepoch 3300 loss=0.0010213557397946715 lr=0.0071740285493433475\nepoch 3400 loss=0.001257327850908041 lr=0.0071229999884963036\nepoch 3500 loss=0.0009237469639629126 lr=0.007060170639306307\nepoch 3600 loss=0.0010209324536845088 lr=0.007001352030783892\nepoch 3700 loss=0.0010023850481957197 lr=0.006939604412764311\nepoch 3800 loss=0.0011092709610238671 lr=0.006892509292811155\nepoch 3900 loss=0.0008320825872942805 lr=0.006856516934931278\nepoch 4000 loss=0.0015411889180541039 lr=0.006792154163122177\nepoch 4100 loss=0.000866543676238507 lr=0.00673663429915905\nepoch 4200 loss=0.0009100940660573542 lr=0.00666508823633194\nepoch 4300 loss=0.0008933907374739647 lr=0.006604321300983429\nepoch 4400 loss=0.0006370239425450563 lr=0.00655223848298192\nepoch 4500 loss=0.0007405806099995971 lr=0.006423715502023697\nepoch 4600 loss=0.0006541237817145884 lr=0.006394413765519857\nepoch 4700 loss=0.0005402541719377041 lr=0.006318734493106604\nepoch 4800 loss=0.0009636411559768021 lr=0.00621804129332304\nepoch 4900 loss=0.0007088639540597796 lr=0.006063143257051706\nepoch 5000 loss=0.0007585420971736312 lr=0.006008874159306288\n\n\n\n\n\n\nprint(\"MLPRegressor \", mean_squared_error(clf.predict(X), y))\nprint(\"PyTorch \", mean_squared_error(torch_model(torch_X).detach().numpy(), y))\nprint(\"Flax \", mean_squared_error(flax_model.apply(flax_theta, X), y))\nprint(\"Flax: meta-learning\", mean_squared_error(flax_model.apply(flax_ml_theta, X), y))\n\nMLPRegressor 0.0011639113\nPyTorch 0.00062100013\nFlax 0.0005836923\nFlax: meta-learning 0.0007262042\n\n\n\ndraw(lambda x: flax_model.apply(flax_ml_theta, x))\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nfrom typing import Callable, Iterable, Any\nfrom types import SimpleNamespace\nimport jax\nfrom jax import numpy as jnp\nfrom flax import struct\nfrom flax.core import FrozenDict\nfrom flax import linen as nn\nimport optax\n\n\nclass Model(nn.Module):\n sizes: Iterable[int] # Number of hidden neurons\n nonlin: Callable\n dropout_rate: float = 0.0\n\n @nn.compact # Tells Flax to look for defined submodules\n def __call__(self, x, train):\n for size in self.sizes:\n x = nn.Dense(size)(x)\n x = self.nonlin(x)\n x = nn.Dropout(rate=self.dropout_rate, deterministic=not train)(x)\n return nn.Dense(1, use_bias=False)(x).flatten()\n\n\nclass TrainState(struct.PyTreeNode):\n dropout_key: jax.Array\n params: FrozenDict[str, Any] = struct.field(pytree_node=True)\n opt_state: optax.OptState = struct.field(pytree_node=True)\n meta_params: float\n meta_opt_state: optax.OptState = struct.field(pytree_node=True)\n\n\ndef predict(model, theta, x):\n y = model.apply({\"params\": theta}, x, train=False)\n with np.errstate(over=\"ignore\"):\n return 1 / (1 + np.exp(-y))\n\n\ndef train(config, x, k, n):\n x = jax.device_put(x)\n k = jax.device_put(k)\n n = jax.device_put(n)\n\n model = Model(config.layers, config.nonlin, config.dropout_rate)\n\n root_key = jax.random.PRNGKey(0)\n params_key, dropout_key, perm_key = jax.random.split(root_key, num=3)\n variables = model.init(params_key, x, train=False)\n\n opt = optax.inject_hyperparams(optax.rmsprop)(learning_rate=config.learning_rate)\n meta_opt = optax.adabelief(learning_rate=config.meta_learning_rate)\n eta = -np.log(1.0 / config.learning_rate - 1)\n\n state = TrainState(\n dropout_key=dropout_key,\n params=variables[\"params\"],\n opt_state=opt.init(variables[\"params\"]),\n meta_params=eta,\n meta_opt_state=meta_opt.init(eta),\n )\n\n @jax.jit\n def loss_fn(params, state, x, k, n):\n def fn(x, k, n):\n y = model.apply(\n {\"params\": params},\n x,\n train=True,\n rngs={\"dropout\": state.dropout_key},\n )\n p_sat = jnp.clip(k / (n + 1e-6), 1e-6, 1 - 1e-6)\n y_sat = jnp.log(p_sat / (1 - p_sat))\n l = k * (y - y_sat) - n * (jnp.logaddexp(0, y) - jnp.logaddexp(0, y_sat))\n return l\n\n l = jax.vmap(fn, in_axes=0)(x, k, n)\n return -2 * jnp.mean(l)\n\n @jax.jit\n def inner_step(state, x, k, n):\n loss, grad = jax.value_and_grad(loss_fn)(state.params, state, x, k, n)\n updates, opt_state = opt.update(grad, state.opt_state)\n params = optax.apply_updates(state.params, updates)\n state = state.replace(params=params, opt_state=opt_state)\n return loss, state\n\n @jax.jit\n def outer_loss(eta, state, x, k, n):\n state.opt_state.hyperparams[\"learning_rate\"] = jax.nn.sigmoid(eta)\n loss, state = inner_step(state, x[:-1], k[:-1], n[:-1])\n return loss_fn(state.params, state, x[-1:], k[-1:], n[-1:]), (loss, state)\n\n @jax.jit\n def outer_step(state, x, k, n):\n grad, (loss, state) = jax.grad(outer_loss, has_aux=True)(\n state.meta_params, state, x, k, n\n )\n meta_updates, meta_opt_state = meta_opt.update(grad, state.meta_opt_state)\n meta_params = optax.apply_updates(state.meta_params, meta_updates)\n state = state.replace(meta_params=meta_params, meta_opt_state=meta_opt_state)\n return loss, state\n\n learning_rates = []\n losses = []\n\n for epoch in range(1, config.max_epoch + 1):\n perm_train_key = jax.random.fold_in(key=perm_key, data=epoch)\n state = state.replace(\n dropout_key=jax.random.fold_in(key=dropout_key, data=epoch),\n )\n perm = jax.random.permutation(perm_train_key, len(x))\n loss, state = outer_step(state, x[perm], k[perm], n[perm])\n learning_rate = jax.nn.sigmoid(state.meta_params)\n losses.append(loss)\n learning_rates.append(learning_rate)\n\n if epoch == 1 or epoch % config.print_freq == 0:\n print(f\"epoch {epoch} loss={loss} lr={learning_rate}\")\n\n return model, state.params, losses, learning_rates\n\n\nconfig = SimpleNamespace()\nconfig.layers = (2**6,) * 4\nconfig.nonlin = nn.relu\nconfig.dropout_rate = 0.01\nconfig.max_epoch = 5000\nconfig.learning_rate = 1e-2\nconfig.meta_learning_rate = 0.03\nconfig.print_freq = 100\n\nflax_model_2, flax_theta_2, losses, learning_rates = train(config, X, data_k, data_n)\n\nfig, ax = plt.subplots(2, 1, sharex=True)\nax[0].plot(losses)\nax[0].set(ylabel=\"loss\", yscale=\"log\")\nax[1].plot(learning_rates)\nax[1].set(xlabel=\"epoch\", ylabel=\"lr\", yscale=\"log\");\n\nepoch 1 loss=8698.1416015625 lr=0.00970732606947422\nepoch 100 loss=879.1326904296875 lr=0.006159865763038397\nepoch 200 loss=270.08587646484375 lr=0.004554468207061291\nepoch 300 loss=317.17718505859375 lr=0.0036662158090621233\nepoch 400 loss=91.02045440673828 lr=0.0032094777561724186\nepoch 500 loss=47.988033294677734 lr=0.0026605320163071156\nepoch 600 loss=49.83475112915039 lr=0.0023816365282982588\nepoch 700 loss=116.31311798095703 lr=0.0021545598283410072\nepoch 800 loss=35.58114242553711 lr=0.0017672122921794653\nepoch 900 loss=64.84205627441406 lr=0.0015509785152971745\nepoch 1000 loss=10.535590171813965 lr=0.001485868007875979\nepoch 1100 loss=26.29697036743164 lr=0.0013874215073883533\nepoch 1200 loss=16.937255859375 lr=0.0013577654026448727\nepoch 1300 loss=15.62310791015625 lr=0.0013214262435212731\nepoch 1400 loss=13.242799758911133 lr=0.001257105148397386\nepoch 1500 loss=11.61507797241211 lr=0.001235545496456325\nepoch 1600 loss=14.939531326293945 lr=0.0012049111537635326\nepoch 1700 loss=27.3996639251709 lr=0.001163122127763927\nepoch 1800 loss=21.23556900024414 lr=0.0010698930127546191\nepoch 1900 loss=32.21183776855469 lr=0.0010345007758587599\nepoch 2000 loss=9.131669044494629 lr=0.001013571978546679\nepoch 2100 loss=13.851628303527832 lr=0.0009823944419622421\nepoch 2200 loss=10.54258918762207 lr=0.0009545115754008293\nepoch 2300 loss=9.792181968688965 lr=0.000919334648642689\nepoch 2400 loss=4.94473123550415 lr=0.000911602983251214\nepoch 2500 loss=46.33889389038086 lr=0.0008922962588258088\nepoch 2600 loss=7.025729179382324 lr=0.0008748381515033543\nepoch 2700 loss=15.959792137145996 lr=0.0008621393935754895\nepoch 2800 loss=48.963592529296875 lr=0.0008530952618457377\nepoch 2900 loss=5.237392902374268 lr=0.0008423625258728862\nepoch 3000 loss=15.789849281311035 lr=0.0008185050683096051\nepoch 3100 loss=12.49224853515625 lr=0.0007375497953034937\nepoch 3200 loss=5.135715961456299 lr=0.0007307584746740758\nepoch 3300 loss=3.9630870819091797 lr=0.0006970498943701386\nepoch 3400 loss=9.80554485321045 lr=0.0006649661809206009\nepoch 3500 loss=4.468692302703857 lr=0.000663100800011307\nepoch 3600 loss=48.7807502746582 lr=0.0006592243444174528\nepoch 3700 loss=3.8573150634765625 lr=0.000651187205221504\nepoch 3800 loss=5.408437728881836 lr=0.0006451716180890799\nepoch 3900 loss=4.103397846221924 lr=0.0006489546503871679\nepoch 4000 loss=7.220167636871338 lr=0.0006393020157702267\nepoch 4100 loss=8.81872272491455 lr=0.0006356375524774194\nepoch 4200 loss=3.7671053409576416 lr=0.0006279195658862591\nepoch 4300 loss=3.080355167388916 lr=0.0006261372473090887\nepoch 4400 loss=26.2066650390625 lr=0.0006219103815965354\nepoch 4500 loss=2.989922046661377 lr=0.0006011227960698307\nepoch 4600 loss=5.964871406555176 lr=0.00058519042795524\nepoch 4700 loss=2.8790411949157715 lr=0.0005803073872812092\nepoch 4800 loss=3.9400908946990967 lr=0.0005809014546684921\nepoch 4900 loss=10.3778715133667 lr=0.0005587802152149379\nepoch 5000 loss=5.031178951263428 lr=0.0005468810559250414\n\n\n\n\n\n\nprint(\"MLPRegressor \", mean_squared_error(clf.predict(X), y))\nprint(\"PyTorch \", mean_squared_error(torch_model(torch_X).detach().numpy(), y))\nprint(\"Flax \", mean_squared_error(flax_model.apply(flax_theta, X), y))\nprint(\" + meta-learning \", mean_squared_error(flax_model.apply(flax_ml_theta, X), y))\nprint(\" + binomial loss, dropout \", mean_squared_error(predict(flax_model_2, flax_theta_2, X), y))\n\nMLPRegressor 0.0011639113\nPyTorch 0.00062100013\nFlax 0.0005836923\n + meta-learning 0.0007262042\n + binomial loss, dropout 0.001051374\n\n\n\ndraw(lambda x: predict(flax_model_2, flax_theta_2, x))\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn a well-trained model, the parameters should not deviate too much from a normal distribution. We plot the parameter distributions for the last model.\n\nfor i, layer in enumerate(flax_theta_2.values()):\n fig, ax = plt.subplots(1, len(layer), figsize=(10, 4))\n ax = np.atleast_1d(ax)\n for axi, (key, val) in zip(ax, layer.items()):\n axi.hist(np.ravel(val), bins=20)\n axi.set_title(f\"layer {i+1}: {key}\")" + }, + { + "objectID": "posts/benchmark of building an array with numba.html", + "href": "posts/benchmark of building an array with numba.html", + "title": "Benchmark of building an array with numba", + "section": "", + "text": "Here I try out different ways of making arrays of tuples with Numba where the number of tuples that need to be produced is not known in advance. The fastest way is to make a list of tuples, and then convert this into a Numpy array before returning. The baseline shown in the last plot is the fastest possible way of pre-allocating a Numpy array with the right size upfront, but this cannot be used if the number of tuples is not known in advance.\n\nimport numba as nb\nimport numpy as np\nimport awkward as ak\nprint(f\"{nb.__version__=}\")\nprint(f\"{ak.__version__=}\")\n\nnb.__version__='0.53.1'\nak.__version__='1.1.2'\n\n\n\n@nb.njit\ndef make0(n):\n r = np.empty((n, 4))\n for i in range(n):\n # simulate some work\n x = np.random.rand()\n y = np.random.rand()\n z = np.random.rand()\n t = np.random.rand()\n r[i,0] = x\n r[i,1] = y\n r[i,2] = z\n r[i,3] = t\n return r\n\nmake0(3)\n\narray([[0.25473852, 0.47593451, 0.04606011, 0.55805331],\n [0.77287874, 0.52879803, 0.91944361, 0.98775869],\n [0.01221509, 0.30608935, 0.90908168, 0.07629862]])\n\n\n\n%timeit make0(1000)\n\n26.1 µs ± 1.51 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n\n\n\n@nb.njit\ndef make1(n):\n r = []\n for i in range(n):\n # simulate some work\n x = np.random.rand()\n y = np.random.rand()\n z = np.random.rand()\n t = np.random.rand()\n r.append((x, y, z, t))\n return np.asarray(r)\n\nmake1(3)\n\narray([[0.40883519, 0.14577046, 0.20275019, 0.24327066],\n [0.7325605 , 0.76214408, 0.47819782, 0.93920595],\n [0.24872141, 0.92187207, 0.6691103 , 0.5643187 ]])\n\n\n\n%timeit make1(1000)\n\n36.9 µs ± 1.55 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n\n\n\n@nb.njit\ndef make2(n):\n r = []\n for i in range(n):\n ri = []\n # simulate some work\n x = np.random.rand()\n y = np.random.rand()\n z = np.random.rand()\n t = np.random.rand()\n ri.append(x)\n ri.append(y)\n ri.append(z)\n ri.append(t) \n r.append(ri)\n return np.asarray(r)\n\nmake2(3)\n\narray([[0.13088925, 0.25692562, 0.48062596, 0.39696723],\n [0.3552985 , 0.99054454, 0.38778594, 0.94289909],\n [0.76625383, 0.01588522, 0.43885749, 0.73488083]])\n\n\n\n%timeit make2(1000)\n\n419 µs ± 18.6 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n\n\n\n@nb.njit\ndef make3(n):\n r = []\n for i in range(n):\n # simulate some work\n x = np.random.rand()\n y = np.random.rand()\n z = np.random.rand()\n t = np.random.rand()\n r.append(x)\n r.append(y)\n r.append(z)\n r.append(t)\n return np.array(r).reshape((n, 4))\n\nmake3(3)\n\narray([[0.60766162, 0.66840636, 0.2019663 , 0.92669818],\n [0.69739471, 0.33933288, 0.9356699 , 0.61572307],\n [0.43173953, 0.40320305, 0.80278753, 0.46843452]])\n\n\n\n%timeit make3(1000)\n\n49.8 µs ± 2.36 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n\n\n\n@nb.njit\ndef _make4(b, n):\n for i in range(n):\n x = np.random.rand()\n y = np.random.rand()\n z = np.random.rand()\n t = np.random.rand()\n b.begin_tuple(4)\n b.index(0).append(x)\n b.index(1).append(y)\n b.index(2).append(z)\n b.index(3).append(t)\n b.end_tuple()\n return b\n\ndef make4(n):\n return _make4(ak.ArrayBuilder(), n).snapshot()\n\nmake4(3)\n\n<Array [(0.376, 0.269, 0.149, ... 0.409, 0.38)] type='3 * (float64, float64, flo...'>\n\n\n\n%timeit make4(1000)\n\n356 µs ± 11.1 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n\n\n\n@nb.njit\ndef _make5(b, n):\n for i in range(n):\n x = np.random.rand()\n y = np.random.rand()\n z = np.random.rand()\n t = np.random.rand()\n b.begin_list()\n b.append(x)\n b.append(y)\n b.append(z)\n b.append(t)\n b.end_list()\n return b\n\ndef make5(n):\n return _make5(ak.ArrayBuilder(), n).snapshot()\n\nmake5(3)\n\n<Array [[0.216, 0.481, ... 0.247, 0.0629]] type='3 * var * float64'>\n\n\n\n%timeit make5(1000)\n\n263 µs ± 10.9 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n\n\n\nimport timeit\nimport matplotlib.pyplot as plt\n\nn = np.geomspace(1, 1000000, 20).astype(int)\n\nys = []\nfor fn in (make0, make1, make2, make3, make4, make5):\n y = []\n for ni in n:\n m, t = timeit.Timer(lambda : fn(ni)).autorange()\n y.append(t/m)\n ys.append(y)\n\n\nfor y, label in zip(ys, (\"baseline\",\n \"list+tuple+asarray\",\n \"list+list+asarray\",\n \"list+array+reshape\",\n \"ArrayBuilder+tuple\",\n \"ArrayBuilder+list\")):\n plt.plot(n, np.divide(y, 1e-6), label=label)\nplt.legend()\nplt.xlabel(\"array length\")\nplt.ylabel(\"time / μs\")\nplt.loglog();" + }, + { + "objectID": "posts/hyperon_feed_down.html", + "href": "posts/hyperon_feed_down.html", + "title": "Non-prompt fraction of KS and Lambda particles from hyperon decays", + "section": "", + "text": "A particle is prompt, if it does not have particles in its decay history with life-times larger than 30 ps. According to this definition, some Lambda particles are not prompt, since a fraction of Xi and Omega decays produce Lambda particles, and Xi and Omega particles have life-times larger than 30 ps. We compute the fraction of KS and Lambda particles that are non-prompt. For the KS, we do not find any, but for Lambdas the feed-down is quite large.\nWe select KS and Lambdas only if they decay into two pions or one pions and one protons, respectively, and if those daughter particles are in the LHCb acceptance 2 < η < 5 and \\(p > 2\\)GeV/c. Since the LHCb acceptance also implicitly places a cut on the position of the decay vertex of the KS or Lambda, we also record it.\n\nimport matplotlib.pyplot as plt\nimport boost_histogram as bh\nfrom tqdm import tqdm\nfrom particle import literals as lp, Particle\nimport joblib # only joblib works reliably in notebooks\nimport numpy as np\n\nfrom chromo.kinematics import CenterOfMass, TeV, MeV, GeV\nfrom chromo.constants import long_lived\nimport chromo.models as im\nfrom chromo.util import pdg2name\nimport numba as nb\nimport gzip\nimport pickle\nfrom pathlib import Path\n\n\n# The event kinematics objects defines the collision and its frame.\nekin = CenterOfMass(\n 2 * 6.8 * TeV,\n \"proton\",\n \"proton\"\n)\n\n\npid_axis = bh.axis.IntCategory([x.pdgid for x in (lp.K_S_0, lp.Lambda, lp.Lambda_bar)])\nparent_axis = bh.axis.IntCategory([], growth=True)\npt_axis = bh.axis.Variable([150, 500, 650, 800, 1000, 1200, 2500])\ny_axis = bh.axis.Regular(5, 2, 4.5)\nz_axis = bh.axis.Regular(100, 0, 2000)\n\nmodels = [\n im.Sibyll23d,\n im.Pythia8,\n im.EposLHC\n]\n\nlog_10_n_events = 6\n\n\n@nb.njit\ndef select(pid, parents, eta, ptot, vz):\n keep = (310, 3122, -3122)\n\n keep_daughters = (211, 2212)\n with_accepted_daughters = {}\n decay_z = {}\n for i, pidi in enumerate(pid):\n if abs(pidi) not in keep_daughters:\n continue\n k = parents[i][0] - 1\n if pid[k] not in keep:\n continue\n if 2 < eta[i] < 5 and ptot[i] > 2 * GeV:\n if k not in with_accepted_daughters:\n with_accepted_daughters[k] = 1\n decay_z[k] = vz[i]\n else:\n with_accepted_daughters[k] += 1\n\n r = np.zeros_like(pid, dtype=\"bool\")\n z = []\n for k, n in with_accepted_daughters.items():\n if n == 2:\n r[k] = True\n z.append(decay_z[k])\n return r, np.array(z)\n\n\n@nb.njit\ndef pid_of_parent(parents, pids, status):\n r = np.zeros_like(pids)\n for i, pa in enumerate(parents):\n k = i\n # we expect that history is not deeper than 100 ancestors\n for _ in range(100):\n k = parents[k][0] - 1\n if k < 0 or status[k] == 4:\n break\n pid = pids[k]\n if pid in long_lived and status[k] != 4:\n r[i] = pid\n break\n else:\n # should never happen\n assert False\n return r\n\n\n@joblib.delayed\ndef run(Model):\n m = Model(ekin, seed=1)\n\n target = [lp.pi_0, lp.K_S_0, lp.Lambda]\n target_mass = [p.mass for p in target]\n # let long-lived particles decay which are heavier than the lightest target particle\n for pid in long_lived:\n mass = Particle.from_pdgid(pid).mass\n if any(mass > m for m in target_mass):\n m.set_stable(pid, False)\n # let target particles decay\n for p in target:\n m.set_stable(p.pdgid, False)\n\n h = bh.Histogram(y_axis, pt_axis, pid_axis, parent_axis, z_axis)\n\n n_events = int(10 ** log_10_n_events)\n with np.errstate(divide=\"ignore\", invalid=\"ignore\"):\n for event in tqdm(m(n_events), total=n_events):\n mother_pid = pid_of_parent(event.parents, event.pid, event.status)\n mask, vz = select(event.pid, event.parents, event.eta, event.p_tot, event.vz)\n event = event[mask]\n mother_pid = mother_pid[mask]\n h.fill(event.y, event.pt / MeV, event.pid, mother_pid, vz)\n return m.label, h\n\nfn = Path(\"feed_down_histograms.pkl.gz\")\nif not fn.exists():\n with joblib.Parallel(n_jobs=10, batch_size=1) as pool:\n results = pool(run(m) for m in models)\n\n results = {label: h for (label, h) in results}\n\n with gzip.open(fn, \"w\") as f:\n pickle.dump(results, f)\nelse:\n with gzip.open(fn) as f:\n results = pickle.load(f)\n\n###################################################################\n# EPOS LHC K. WERNER, T. PIEROG #\n# Contact: tanguy.pierog@kit.edu #\n###################################################################\n# WARNING: This is a special retuned version !!! #\n ====================================================\n | |\n# Do not publish results without contacting the authors. #\n | S I B Y L L 2.3d |\n | |\n | HADRONIC INTERACTION MONTE CARLO |\n | BY |\n | Eun-Joo AHN, Felix RIEHN |\n | R. ENGEL, A. FEDYNITCH, R.S. FLETCHER, |\n | T.K. GAISSER, P. LIPARI, T. STANEV |\n | |\n | Publication to be cited when using this program: |\n | Eun-Joo AHN et al., Phys.Rev. D80 (2009) 094003 |\n###################################################################\n | F. RIEHN et al., hep-ph: 1912.03300 |\n | last modifications: F. Riehn (05/20/2020) |\n ====================================================\n\nread from /usr/local/lib/python3.11/site-packages/chromo/iamdata/epos/epos.iniev ...\nread from /usr/local/lib/python3.11/site-packages/chromo/iamdata/epos/epos.initl ...\nread from /usr/local/lib/python3.11/site-packages/chromo/iamdata/epos/epos.inirj.lhc ...\nread from /usr/local/lib/python3.11/site-packages/chromo/iamdata/epos/epos.inics.lhc ...\n Compute Cross-section (can take a while...)\n\n *------------------------------------------------------------------------------------* \n | | \n | *------------------------------------------------------------------------------* | \n | | | | \n | | | | \n | | PPP Y Y TTTTT H H III A Welcome to the Lund Monte Carlo! | | \n | | P P Y Y T H H I A A This is PYTHIA version 8.308 | | \n | | PPP Y T HHHHH I AAAAA Last date of change: 16 Nov 2022 | | \n | | P Y T H H I A A | | \n | | P Y T H H III A A Now is 05 Sep 2023 at 16:36:48 | | \n | | | | \n | | Program documentation and an archive of historic versions is found on: | | \n | | | | \n | | https://pythia.org/ | | \n | | | | \n | | PYTHIA is authored by a collaboration consisting of: | | \n | | | | \n | | Christian Bierlich, Nishita Desai, Leif Gellersen, Ilkka Helenius, Philip | | \n | | Ilten, Leif Lonnblad, Stephen Mrenna, Stefan Prestel, Christian Preuss, | | \n | | Torbjorn Sjostrand, Peter Skands, Marius Utheim and Rob Verheyen. | | \n | | | | \n | | The complete list of authors, including contact information and | | \n | | affiliations, can be found on https://pythia.org/. | | \n | | Problems or bugs should be reported on email at authors@pythia.org. | | \n | | | | \n | | The main program reference is C. Bierlich et al, | | \n | | 'A comprehensive guide to the physics and usage of Pythia 8.3', | | \n | | SciPost Phys. Codebases 8-r8.3 (2022) [arXiv:2203.11601 [hep-ph]] | | \n | | | | \n | | PYTHIA is released under the GNU General Public Licence version 2 or later.| | \n | | Please respect the MCnet Guidelines for Event Generator Authors and Users. | | \n | | | | \n | | Disclaimer: this program comes without any guarantees. | | \n | | Beware of errors and use common sense when interpreting results. | | \n | | | | \n | | Copyright (C) 2022 Torbjorn Sjostrand | | \n | | | | \n | | | | \n | *------------------------------------------------------------------------------* | \n | | \n *------------------------------------------------------------------------------------* \n\n SIG_AIR_INI: initializing target: (i,A) 1 0 air..\nseedj: 2 0.2378239512329647-313\n EPOS used with FUSION option\n SIG_AIR_INI: initializing target: (i,A) 2 14 nit..\n SIG_AIR_INI: initializing target: (i,A) 3 16 oxy..\n\n\n 29%|██▉ | 288315/1000000 [03:02<06:51, 1730.64it/s]\n\n\n\nmothers = set()\nfor model, h in results.items():\n for pid in h.axes[3]:\n mothers.add(pid)\n\nfor model, h in results.items():\n fig, ax = plt.subplots(1, 3, figsize=(10, 4), sharex=True, sharey=True)\n val = h.project(2, 3).values()\n plt.suptitle(model)\n for i, pid in enumerate(h.axes[2]):\n plt.sca(ax[i])\n plt.title(pdg2name(pid))\n v = np.zeros(len(mothers))\n for pid2 in mothers:\n try:\n k = h.axes[3].index(pid2)\n v[k] = val[i, k]\n except KeyError:\n pass\n y = np.arange(len(v))\n plt.barh(y, v)\n plt.yticks(y, [pdg2name(pid) if pid != 0 else \"prompt\" for pid in mothers])\n plt.xlim(0.5, None)\n plt.semilogx();\n\n\n\n\n\n\n\n\n\n\n\nfor model, h in results.items():\n val = h.project(2, 4).values()\n plt.figure()\n plt.suptitle(model)\n for i, pid in enumerate(h.axes[2]):\n plt.stairs(val[i], h.axes[4].edges, label=pdg2name(pid))\n plt.legend()\n\n\n\n\n\n\n\n\n\n\n\nfor model, h in results.items():\n fig, ax = plt.subplots(1, 3, figsize=(10, 4), \n sharex=True, sharey=True, layout=\"compressed\")\n plt.suptitle(model)\n for i, pid in enumerate(h.axes[2]):\n plt.sca(ax[i])\n plt.title(pdg2name(pid))\n val = np.sum(h.values()[:, :, i, :, :], axis=-1)\n n_total = np.sum(val, axis=-1)\n n_prompt = val[..., h.axes[3].index(0)]\n feed_down_fraction = 1 - n_prompt / n_total\n feed_down_fraction *= 100 # make percent\n plt.pcolormesh(h.axes[0].edges, h.axes[1].edges, feed_down_fraction.T)\n for i, x in enumerate(h.axes[0].centers):\n for j, y in enumerate(h.axes[1].centers):\n f = feed_down_fraction[i, j]\n plt.text(x, y, f\"{f:.0f}\", color=\"w\", ha=\"center\", va=\"center\")\n plt.clim(0, 100)\n plt.colorbar()\n\n/var/folders/tl/pv6mt7z17tz0stm1fjfg01cc0000gn/T/ipykernel_31759/777406361.py:11: RuntimeWarning: invalid value encountered in divide\n feed_down_fraction = 1 - n_prompt / n_total\n\n\n\n\n\n\n\n\n\n\n\n\nfig, ax = plt.subplots(1, 3, figsize=(10, 4), \n sharex=True, sharey=True, layout=\"compressed\")\n\nfor model, h in results.items():\n for i, pid in enumerate(h.axes[2]):\n plt.sca(ax[i])\n plt.title(pdg2name(pid))\n val = np.sum(h.values()[:, :, i, :], axis=(0, -1))\n n_total = np.sum(val, axis=-1)\n n_prompt = val[..., h.axes[3].index(0)]\n feed_down_fraction = 1 - n_prompt / n_total\n feed_down_fraction *= 100 # make percent\n plt.stairs(feed_down_fraction, h.axes[1].edges, label=model)\nplt.sca(ax[0])\nplt.ylabel(\"non-prompt fraction in percent\")\nfig.supxlabel(\"$p_\\mathrm{T}$ / MeV$c^{-1}$\")\nplt.legend();\n\n\n\n\n\nfig, ax = plt.subplots(1, 3, figsize=(10, 4), \n sharex=True, sharey=True, layout=\"compressed\")\n\nfor model, h in results.items():\n for i, pid in enumerate(h.axes[2]):\n plt.sca(ax[i])\n plt.title(pdg2name(pid))\n val = np.sum(h.values()[:, :, i, :], axis=(1, -1))\n n_total = np.sum(val, axis=-1)\n n_prompt = val[..., h.axes[3].index(0)]\n feed_down_fraction = 1 - n_prompt / n_total\n feed_down_fraction *= 100 # make percent\n plt.stairs(feed_down_fraction, h.axes[0].edges, label=model)\nplt.sca(ax[0])\nplt.ylabel(\"non-prompt fraction in percent\")\nfig.supxlabel(\"$y$\")\nplt.legend();\n\n\n\n\n\n# look feed-down fraction as function of z cut: 0 < z < 2m\n\nfig, ax = plt.subplots(1, 3, figsize=(10, 4), \n sharex=True, sharey=True, layout=\"compressed\")\n\nfor model, h in results.items():\n for i, pid in enumerate(h.axes[2]):\n plt.sca(ax[i])\n plt.title(pdg2name(pid))\n h2 = h[..., bh.rebin(10)]\n val = np.sum(h2.values()[:, :, i, :], axis=(0, 1))\n n_total = np.sum(val, axis=0)\n n_prompt = val[h2.axes[3].index(0)]\n feed_down_fraction = 1 - n_prompt / n_total\n feed_down_fraction *= 100 # make percent\n plt.stairs(feed_down_fraction, h2.axes[4].edges, label=model)\nplt.sca(ax[0])\nplt.ylabel(\"non-prompt fraction in percent\")\nfig.supxlabel(\"$y$\")\nplt.legend();\n\n/var/folders/tl/pv6mt7z17tz0stm1fjfg01cc0000gn/T/ipykernel_31759/729878470.py:14: RuntimeWarning: invalid value encountered in divide\n feed_down_fraction = 1 - n_prompt / n_total" + }, + { + "objectID": "posts/RooFit.html", + "href": "posts/RooFit.html", + "title": "RooFit with Chebyshev and Bernstein polynomials", + "section": "", + "text": "This is a little demonstration of how to fit a peak plus smooth background with either an empirical Chebyshev polynomial for the background or a Bernstein polynomial. The Bernstein polynomial is technically superior, because it allows us to ensure that the background density is always positive with a simple box constrain on its parameters. That is not possible with Chebyshev.\nThis example is based on the RooFit Tutorial. I generate a toy model which is an additive mixture of a normal distribution for a peak and a polynomial background. I draw a sample, and fit two variations of the toy model, one with either polynomial. The fitted lines look identical after the fit, as expected. The result of the maximum likelihood method is invariant to reparametrisations.\n\nimport ROOT as R\n\nWelcome to JupyROOT 6.30/02\n\n\n\nx = R.RooRealVar(\"x\", \"x\", -10, 10)\n\nmu1 = R.RooRealVar(\"mu1\", \"mu\", 0, -10, 10)\nsigma1 = R.RooRealVar(\"sigma1\", \"sigma\", 1, 0.01, 1e2)\ngauss1 = R.RooGaussian(\"gauss1\", \"Gauss\", x, mu1, sigma1)\n\np0 = R.RooRealVar(\"p0\", \"p_0\", 1, 0, 1e2)\np1 = R.RooRealVar(\"p1\", \"p_1\", 0.2, 0, 1e2)\ncheb = R.RooChebychev(\"cheb\", \"Chebyshev\", x, R.RooArgList(p0, p1))\n\nmu2 = R.RooRealVar(\"mu2\", \"mu\", 0, -10, 10)\nsigma2 = R.RooRealVar(\"sigma2\", \"sigma\", 1, 0.01, 1e2)\ngauss2 = R.RooGaussian(\"gauss2\", \"Gauss\", x, mu2, sigma2)\n\nfix = R.RooRealVar(\"one\", \"one\", 0.1)\nq0 = R.RooRealVar(\"q0\", \"q_0\", 1, 0, 1e2)\nq1 = R.RooRealVar(\"q1\", \"q_1\", 0.2, 0, 1e2)\nbern = R.RooBernstein(\"bern\", \"Bernstein\", x, R.RooArgList(fix, q0, q1))\n\nf1 = R.RooRealVar(\"f1\", \"signal fraction\", 0.3, 0, 1)\nmodel1 = R.RooAddPdf(\"sum1\", \"S+B model\", R.RooArgList(gauss1, cheb), f1)\n\ndata = model1.generate(R.RooArgSet(x), 1000)\n\nmodel1.fitTo(data)\n\nf2 = R.RooRealVar(\"f2\", \"signal fraction\", 0.3, 0, 1)\nmodel2 = R.RooAddPdf(\"sum2\", \"S+B model\", R.RooArgList(gauss2, bern), f2)\nmodel2.fitTo(data, R.RooFit.PrintLevel(-1))\n\nc = R.TCanvas()\nframe = x.frame()\ndata.plotOn(frame)\nmodel1.plotOn(frame)\nmodel2.plotOn(frame, R.RooFit.LineStyle(R.kDashed), R.RooFit.LineColor(R.kRed))\nmodel2.paramOn(frame)\nframe.Draw()\n\n[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization\n[#1] INFO:Minimization -- The following expressions will be evaluated in cache-and-track mode: (gauss1,cheb)\nMinuit2Minimizer: Minimize with max-calls 2500 convergence for edm < 1 strategy 1\nMinuit2Minimizer : Valid minimum - status = 0\nFVAL = 2737.75116242620834\nEdm = 0.000269023969144123421\nNfcn = 100\nf1 = 0.275394 +/- 0.0242084 (limited)\nmu1 = -0.0175247 +/- 0.091251 (limited)\np0 = 0.990493 +/- 0.0455302 (limited)\np1 = 0.103726 +/- 0.0587728 (limited)\nsigma1 = 1.01262 +/- 0.0914058 (limited)\n[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization\n[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization\n[#1] INFO:Minimization -- The following expressions will be evaluated in cache-and-track mode: (gauss2,bern)\n[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization\n\n\nInfo in <Minuit2>: MnSeedGenerator Computing seed using NumericalGradient calculator\nInfo in <Minuit2>: MnSeedGenerator Initial state: FCN = 2739.667437 Edm = 1.767793129 NCalls = 21\nInfo in <Minuit2>: MnSeedGenerator Initial state \n Minimum value : 2739.667437\n Edm : 1.767793129\n Internal parameters: [ -0.4115168461 0 -1.370461484 -1.481323767 -1.371459022] \n Internal gradient : [ 15.50719084 34.67327852 -68.9043889 127.4588967 -139.7445175] \n Internal covariance matrix:\n[[ 0.0035221149 0 0 0 0]\n [ 0 0.0001480147 0 0 0]\n [ 0 0 4.3539642e-05 0 0]\n [ 0 0 0 0.00023511592 0]\n [ 0 0 0 0 0.00010343293]]]\nInfo in <Minuit2>: VariableMetricBuilder Start iterating until Edm is < 0.001 with call limit = 2500\nInfo in <Minuit2>: VariableMetricBuilder 0 - FCN = 2739.667437 Edm = 1.767793129 NCalls = 21\nInfo in <Minuit2>: VariableMetricBuilder 1 - FCN = 2738.073461 Edm = 0.1061417478 NCalls = 33\nInfo in <Minuit2>: VariableMetricBuilder 2 - FCN = 2737.800519 Edm = 0.03895269349 NCalls = 46\nInfo in <Minuit2>: VariableMetricBuilder 3 - FCN = 2737.753549 Edm = 0.002339429787 NCalls = 58\nInfo in <Minuit2>: VariableMetricBuilder 4 - FCN = 2737.751162 Edm = 0.0002368589638 NCalls = 69\nInfo in <Minuit2>: VariableMetricBuilder After Hessian\nInfo in <Minuit2>: VariableMetricBuilder 5 - FCN = 2737.751162 Edm = 0.0002690239691 NCalls = 100\nInfo in <Minuit2>: Minuit2Minimizer::Hesse Using max-calls 2500\nInfo in <Minuit2>: Minuit2Minimizer::Hesse Hesse is valid - matrix is accurate\n\n\n\nmodel1.getParameters(data).Print(\"v\")\n\n 1) 0x7f7e8f6c9310 RooRealVar:: f1 = 0.275394 +/- 0.0241924 L(0 - 1) \"signal fraction\"\n 2) 0x7f7e8eff2700 RooRealVar:: mu1 = -0.0175247 +/- 0.0912425 L(-10 - 10) \"mu\"\n 3) 0x7f7e8f2b7960 RooRealVar:: p0 = 0.990493 +/- 0.0454892 L(0 - 100) \"p_0\"\n 4) 0x7f7e8f692e30 RooRealVar:: p1 = 0.103726 +/- 0.058683 L(0 - 100) \"p_1\"\n 5) 0x7f7e8eff2a90 RooRealVar:: sigma1 = 1.01262 +/- 0.0913868 L(0.01 - 100) \"sigma\"\n\n\n\nmodel2.getParameters(data).Print(\"v\")\n\n 1) 0x7f7e8fda7940 RooRealVar:: f2 = 0.275478 +/- 0.0241593 L(0 - 1) \"signal fraction\"\n 2) 0x7f7e8fd050e0 RooRealVar:: mu2 = -0.0183764 +/- 0.0911997 L(-10 - 10) \"mu\"\n 3) 0x7f7e8fd08f60 RooRealVar:: one = 0.1 C L(-INF - +INF) \"one\"\n 4) 0x7f7e8fd092f0 RooRealVar:: q0 = 0.604314 +/- 0.403307 L(0 - 100) \"q_0\"\n 5) 0x7f7e8fd15aa0 RooRealVar:: q1 = 1.84239 +/- 0.845585 L(0 - 100) \"q_1\"\n 6) 0x7f7e8fd08bd0 RooRealVar:: sigma2 = 1.01227 +/- 0.0912609 L(0.01 - 100) \"sigma\"" + }, + { + "objectID": "posts/look_elsewhere_effect.html", + "href": "posts/look_elsewhere_effect.html", + "title": "Look-elsewhere effect", + "section": "", + "text": "When searching for an excess (peak) in some phase-space which has an unknown location, one has to correct the local significance for the fact that one would have reported random peaks produced by chance in the entire phase-space if these peaks were as large as the observed one. The result is called global significance. It is possible to compute the global significance with Monte-Carlo simulation, but this is expensive.\nGross and Vitells give a recipe for a surprisingly simple approximate calculation of global significance, which in its simplest form requires no Monte-Carlo simulation. One merely has to scan over the phase-space and count the number of upcrossings of the local significance of a small threshold. The accuracy of the method can be improved by running a small number of simulations.\nThis notebook applies the results from these two papers to a toy example\nand includes small refinements discussed in this presentation\nThe toy example is a search for a resonance in a wide mass window filled with background. Let’s assume that an experiment has found a peak in this window that does not correspond to a known particle. Of interest is the global significance of this peak. The model for the peak is a voigtian, which has two unknown parameters: the location of the peak, equal to the mass of the particle, and the width of the Cauchy distribution, related to the unknown life-time of the particle. The third parameter of the voigtian, the width of the Gaussian part, is assumed to be known, since it corresponds to the resolution of the experiment.\nfrom numba_stats import truncexpon, voigt\nfrom scipy.stats import chi2, norm as scipy_norm\nimport numpy as np\nfrom iminuit import Minuit\nfrom iminuit.cost import ExtendedBinnedNLL\nfrom matplotlib import pyplot as plt\nfrom scipy.optimize import NonlinearConstraint\nfrom jacobi import propagate\nfrom resample.bootstrap import resample, variance\nfrom skimage.measure import euler_number\nfrom joblib import delayed, Parallel\nfrom pathlib import Path\nimport pickle\nimport gzip\nfrom rich.progress import Progress" + }, + { + "objectID": "posts/look_elsewhere_effect.html#prelude-technical-obstacles-and-their-solutions", + "href": "posts/look_elsewhere_effect.html#prelude-technical-obstacles-and-their-solutions", + "title": "Look-elsewhere effect", + "section": "Prelude: technical obstacles and their solutions", + "text": "Prelude: technical obstacles and their solutions\n\nNumerical issues when converting between large sigmas and small probabilities\nIn the calculation, the chi2 distribution is evaluated in its upper tail (and the standard normal, where the same issue occurs). Computing the p-value as 1 - chi2(...).cdf(z ** 2) becomes inaccurate when z is large. To fix this, one has to use the survival function implemented in scipy.stats, which calculates 1 - cdf accurately. Likewise, chi2(...).isf is used to compute the inverse of the survival function.\n\nfig, ax = plt.subplots(1, 2, figsize=(12, 4))\nplt.sca(ax[0])\nz = np.linspace(0, 100)\nplt.title(\"chi2(1)\")\nplt.plot(z, chi2(1).sf(z**2), label=\"sf\")\nplt.plot(z, 1 - chi2(1).cdf(z**2), label=\"1-cdf\", ls=\"--\")\nplt.loglog()\nplt.xlabel(\"z\")\nplt.ylabel(\"p-value\")\nplt.legend()\n\nplt.sca(ax[1])\nplt.title(\"chi2(1)\")\np = np.geomspace(1, 1e-100)\nplt.plot(p, chi2(1).isf(p) ** 0.5, label=\"isf(p)\")\nplt.plot(p, chi2(1).ppf(1 - p) ** 0.5, label=\"ppf(1-p)\", ls=\"--\")\nplt.loglog()\nplt.xlabel(\"p-value\")\nplt.ylabel(\"z\")\nplt.legend();\n\n\n\n\nThe same numerical issue also occurs when p-values are converted to sigmas using the integral of the upper tail of a standard normal distribution. The following two functions convert between p-values and z, the significance in units of standard deviation of a normal distribution. They will be used very often.\n\ndef z_to_pvalue(z):\n return scipy_norm.sf(z)\n\n\ndef pvalue_to_z(pvalue):\n return scipy_norm.isf(pvalue)\n\nThe quantity \\(q = z^2\\) is chi2 distributed with one degree of freedom. Only a positive excess is counted in this study as evidence against the background-only hypothesis, but the chi2 distribution ignores the sign of the deviation. One can nevertheless obtain the same mapping between probabilities and sigmas by taking half of the p-value for the chi2 distribution. This identity will be important later.\n\nfrom scipy.stats import norm\n\nz = 5\nq = z ** 2\n\nnorm().sf(z), 0.5 * chi2(1).sf(q)\n\n(2.8665157187919333e-07, 2.8665157187919375e-07)\n\n\n\n\nLocally negative densities\nIn the scan over the parameter space, one has to fit the model µ s + b, where µ is the signal amplitude, s is the signal shape, and b is the background density. To avoid issues with Wilks’ theorem, fit is allowed to find peaks with µ < 0, even though they are not considered as evidence against the background-only hypothesis. This, however, also opens the technical possibility that the sum of signal and background densities locally becomes negative during the numerical minimization process, which is forbidden, as densities cannot be negative. If this happens, the fit would likely fail.\nTo work around this, a standard minimization is performed first with iminuit using the MIGRAD algorithm, but if that fails, scipy minimizers are called that support additional constrains which prevent the density from becoming locally negative. iminuit wraps these to make that simple. Minimization with MIGRAD is preferred, however, because MIGRAD is faster than the scipy minimizer with constraints. Having a fast and stable fit is beneficial, since millions of fits are run in this notebook, and it is important that the fit converges every time.\n\n\nComputing the test statistic \\(q\\)\nThe computation of global significance is based on the local test statistic \\(q\\), which must be chi2 distributed. This test statistic can be computed in two ways, but the first way presented now is more accurate and is generally recommended.\nThe test statistic \\(q\\) appears in the likelihood ratio test, in which the likelihood of the background only hypothesis, where the signal amplitude is zero, is compared to the likelihood of the alternative hypothesis where the amplitude is non-zero. According to Wilks’ theorem, the quantity \\(q\\) is asymptotically chi2 distributed with one degree of freedom, \\[\nq = -2 \\ln\\frac{\\mathcal{L}(0, \\theta, \\hat \\phi')}{\\mathcal{L}(\\hat \\mu, \\theta, \\hat \\phi)}\\,,\n\\] where \\(\\mathcal{L}(\\mu, \\theta, \\phi)\\) is the likelihood for both hypotheses, \\(\\hat \\mu\\) is the best-fit value of the signal amplitude, \\(\\theta\\) are any number of nuisance parameters fixed in the scan, and \\(\\hat \\phi\\) and \\(\\hat \\phi'\\) are best fit values of the parameters of the background model for the two hypotheses. These can take different values depending on whether \\(\\mu\\) is zero or not, hence they are distinguished as \\(\\hat \\phi\\) and \\(\\hat \\phi'\\).\nNote that \\(q\\) asymptotically follows a chi2 distribution with one degree of freedom for any lengths of the parameter vectors \\(\\theta\\) and \\(\\phi\\), since Wilks’ theorem only counts the difference in the number of free parameters between both hypotheses. This difference is always 1.\nThe second asymptotically equivalent way to compute \\(q\\) is to divide the signal amplitude by the signal uncertainty estimated by the fit, and to square the result: \\[\nq = \\left(\\frac{\\hat \\mu}{\\hat \\sigma_\\mu}\\right)^2\n\\] Note that \\(z = \\hat \\mu / \\hat \\sigma_\\mu\\) is also the local significance in units of sigmas. However, numerical studies showed that this way of calculating \\(q\\) is less accurate and leads to deviations from the expected chi2 distribution in simulated experiments. The reason is that in practice the signal uncertainty is computed numerically from the second derivative of the log-likelihood, and is sometimes underestimated, leading to \\(q\\) values that are too large." + }, + { + "objectID": "posts/look_elsewhere_effect.html#construction-of-the-dataset", + "href": "posts/look_elsewhere_effect.html#construction-of-the-dataset", + "title": "Look-elsewhere effect", + "section": "Construction of the dataset", + "text": "Construction of the dataset\nThe toy distribution is constructed. It is a background-only sample from a truncated exponential distribution. This random sample is called “data”, it shall represent the actual measurement which is scanned over for a peak.\n\nlambd_true = 2000\nbackground_true = 10000\n\ndata = truncexpon.rvs(1000, 4000, 0, lambd_true, background_true, 1)\nplt.hist(data, bins=100)\nplt.semilogy();" + }, + { + "objectID": "posts/look_elsewhere_effect.html#d-case", + "href": "posts/look_elsewhere_effect.html#d-case", + "title": "Look-elsewhere effect", + "section": "1D case", + "text": "1D case\nFor the 1D case, I consider only the mass of the particle as unknown, but the width of the Cauchy distribution as known. I scan over the unknown mass and compute the local significance at each location. This requires fitting a signal + background model. I use iminuit and numba_stats, which works very well and is very fast, but other fitting frameworks can do the same job.\nIn case of iminuit, I use Minuit.fval to compute \\(q\\) which returns the value \\(-2 \\ln\\mathcal{L}\\) for the built-in cost function ExtendedBinnedNLL.\n\ndef model(xe, signal, mass, width, background, lambd):\n b = background * truncexpon.cdf(xe, 1000, 4000, 0, lambd)\n\n sigma_resolution = 50 # some fixed value\n\n # poor man's integration of voigt.pdf\n dx = np.diff(xe)\n ds = signal * voigt.pdf(xe[:-1] + 0.5 * dx, width, mass, sigma_resolution) * dx\n s = np.append(0, np.cumsum(ds))\n\n return s + b\n\n\ndef fit_background(mass):\n n, xe = np.histogram(mass, bins=100)\n c = ExtendedBinnedNLL(n, xe, model)\n minuit = Minuit(\n c, signal=0, mass=0, width=1, background=background_true, lambd=lambd_true\n )\n minuit.fixed[\"signal\", \"width\", \"mass\"] = True\n minuit.limits[\"lambd\", \"background\"] = (0, None)\n minuit.migrad()\n return minuit.fval, minuit\n\n\ndef fit(mass, m, w, chi2_background):\n n, xe = np.histogram(mass, bins=100)\n c = ExtendedBinnedNLL(n, xe, model)\n minuit = Minuit(\n c, signal=0, mass=m, width=w, background=background_true, lambd=lambd_true\n )\n minuit.fixed[\"width\", \"mass\"] = True\n minuit.limits[\"lambd\", \"background\"] = (0, None)\n minuit.migrad()\n # The pdf must not be negative. If MIGRAD fails, then usually because of that.\n # If that happens, I try again with with a constrained minimization,\n # see explaination above.\n if not minuit.valid:\n minuit.reset()\n minuit.scipy(\n constraints=[\n NonlinearConstraint(lambda *args: np.diff(model(xe, *args)), 0, np.inf)\n ]\n )\n if not minuit.valid:\n return np.nan, minuit\n\n # In theory, q is always positive, but it can take a tiny negative value in practice\n # due to numerical errors. I guard against that with the max function.\n q = max(chi2_background - minuit.fval, 0)\n\n # I want to return z instead of q, which is not necessary, but instructive. So I\n # take the square root here. The square-root of q is always positive, but I also\n # want to know the sign of the fitted amplitude for further analysis, so I copy it\n # over from the fitted signal.\n z = np.copysign(q ** 0.5, minuit.values[\"signal\"])\n\n # This alternative calculation of z is more straight-forward but less accurate:\n # z = (minuit.values[\"signal\"] / minuit.errors[\"signal\"])\n return z, minuit\n\n\ndef z_scan(mass, m_grid, rw_grid):\n chi2_background = fit_background(mass)[0]\n X, Y = np.meshgrid(m_grid, rw_grid)\n fn = np.vectorize(lambda m, rw: fit(mass, m, m * rw, chi2_background)[0])\n Z = fn(X, Y).T\n return np.squeeze(Z)\n\n\nm_grid = np.linspace(1200, 3800, 200)\n# relative width of Cauchy part is fixed in 1D case, will be scanned over in 2D case\nrw_grid = 0.05\n\nz = z_scan(data, m_grid, rw_grid)\n\nLet’s plot the most significant peak candidate.\n\nimax = np.argmax(z**2)\nzmax, minuit = fit(\n data, m_grid[imax], rw_grid * m_grid[imax], fit_background(data)[0]\n)\nminuit.visualize()\nplt.title(f\"z = {zmax:.2f}\")\nplt.xlabel(\"mass\");\n\n\n\n\nIt has a local significance of \\(z_\\mathrm{local} = 2.45\\).\nNext, I plot the \\(z\\) values and the p-values computed from \\(q = z^2\\) and a chi2 distribution with one degree of freedom. From this I need to derive an excursion set, how many connected stretches (“upcrossings”) there are for which these p-values fall below threshold values computed from the one-sided standard normal distribution (these thresholds are given by the function z_to_pvalue). I emphasize that the p-value is computed from \\(q\\) using the chi2(1) distribution, but for the thresholds I use z_to_pvalue which gives half of chi2(1).sf(z ** 2).\nThis accounts for the fact that only positive \\(z\\) provide evidence against the background-only hypothesis. It also allows one to count the upcrossings at the lowest possible level of \\(0\\sigma\\), which is mapped to the threshold P=0.5. These upcrossings occur very frequently and thus have a small relative counting error. This nice trick is not documented in the papers, but described in a presentation by Eilam Gross given at the ATLAS Stat Forum, 3/2010.\n\n# p-value from chi2(1) for an excess in q=z^2\npvalue_q = chi2(1).sf(z ** 2)\n\nfig, ax = plt.subplots(1, 2, figsize=(12, 4))\nplt.sca(ax[0])\nplt.axhline(0, ls=\"--\", color=\"0.5\")\nplt.xlabel(\"mass / MeV\")\nplt.ylabel(r\"$z_\\mathrm{local}$\")\nplt.plot(m_grid, z)\nplt.sca(ax[1])\nplt.plot(m_grid, pvalue_q)\nplt.semilogy()\nplt.ylim(None, 1)\nfor sigma in range(5):\n # for thresholds, I use z_to_pvalue, see text\n p = z_to_pvalue(sigma)\n if p > np.min(pvalue_q):\n plt.axhline(p, ls=\"--\", color=\"k\")\n plt.text(\n 3900, p, f\"${sigma}\\sigma$\", fontsize=\"x-large\", va=\"bottom\", ha=\"right\"\n )\nplt.ylabel(\"p-value from chi2(1)\")\nplt.xlabel(\"mass / MeV\")\n\nz_local = z[imax]\nprint(f\"z_local = {z_local:.2f} P_local = {z_to_pvalue(z_local):.3f}\")\n\nz_local = 2.45 P_local = 0.007\n\n\n\n\n\nThe following functions compute the number of upcrossings, the global significance, and the global p-value from the local significance values.\n\ndef compute_p_global(z, z_threshold, n_upcrossings):\n u = z**2\n u_threshold = z_threshold**2\n phalf = 0.5 * chi2(1).sf(u) # equal to p_local = z_to_pvalue(z_local)\n return phalf + n_upcrossings * np.exp(0.5 * (u_threshold - u))\n\n\ndef compute_z_global(z, z_threshold, n_upcrossings):\n return pvalue_to_z(compute_p_global(z, z_threshold, n_upcrossings))\n\n\ndef compute_number_of_upcrossings(z, z_threshold):\n # I use euler_number from scikit-image to compute the number of upcrossings.\n # It is designed for 2D data, but also works on 1D data.\n excursion_set = chi2(1).sf(z ** 2) < z_to_pvalue(z_threshold)\n excursion_set = np.atleast_2d(excursion_set)\n return euler_number(excursion_set)\n\n\nz_threshold = 0 # chosen threshold for upcrossings in units of sigma\nn_upcrossings = compute_number_of_upcrossings(z, z_threshold)\nprint(f\"upcrossings = {n_upcrossings}\")\n\n# propagate uncertainty of n_upcrossing, which is a Poisson distributed number\nz_global, z_global_cov = propagate(\n lambda n: compute_z_global(z_local, z_threshold, n), n_upcrossings, n_upcrossings\n)\n\nprint(f\"z_global = {z_global:.2f} ± {z_global_cov ** 0.5:.2f}\")\n\nupcrossings = 5\nz_global = 0.66 ± 0.35\n\n\nThe uncertainty of the computed global significance is statistical: the expected number of upcrossings was replaced with an estimate based on the observed number of upcrossings in only one sample.\nThis error can be reduced to any desired value by running more simulations and computing the average of the number of upcrossings from these simulations. With 100 simulations, the error can be reduced by a factor of 10 to 0.02, which is already almost negligible. If a larger local significance is extrapolated, the error of the global significance measured in units of sigma becomes smaller.\nI now confirm the calculation with a toy simulation, in which I repeatedly simulate independent experiments, do the local scan, and record the largest \\(q\\) value found. This produces a sample of \\(q\\) values.\n\n# I cache this calculation, since it takes a while\ncache_file = Path(\"q_samples.pkl.gz\")\n\nif cache_file.exists():\n with gzip.open(cache_file) as f:\n q = pickle.load(f)\nelse:\n q = []\n\n seeds = 10000\n\n @delayed\n def run(iseed):\n mass = truncexpon.rvs(1000, 4000, 0, lambd_true, background_true, iseed)\n z = z_scan(mass, m_grid, rw_grid)\n return np.max(z**2)\n\n with Progress() as progress:\n q = Parallel(n_jobs=8)(run(iseed) for iseed in progress.track(range(seeds)))\n q = np.array(q)\n\n with gzip.open(cache_file, \"w\") as f:\n pickle.dump(q, f)\n\n# check that all fits converged properly\nnp.mean(np.isnan(q))\n\n0.0\n\n\nHere is the distribution of the test statistic \\(q\\) over many independent experiments. For large \\(q\\) values it becomes an exponential distribution, as predicted by the Gross & Vitells formula.\n\nplt.hist(q, bins=100)\nplt.semilogy()\nplt.xlabel(\"q\");\n\n\n\n\nThe global p-value is given by the fraction of \\(q\\) values in simulated experiments which exceed the value found in the original data sample. This probability is converted into sigmas with the usual conversion formula. The uncertainty is readily obtained with the bootstrap.\nThe simulation result is in agreement with the Gross & Vitells calculation.\n\np_global_sim = np.mean(q > z_local**2)\nz_global_sim = pvalue_to_z(p_global_sim)\n\nz_global_sim_cov = variance(lambda q: pvalue_to_z(np.mean(q > z_local**2)), q)\nprint(f\"z_global_sim = {z_global_sim:.2f} +/- {z_global_sim_cov ** 0.5:.2f}\")\n\nz_global_sim = 0.73 +/- 0.01\n\n\nOut of curiosity, let’s plot the experiment with the largest \\(q\\). It turns out to be a negative deviation.\n\niseed = np.argmax(q)\nmass = truncexpon.rvs(1000, 4000, 0, lambd_true, background_true, iseed)\nz_m = []\nfor m in m_grid:\n z_m.append(fit(mass, m, rw_grid, fit_background(mass)[0]))\nplt.plot(m_grid, [x[0] for x in z_m])\nplt.ylabel(\"$z_\\mathrm{local}$\")\nplt.xlabel(\"mass / MeV\");\n\n\n\n\nThis is the corresponding fit of the maximum deviation:\n\nz_m[np.argmax([x[0] ** 2 for x in z_m])][1]\n\n\n\n\nMigrad\n\n\nFCN = 93.4 (χ²/ndof = 1.0)\nNfcn = 51\n\n\nEDM = 1.26e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nsignal\n-114\n26\n\n\n\n\n\n\n\n1\nmass\n1.919e3\n0.019e3\n\n\n\n\nyes\n\n\n2\nwidth\n50.0e-3\n0.5e-3\n\n\n\n\nyes\n\n\n3\nbackground\n10.12e3\n0.10e3\n\n\n0\n\n\n\n\n4\nlambd\n1.97e3\n0.05e3\n\n\n0\n\n\n\n\n\n\n\n\n\nsignal\nmass\nwidth\nbackground\nlambd\n\n\nsignal\n655\n0\n0\n-0.8e3 (-0.288)\n0.1e3 (0.078)\n\n\nmass\n0\n0\n0\n0\n0\n\n\nwidth\n0\n0\n0\n0\n0\n\n\nbackground\n-0.8e3 (-0.288)\n0\n0\n1.09e+04\n-0.1e3 (-0.019)\n\n\nlambd\n0.1e3 (0.078)\n0\n0\n-0.1e3 (-0.019)\n2.22e+03\n\n\n\n\n\n\n \n \n \n \n 2023-10-17T17:52:30.320907\n image/svg+xml\n \n \n Matplotlib v3.7.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nTo verify the result of the Gross & Vitells formula more broadly, I compare the global p-values from the simulation and the computed ones from the formula for various values of \\(q\\). I use the bootstrap method to estimate the uncertainty of the simulation result, and numerical error propagation to compute the uncertainty of the Gross & Vitells formula.\nBoth results are in very good agreement above \\(q > 7\\), corresponding to \\(z \\approx 2.6\\). The beauty of the Gross & Vitells formula is that it performs the best if \\(z_\\mathrm{local}\\) is large, which is exactly the case where the classic simulation strategy becomes infeasible. The error on the Gross & Vitells formula can be easily reduced by a factor of 10, if the number of excursions is computed from an average of 100 background-only simulations (I don’t do that here).\n\nq_grid = np.linspace(0, 25, 1000)\n\np = [np.mean(q > qi) for qi in q_grid]\nplt.plot(q_grid, p, label=\"simulation\")\n\np_err = variance(lambda q: [np.mean(q > qi) for qi in q_grid], q) ** 0.5\nplt.fill_between(q_grid, p - p_err, p + p_err, fc=\"C0\", alpha=0.5)\n\np2, p2cov = propagate(\n lambda n: compute_p_global(q_grid**0.5, z_threshold, n),\n n_upcrossings, n_upcrossings, diagonal=True\n)\np2err = p2cov ** 0.5\nplt.plot(q_grid, p2, label=\"Gross & Vitells\")\nplt.fill_between(q_grid, p2 - p2err, p2 + p2err, fc=\"C1\", alpha=0.5)\nplt.semilogy()\nplt.ylabel(\"global p-value\")\nplt.xlabel(\"q = $z_\\\\mathrm{local}^2$\")\nplt.legend();" + }, + { + "objectID": "posts/look_elsewhere_effect.html#d-case-1", + "href": "posts/look_elsewhere_effect.html#d-case-1", + "title": "Look-elsewhere effect", + "section": "2D case", + "text": "2D case\nIn HEP, one often has to scan over more than just the mass, the natural width of the peak is also unknown unless the life-time of the particle can be measured as well.\nIn this toy example, I scan over the relative width from 0 % to 10 % of the mass of the particle. One could equally well implement a scan where the width is independent of the mass.\nI already wrote the scan function for the 1D case so that it also works for the 2D case. It is reused unmodified here.\n\nm_grid = np.linspace(1200, 3800, 100)\nrw_grid = np.linspace(0, 0.1, 20)\n\nz = z_scan(data, m_grid, rw_grid)\nz_local = np.max(z)\nprint(f\"z_local = {z_local:.2f}\")\n\nz_local = 3.14\n\n\nA local excess of \\(3.1\\sigma\\) is found, higher than the previous local excess found when the relative width was fixed. This is expected, it is more likely to find a large local deviation if the scanned parameter space is larger.\nI visualize the result of the scan.\n\nfrom matplotlib.colors import BoundaryNorm\n\ncmap = plt.colormaps[\"bwr\"]\nnorm = BoundaryNorm(np.linspace(-5, 5, 11), ncolors=cmap.N, clip=True)\nplt.pcolormesh(m_grid, rw_grid, z.T, norm=norm, cmap=cmap)\nplt.colorbar().set_label(\"z\")\nplt.title(f\"z_local = {z_local:.2f}\")\nplt.xlabel(\"mass\")\nplt.ylabel(\"relative width\");\n\n\n\n\nTo apply the Gross & Vitells formula for the 2D case, excursion sets need to be computed again. An excursion set is obtained as before: the p-value is computed from \\(q = z^2\\) using the upper tail of the chi2(1) distribution, but the threshold probabilities are computed using the upper tail of a standard normal.\nHowever, in a 2D plane, the excursions are counted differently. They are given by the number of locally connected excursions minus the number of holes. This difference is called the Euler characteristic.\nGross & Vitells describe a general algorithm to compute the Euler characteristic for any number of dimensions, which would take time to implement and test. Fortunately, I found an algorithm for the 2D case in the scikit-image Python library, which is a great time-saver.\nShown below are the excursion sets for \\(0\\sigma\\) and \\(1\\sigma\\) and the computed Euler characteristic for these two binary images.\n\nz_threshold = [0, 1]\neuler_characteristic = []\nfor zt in z_threshold:\n excursions = chi2(1).sf(z**2) < z_to_pvalue(zt)\n ec = euler_number(excursions) # euler_number is a misnomer by scikit-image\n euler_characteristic.append(ec)\n\n plt.figure()\n plt.pcolormesh(m_grid, rw_grid, excursions.T, cmap=cmap)\n plt.title(f\"u = {zt} Euler characteristic = {ec}\")\n\n\n\n\n\n\n\nWith these numbers I can again compute the extrapolation from local to global significance. I need to first solve a system of equations, for which one row is given by \\[\nN[\\phi_u] = 1/2 P(\\chi^2 > u) + (\\mathcal{N}_1 + \\mathcal{N}_2 \\sqrt{u}) e^{-u/2},\n\\] where \\(N[\\phi_u]\\) is the expected Euler characteristic for excursions above \\(u = z_\\mathrm{threshold}^2\\), \\(P(\\chi^2 > u)\\) is the upper tail integral of a chi2(1) distribution, and \\(\\mathcal{N}_1\\), \\(\\mathcal{N}_2\\) are unknown parameters for which I solve the system, using the estimated values \\(N[\\phi_u]\\) for two chosen thresholds \\(u_0 = 0\\) and \\(u_1 = 1\\).\nThe system is linear in \\(\\mathcal{N}_1\\) and \\(\\mathcal{N}_2\\) and the solution readily obtained with standard algorithms. Once \\(\\mathcal{N}_1\\) and \\(\\mathcal{N}_2\\) are given, I can use the same formula to compute the global p-value (equal to \\(N[\\phi_u]\\) for large \\(u\\)) from the local p-value (equal to \\(1/2 P(\\chi^2 > u)\\)).\n\ndef compute_p_global(z_local, z_threshold, euler_characteristic):\n z_threshold = np.array(z_threshold)\n euler_characteristic = np.array(euler_characteristic)\n u = z_threshold ** 2\n p_chi2 = chi2(1).sf(u)\n\n # Solve system of linear equations\n # N[𝜙ᵤ] = 1/2 P(𝜒² > u) + (𝒩₁ + 𝒩₂ √𝑢) exp(−𝑢/2)\n # to get parameters 𝒩₁ and 𝒩₂.\n b = euler_characteristic - 0.5 * p_chi2\n f = np.exp(-u / 2)\n A = np.transpose((f, f * np.sqrt(u)))\n n1, n2 = np.linalg.solve(A, b)\n\n # I now apply\n # N[𝜙ᵤ] = 1/2 P(𝜒² > u) + (𝒩₁ + 𝒩₂ √𝑢) exp(−𝑢/2)\n # to the observed value of u = z_local^2.\n u = z_local**2\n f = np.exp(-u / 2)\n p_chi2 = chi2(1).sf(u)\n return 0.5 * p_chi2 + (n1 + n2 * np.sqrt(u)) * f\n\n\ndef compute_z_global(z_local, z_threshold, euler_characteristic):\n p_global = compute_p_global(z_local, z_threshold, euler_characteristic)\n return pvalue_to_z(p_global)\n\n\n# Error propagation assumes that both Euler characteristics are 100% correlated.\n# This is an approximation, but they must be strongly positively correlated.\nz_global, z_global_cov = propagate(lambda n : compute_z_global(z_local, z_threshold, n),\n euler_characteristic, \n [euler_characteristic, euler_characteristic])\n\nprint(f\"z_local = {z_local:.2f}\")\nprint(f\"z_global = {z_global:.2f} +/- {z_global_cov ** 0.5:.2f}\")\n\nz_local = 3.14\nz_global = 1.23 +/- 0.26\n\n\nI check this result again using simulations of independent experiments as in the previous case. The simulation takes a long time now since each iteration is a scan over a 2D plane.\n\n# I cache this calculation, since it takes a while\ncache_file = Path(\"q_samples_2d.pkl.gz\")\n\nif cache_file.exists():\n with gzip.open(cache_file) as f:\n q = pickle.load(f)\nelse:\n q = []\n\n seeds = 10000\n\n @delayed\n def run(iseed):\n mass = truncexpon.rvs(1000, 4000, 0, lambd_true, background_true, iseed)\n z = z_scan(mass, m_grid, rw_grid)\n return np.max(z**2)\n\n with Progress() as progress:\n q = Parallel(n_jobs=8)(run(iseed) for iseed in progress.track(range(seeds)))\n q = np.array(q)\n\n with gzip.open(cache_file, \"w\") as f:\n pickle.dump(q, f)\n\n# check that all fits converged properly\nnp.mean(np.isnan(q))\n\n0.0\n\n\nAs expected, the distribution of \\(q\\) values approaches an exponential distribution for large \\(q\\).\n\nplt.hist(q, bins=100)\nplt.semilogy()\nplt.xlabel(\"q\");\n\n\n\n\nThe simulation result is once again in excellent agreement.\n\np_global_sim = np.mean(q > z_local**2)\nz_global_sim = pvalue_to_z(p_global_sim)\n\nz_global_sim_cov = variance(lambda q: pvalue_to_z(np.mean(q > z_local**2)), q)\nprint(f\"z_global_sim = {z_global_sim:.2f} +/- {z_global_sim_cov ** 0.5:.2f}\")\n\nz_global_sim = 1.25 +/- 0.02\n\n\nI plot again the scan with the most extreme local significance. It is a positive excess this time.\n\niseed = np.argmax(q)\nmass = truncexpon.rvs(1000, 4000, 0, lambd_true, background_true, iseed)\nz_m = []\nfor m in m_grid:\n z_m.append(fit(mass, m, rw_grid[0], fit_background(mass)[0]))\nplt.plot(m_grid, [x[0] for x in z_m])\nplt.ylabel(\"$z_\\mathrm{local}$\")\nplt.xlabel(\"mass / MeV\");\n\n\n\n\n\nz_m[np.argmax([x[0] ** 2 for x in z_m])][1]\n\n\n\n\nMigrad\n\n\nFCN = 95.07 (χ²/ndof = 1.0)\nNfcn = 52\n\n\nEDM = 3.11e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nsignal\n180\n40\n\n\n\n\n\n\n\n1\nmass\n1.410e3\n0.014e3\n\n\n\n\nyes\n\n\n2\nwidth\n0.0\n0.1\n\n\n\n\nyes\n\n\n3\nbackground\n9.82e3\n0.10e3\n\n\n0\n\n\n\n\n4\nlambd\n2.05e3\n0.05e3\n\n\n0\n\n\n\n\n\n\n\n\n\nsignal\nmass\nwidth\nbackground\nlambd\n\n\nsignal\n1.31e+03\n0\n0\n-1.1e3 (-0.298)\n0.5e3 (0.270)\n\n\nmass\n0\n0\n0\n0\n0\n\n\nwidth\n0\n0\n0\n0\n0\n\n\nbackground\n-1.1e3 (-0.298)\n0\n0\n1.1e+04\n-0.5e3 (-0.093)\n\n\nlambd\n0.5e3 (0.270)\n0\n0\n-0.5e3 (-0.093)\n2.9e+03\n\n\n\n\n\n\n \n \n \n \n 2023-10-17T17:52:46.534849\n image/svg+xml\n \n \n Matplotlib v3.7.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nThis indeed looks very much like a local resonance, but it is a random coincidence.\nAs before, I compute the global p-value using simulation and with the Gross & Vitells formula for various q-values to get a better picture of their agreement. The agreement is again very good above \\(q > 7\\), which corresponds to about \\(2.6\\sigma\\).\n\nq_grid = np.linspace(0, 25, 1000)\n\np = [np.mean(q > qi) for qi in q_grid]\nplt.plot(q_grid, p, label=\"simulation\")\n\np_err = variance(lambda q: [np.mean(q > qi) for qi in q_grid], q) ** 0.5\nplt.fill_between(q_grid, p - p_err, p + p_err, fc=\"C0\", alpha=0.5)\n\n# We assume 100% correlation for both values of the Euler characteristic.\np2, p2cov = propagate(\n lambda n: compute_p_global(q_grid**0.5, z_threshold, n),\n euler_characteristic,\n [euler_characteristic, euler_characteristic],\n)\np2err = np.diag(p2cov) ** 0.5\nplt.plot(q_grid, p2, label=\"Gross & Vitells\")\nplt.fill_between(q_grid, p2 - p2err, p2 + p2err, fc=\"C1\", alpha=0.5)\nplt.semilogy()\nplt.ylabel(\"global p-value\")\nplt.xlabel(\"q = $z_\\\\mathrm{local}^2$\")\nplt.legend();" + }, + { + "objectID": "posts/look_elsewhere_effect.html#conclusion", + "href": "posts/look_elsewhere_effect.html#conclusion", + "title": "Look-elsewhere effect", + "section": "Conclusion", + "text": "Conclusion\nI explored the application of the Gross & Vitells formulas for signal models with one or two nuisance parameters (apart from the amplitude), which extrapolate a local significance for a signal found in 1D or 2D scans to a global significance.\nUsing iminuit and numba-stats, I implemented a fast and robust fit for a toy example, which consists of a model that mixes exponential background with a voigtian signal. The fits run fast and converge successfully every time, even though the total amount of fits run in this notebook is in the millions.\nThe Gross & Vitells formulas do not need my verification (they hold generally), but I needed to verified my implementation of them. As source material I used the respective papers and the slides listed at the beginning of this notebook. The verification is based on a comparison to the result obtained with classical simulation. I found excellent agreement between both types of calculation.\nTo propagate statistical errors in each calculation, I used the jacobi and resample libraries, which implement numerical error propagation and the bootstrap method, respectively.\nThe source material also describes how the formulas can be further generalized to arbitrary dimensions. It should be easy to generalize my implementation for the 2D case to the 3D case, since the scikit-image algorithm to compute the Euler characteristic is also applicable to 3D sets, but I do not attempt this here. A generalization to even higher dimensions would require an implementation of the algorithm to compute the Euler characteristic in N dimensions." + }, + { + "objectID": "posts/Simple parallelization in Jupyter Notebooks.html", + "href": "posts/Simple parallelization in Jupyter Notebooks.html", + "title": "Simple parallelization in Jupyter Notebooks", + "section": "", + "text": "The usual modules concurrent.futures and multiprocessing do not work correctly in notebooks on all platforms (notably on OSX there are issues). What does work is joblib, which also offers a simple way to run embarrassingly parallel jobs.\n\nfrom joblib import Parallel, delayed\nimport numpy as np\n\n\n# silly function that simulates numerical workload\ndef run(n):\n r = 0\n rng = np.random.default_rng(1)\n for itry in range(1000):\n x = rng.normal(size=n)\n r += np.sum(x)\n return r\n\nn = np.full((4,), 100000)\ny = Parallel(8)(delayed(run)(ni) for ni in n)\n\n\nimport time\nimport matplotlib.pyplot as plt\n\n\nnjob = np.arange(1, 9)\nn = (10 ** np.arange(2, 6, 0.5)).astype(int)\ndata = np.empty((len(njob), len(n)))\nfor i, njobi in enumerate(njob):\n for j, nj in enumerate(n):\n t = time.perf_counter()\n n_per_job = nj // njobi\n Parallel(njobi)(delayed(run)(ni) for ni in np.full(njobi, n_per_job))\n t = time.perf_counter() - t\n data[i, j] = t\n print(njobi, nj, t)\n\n1 100\n1 316\n1 1000\n1 3162\n1 10000\n1 31622\n1 100000\n1 316227\n2 100\n2 316\n2 1000\n2 3162\n2 10000\n2 31622\n2 100000\n2 316227\n3 100\n3 316\n3 1000\n3 3162\n3 10000\n3 31622\n3 100000\n3 316227\n4 100\n4 316\n4 1000\n4 3162\n4 10000\n4 31622\n4 100000\n4 316227\n5 100\n5 316\n5 1000\n5 3162\n5 10000\n5 31622\n5 100000\n5 316227\n6 100\n6 316\n6 1000\n6 3162\n6 10000\n6 31622\n6 100000\n6 316227\n7 100\n7 316\n7 1000\n7 3162\n7 10000\n7 31622\n7 100000\n7 316227\n8 100\n8 316\n8 1000\n8 3162\n8 10000\n8 31622\n8 100000\n8 316227\n\n\n\nfor i, njobi in enumerate(njob):\n plt.plot(n, data[0] / data[i], label=f\"{njobi}\")\nplt.loglog()\nplt.legend();" + }, + { + "objectID": "posts/exceptions.html", + "href": "posts/exceptions.html", + "title": "C++ exceptions in high-performance code", + "section": "", + "text": "Here is a collection of advice on using exceptions in high-performance libraries. There has been a lot of discussion in the Boost community about exceptions lately, since some people want to improve error reporting in C++, which led to the development of Boost.Outcome and similar libraries. Here we deal with classic C++ exceptions.\nFurther reading:\n\nhttps://www.boost.org/community/error_handling.html\nScott Meyers, Effective Modern C++, O’Reilly\nhttps://isocpp.github.io/CppCoreGuidelines/CppCoreGuidelines\nhttps://www.boost.org/community/exception_safety.html\nhttps://www.boost.org/doc/libs/1_72_0/libs/exception/doc/boost-exception.html\nhttp://open-std.org/JTC1/SC22/WG21/docs/papers/2018/p0709r0.pdf\nhttp://www.open-std.org/jtc1/sc22/wg21/docs/papers/2018/p1095r0.pdf\nhttps://stackoverflow.com/questions/13835817/are-exceptions-in-c-really-slow\nhttps://stackoverflow.com/questions/26079903/noexcept-stack-unwinding-and-performance\nhttps://en.cppreference.com/w/cpp/language/noexcept_spec\nhttps://gcc.gnu.org/onlinedocs/libstdc++/manual/using_exceptions.html\n\nI thank users on the cpplang boost channel for feedback and additional links. Further contributions and corrections came from Github users degski, max0x7ba, and expnkx.\n\n\nExceptions are the official language feature of C++ for reporting errors. There are two cases in C++ where other error reporting systems based on return values do not work: in constructors (which do not return) and in operators (which must use the return value for something else). There are workarounds for these cases, but they lead to less idiomatic C++. Of course, there are also reasons to avoid exceptions. Herb Sutter gives a comprehensive overview of the various pros and cons of exceptions and their alternatives.\n\n\n\n\nRun time:\n\nExceptions have zero run-time cost if they do not trigger, but reduce optimisation opportunities (read on for details)\nExceptions that trigger have a large cost (thousands of CPU cycles)\n\nCompile time: small cost\nCode size increases by about +7 % to +40 %, depending on compiler and operating system\n\nExceptions in C++ were designed to have zero run time cost when they do not trigger. Zero cost is even less cost than an if-else-branch. In theory, this makes C++ exceptions more performant than the C style alternative of returning an error code on the happy path (but read on). C++ exceptions cost a lot of cycles when they trigger. Therefore, exceptions should never be used for normal control flow, where both alternatives happen regularly. Exceptions are for exceptional events only, faults that occur rarely during the run of a program. A good example is wrong user input and unexpected I/O errors.\nSo exceptions seem pretty great, but Google turns off exceptions in their builds (-fno-exceptions in gcc and clang) – why? Apart from stylistic issues, exceptions reduce opportunities for the optimiser to make the code faster. The optimiser is allowed to transform code if it can prove that this does not change the visible outcome. A potentially throwing expression can prevent fusing instructions before and after the expression. When the exception triggers, the instructions after the exception are not executed. If this has observable side effects, the optimiser cannot fuse the instructions to increase performance.\nAll in all, this has a noticeable effect even in carefully written libraries that use exceptions. In Boost.Histogram, benchmarks run 10-15 % faster when I deactivate exceptions with -fno-exceptions, even though no exceptions are thrown in these benchmarks.\n\n\n\nThe noexcept specifier marks a function or method as not throwing any exception ever. This is great for the optimiser.\nThe compiler trusts this declaration. You won’t get a compile-time error if code that was declared noexcept throws an exception anyway. If that happens at run-time, the program simply terminates. Compilers may emit a warning about this, but at the time of this writing, they only do this in obvious cases1. The developer must make sure to not lie to the compiler when declaring something as noexcept.\nThere are legitimate reasons to declare a function noexcept which has throwing internal code (which may be third-party code). If all conditions can be anticipated and explicitly handled under which the internal code could throw, the surrounding code can be declared noexcept since no throw will actually occur. Similarly, a function or method that throws and catches all exceptions internally, so that none leave the function, can be declared noexcept. This should be a performance gain in theory, but in practice it depends on the compiler support for noexcept.\nAlso in theory, it should not be necessary to mark every non-throwing function or method as noexcept, the compiler is supposed to detect simple cases. At the time of this writing, however, not even the most recent compiler versions (clang-10, gcc-9.3) detect the noexcept property even for the most trivial functions.\nNote 1: At the time of this writing, neither gcc or clang warn if the throw is nested in another function, even if the implementation of that function is visible to the compiler.\n\n\n\n\n\nIt may be tempting use an assert instead of an exception, because the optimiser is not troubled by an assert, but don’t do that. An assert is usually only checked when the code is compiled in debug mode2, while exceptions work also in production code. Therefore, an assert cannot be used in place of an exception, in particular in code that checks or validates user input.\nRule-of-thumb for using either assert or throwing an exception: - In private interfaces and private implementation code, where you have full control over the input, use assert to check the consistency of your program logic - In user-facing interfaces, use exceptions\nIn other words, users should never see a failing assert. Anything that can go wrong due to external circumstances outside of the control of the program should trigger an exception. An assert is an executable part of the interface documentation: it reminds a developer that this code expects certain inputs and cannot run correctly when these are violated.\nExample: Let us say some code requires some user-defined number to be greater than 10. The user-facing layer should check whether the number is greater than 10 and otherwise throw an exception. The deeper implementation layers should assert on the same condition. This is not redundant, since the assert documents what the implementation layer expects. If the program is not altered, the assert will never be violated, but it is there in case someone refactors the code and forgets to protect the implementation layer from invalid external input.\nNote 2: To be more precise, the assert macro from <cassert> expands to nothing when -DNDEBUG is set, which is usually set in a release build (for example, this is the cmake default).\n\n\n\nIf the implementation allows it at all, destructors, move constructors, and move-assignment operators should not throw exceptions. If they are not declared noexcept (more details on noexcept are given below), the compiler will try to figure this out.\nThrowing destructors are a really bad idea, because destructors are called during the stack-unwinding when another exception was thrown. If the exception is allowed to leave the destructor in this situation, the program will terminate immediately3.\nContainers like std::vector need their elements to have non-throwing move constructors and move-assignment to make efficient use of them. Containers typically want to guarantee that they are in a valid state at all times. For example, std::vector::push_back grants the strong exception guarantee: if an exception is thrown while the method runs, the vector is guaranteed to remain in its original state. This guarantee cannot be given if moves can throw. The vector first increases its size and default constructs a value in the new position. For moves to be efficient, the original value in the container has to be destroyed before the new value is moved into its memory block. If the move operation throws, the original value cannot be restored. Since correctness is more important than performance, std::vector::push_back tries to copy values instead of moving them when moves can throw.\nThe existance of throwing moves caused considerable head ache for developers of type-safe union types, like std::variant and boost::variant2. While std::variant gives up on the strong exception guarantee in this case, boost::variant2 adheres to it at the cost of doubling the size of the variant if any type in the variant set has throwing moves.\nNote 3: It is possible to detect the exception in flight and react, but that is a really unappealing solution.\n\n\n\nGood software is programmed in layers of abstraction. Exceptions often occur in the lowest implementation layer. Sometimes the lowest layer cannot fully report the context of the exception, because the information is not available in that layer.\nHere is an example from the documentation of Boost.Exception:\nvoid\nread_file( FILE * f )\n {\n ....\n const size_t nr = fread(buf, 1, count, f);\n if( ferror(f) )\n throw file_read_error();\n ....\n }\nIf the file cannot be read, read_file throws an exception. Users probably want to know which file could not be read, but this layer does not know the file name. It only got a FILE pointer.\nChanging read_file so that it accepts the filename is breaking modularisation. The author of read_file cannot and should not need to know in which context this function is used.\nA better design is to catch the exception in a higher layer where the context information, like the filename, is available, and then add that information to the exception and re-throw it. Boost.Exception offers boost::exception which derives from std::exception and allows one to add arbitrary information to an exception in flight.\n\n\n\nThrowing an exception in an otherwise small function or method may prevent the optimiser from inlining it. The throw path generates additional instructions which increases the size of the function body, even if it is rarely triggered. The inliner tries to balance the overall increase in code size from inlining against possible performance benefits and may refuse to inline due to the presence of the throw instructions.\nOne can help the optimiser in these situations by wrapping the throw in a small function, e.g. throw_exception(std::exception const& e) and mark it with a compiler-specific attribute so that it is never inlined. Such a function is readily provided by Boost.Exception. The surrounding code is now much leaner since it contains only an instruction to call a function pointer, and the optimiser will find more opportunities to inline it.\n\n\n\nIf your code throws exceptions at all, it will not compile when exceptions are turned off in the compiler (for example, with the flag -fno-exceptions in gcc and clang). As a library developer, you should be interested in supporting compilation without exceptions, since this makes your library useful for more people. At the very least, it helps you to see whether you currently loose performance by using exceptions and whether something has to be done about it (often implementations can be manually optimised to keep the cost small).\nAgain, Boost.Exception has a solution ready: if you consistently use the macro BOOST_THROW_EXCEPTION or the function boost::throw_exception instead of a naked throw (which also has the performance benefits previously mentioned), your code will compile even when exceptions are disabled. The library will detect this and call a user-defined implementation of void throw_exception( std::exception const& e , boost::source_location const& l) instead, which must terminate the program but can run error logging or clean up code before. If the code also catches and rethrows exceptions, the keywords try and catch need to be conditionally hidden, for example, like this\n#ifdef BOOST_NO_EXCEPTIONS\npotentially_throwing(....);\n#else\ntry {\n potentially_throwing(....);\n} catch(....) {\n ....\n}\n#endif\nThe gcc implementation of the C++ stdlib uses a similar approach when you turn off exceptions.\nBoost.Histogram uses Boost.Exception everywhere. This allows me to benchmark it with and without exceptions enabled (and thus I know about the 10-15 % difference in performance). The simple implementation of void throw_exception( std::exception const& e, boost::source_location const& l) in the tests and benchmarks reports where the exception has occured and then aborts the program.\nvoid throw_exception(std::exception const& e, boost::source_location const& l) {\n std::cerr << l.file_name() << \":\" << l.line() << \":\" << l.column() << \": exception in '\"\n << l.function_name() << \" \\\"\" << e.what() << \"\\\"\" << std::endl;\n std::abort();\n}" + }, + { + "objectID": "posts/exceptions.html#why-use-exceptions-and-not-an-alternative", + "href": "posts/exceptions.html#why-use-exceptions-and-not-an-alternative", + "title": "C++ exceptions in high-performance code", + "section": "", + "text": "Exceptions are the official language feature of C++ for reporting errors. There are two cases in C++ where other error reporting systems based on return values do not work: in constructors (which do not return) and in operators (which must use the return value for something else). There are workarounds for these cases, but they lead to less idiomatic C++. Of course, there are also reasons to avoid exceptions. Herb Sutter gives a comprehensive overview of the various pros and cons of exceptions and their alternatives." + }, + { + "objectID": "posts/exceptions.html#cost-of-using-exceptions", + "href": "posts/exceptions.html#cost-of-using-exceptions", + "title": "C++ exceptions in high-performance code", + "section": "", + "text": "Run time:\n\nExceptions have zero run-time cost if they do not trigger, but reduce optimisation opportunities (read on for details)\nExceptions that trigger have a large cost (thousands of CPU cycles)\n\nCompile time: small cost\nCode size increases by about +7 % to +40 %, depending on compiler and operating system\n\nExceptions in C++ were designed to have zero run time cost when they do not trigger. Zero cost is even less cost than an if-else-branch. In theory, this makes C++ exceptions more performant than the C style alternative of returning an error code on the happy path (but read on). C++ exceptions cost a lot of cycles when they trigger. Therefore, exceptions should never be used for normal control flow, where both alternatives happen regularly. Exceptions are for exceptional events only, faults that occur rarely during the run of a program. A good example is wrong user input and unexpected I/O errors.\nSo exceptions seem pretty great, but Google turns off exceptions in their builds (-fno-exceptions in gcc and clang) – why? Apart from stylistic issues, exceptions reduce opportunities for the optimiser to make the code faster. The optimiser is allowed to transform code if it can prove that this does not change the visible outcome. A potentially throwing expression can prevent fusing instructions before and after the expression. When the exception triggers, the instructions after the exception are not executed. If this has observable side effects, the optimiser cannot fuse the instructions to increase performance.\nAll in all, this has a noticeable effect even in carefully written libraries that use exceptions. In Boost.Histogram, benchmarks run 10-15 % faster when I deactivate exceptions with -fno-exceptions, even though no exceptions are thrown in these benchmarks." + }, + { + "objectID": "posts/exceptions.html#how-to-use-noexcept", + "href": "posts/exceptions.html#how-to-use-noexcept", + "title": "C++ exceptions in high-performance code", + "section": "", + "text": "The noexcept specifier marks a function or method as not throwing any exception ever. This is great for the optimiser.\nThe compiler trusts this declaration. You won’t get a compile-time error if code that was declared noexcept throws an exception anyway. If that happens at run-time, the program simply terminates. Compilers may emit a warning about this, but at the time of this writing, they only do this in obvious cases1. The developer must make sure to not lie to the compiler when declaring something as noexcept.\nThere are legitimate reasons to declare a function noexcept which has throwing internal code (which may be third-party code). If all conditions can be anticipated and explicitly handled under which the internal code could throw, the surrounding code can be declared noexcept since no throw will actually occur. Similarly, a function or method that throws and catches all exceptions internally, so that none leave the function, can be declared noexcept. This should be a performance gain in theory, but in practice it depends on the compiler support for noexcept.\nAlso in theory, it should not be necessary to mark every non-throwing function or method as noexcept, the compiler is supposed to detect simple cases. At the time of this writing, however, not even the most recent compiler versions (clang-10, gcc-9.3) detect the noexcept property even for the most trivial functions.\nNote 1: At the time of this writing, neither gcc or clang warn if the throw is nested in another function, even if the implementation of that function is visible to the compiler." + }, + { + "objectID": "posts/exceptions.html#best-practices-when-using-c-exceptions", + "href": "posts/exceptions.html#best-practices-when-using-c-exceptions", + "title": "C++ exceptions in high-performance code", + "section": "", + "text": "It may be tempting use an assert instead of an exception, because the optimiser is not troubled by an assert, but don’t do that. An assert is usually only checked when the code is compiled in debug mode2, while exceptions work also in production code. Therefore, an assert cannot be used in place of an exception, in particular in code that checks or validates user input.\nRule-of-thumb for using either assert or throwing an exception: - In private interfaces and private implementation code, where you have full control over the input, use assert to check the consistency of your program logic - In user-facing interfaces, use exceptions\nIn other words, users should never see a failing assert. Anything that can go wrong due to external circumstances outside of the control of the program should trigger an exception. An assert is an executable part of the interface documentation: it reminds a developer that this code expects certain inputs and cannot run correctly when these are violated.\nExample: Let us say some code requires some user-defined number to be greater than 10. The user-facing layer should check whether the number is greater than 10 and otherwise throw an exception. The deeper implementation layers should assert on the same condition. This is not redundant, since the assert documents what the implementation layer expects. If the program is not altered, the assert will never be violated, but it is there in case someone refactors the code and forgets to protect the implementation layer from invalid external input.\nNote 2: To be more precise, the assert macro from <cassert> expands to nothing when -DNDEBUG is set, which is usually set in a release build (for example, this is the cmake default).\n\n\n\nIf the implementation allows it at all, destructors, move constructors, and move-assignment operators should not throw exceptions. If they are not declared noexcept (more details on noexcept are given below), the compiler will try to figure this out.\nThrowing destructors are a really bad idea, because destructors are called during the stack-unwinding when another exception was thrown. If the exception is allowed to leave the destructor in this situation, the program will terminate immediately3.\nContainers like std::vector need their elements to have non-throwing move constructors and move-assignment to make efficient use of them. Containers typically want to guarantee that they are in a valid state at all times. For example, std::vector::push_back grants the strong exception guarantee: if an exception is thrown while the method runs, the vector is guaranteed to remain in its original state. This guarantee cannot be given if moves can throw. The vector first increases its size and default constructs a value in the new position. For moves to be efficient, the original value in the container has to be destroyed before the new value is moved into its memory block. If the move operation throws, the original value cannot be restored. Since correctness is more important than performance, std::vector::push_back tries to copy values instead of moving them when moves can throw.\nThe existance of throwing moves caused considerable head ache for developers of type-safe union types, like std::variant and boost::variant2. While std::variant gives up on the strong exception guarantee in this case, boost::variant2 adheres to it at the cost of doubling the size of the variant if any type in the variant set has throwing moves.\nNote 3: It is possible to detect the exception in flight and react, but that is a really unappealing solution.\n\n\n\nGood software is programmed in layers of abstraction. Exceptions often occur in the lowest implementation layer. Sometimes the lowest layer cannot fully report the context of the exception, because the information is not available in that layer.\nHere is an example from the documentation of Boost.Exception:\nvoid\nread_file( FILE * f )\n {\n ....\n const size_t nr = fread(buf, 1, count, f);\n if( ferror(f) )\n throw file_read_error();\n ....\n }\nIf the file cannot be read, read_file throws an exception. Users probably want to know which file could not be read, but this layer does not know the file name. It only got a FILE pointer.\nChanging read_file so that it accepts the filename is breaking modularisation. The author of read_file cannot and should not need to know in which context this function is used.\nA better design is to catch the exception in a higher layer where the context information, like the filename, is available, and then add that information to the exception and re-throw it. Boost.Exception offers boost::exception which derives from std::exception and allows one to add arbitrary information to an exception in flight.\n\n\n\nThrowing an exception in an otherwise small function or method may prevent the optimiser from inlining it. The throw path generates additional instructions which increases the size of the function body, even if it is rarely triggered. The inliner tries to balance the overall increase in code size from inlining against possible performance benefits and may refuse to inline due to the presence of the throw instructions.\nOne can help the optimiser in these situations by wrapping the throw in a small function, e.g. throw_exception(std::exception const& e) and mark it with a compiler-specific attribute so that it is never inlined. Such a function is readily provided by Boost.Exception. The surrounding code is now much leaner since it contains only an instruction to call a function pointer, and the optimiser will find more opportunities to inline it.\n\n\n\nIf your code throws exceptions at all, it will not compile when exceptions are turned off in the compiler (for example, with the flag -fno-exceptions in gcc and clang). As a library developer, you should be interested in supporting compilation without exceptions, since this makes your library useful for more people. At the very least, it helps you to see whether you currently loose performance by using exceptions and whether something has to be done about it (often implementations can be manually optimised to keep the cost small).\nAgain, Boost.Exception has a solution ready: if you consistently use the macro BOOST_THROW_EXCEPTION or the function boost::throw_exception instead of a naked throw (which also has the performance benefits previously mentioned), your code will compile even when exceptions are disabled. The library will detect this and call a user-defined implementation of void throw_exception( std::exception const& e , boost::source_location const& l) instead, which must terminate the program but can run error logging or clean up code before. If the code also catches and rethrows exceptions, the keywords try and catch need to be conditionally hidden, for example, like this\n#ifdef BOOST_NO_EXCEPTIONS\npotentially_throwing(....);\n#else\ntry {\n potentially_throwing(....);\n} catch(....) {\n ....\n}\n#endif\nThe gcc implementation of the C++ stdlib uses a similar approach when you turn off exceptions.\nBoost.Histogram uses Boost.Exception everywhere. This allows me to benchmark it with and without exceptions enabled (and thus I know about the 10-15 % difference in performance). The simple implementation of void throw_exception( std::exception const& e, boost::source_location const& l) in the tests and benchmarks reports where the exception has occured and then aborts the program.\nvoid throw_exception(std::exception const& e, boost::source_location const& l) {\n std::cerr << l.file_name() << \":\" << l.line() << \":\" << l.column() << \": exception in '\"\n << l.function_name() << \" \\\"\" << e.what() << \"\\\"\" << std::endl;\n std::abort();\n}" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html", + "href": "2024_LHCb_Starterkit/lecture_1_full.html", + "title": "Statistics I", + "section": "", + "text": "Hans Dembinski, TU Dortmund" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#about-me", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#about-me", + "title": "Statistics I", + "section": "About me", + "text": "About me\n\nWorking with the LHCb experiment since 2016\nBefore astroparticle physicist working with Pierre Auger Observatory and IceCube\nStatistics convener in LHCb (2019-2021, 2024-now)\nInterests\n\nStatistics\nHigh-performance computing\nAPI design: “Make Interfaces Easy to Use Correctly and Hard to Use Incorrectly” Scott Meyers" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#scikit-hep-project", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#scikit-hep-project", + "title": "Statistics I", + "section": "Scikit-HEP project", + "text": "Scikit-HEP project\n\nProvides high-performance data analysis tools in Python\nPartially funded by NSF grants\nMany LHCb developers: Eduardo Rodrigues, Chris Burr, myself, …\nCore values\n\nSpecialized tools that integrate well with existing Python ecosystem\nEasy to use for beginners\nFlexible for power users\nGood documentation\nEasy installation" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#the-power-of-combinatorics", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#the-power-of-combinatorics", + "title": "Statistics I", + "section": "The power of combinatorics", + "text": "The power of combinatorics\nBuild everything from basic components that fit together\n\nImage credit: Toni Zaat on Unsplash" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#important-packages", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#important-packages", + "title": "Statistics I", + "section": "Important packages", + "text": "Important packages\n\nPython scientific stack\n\nNumpy: fast computation with arrays\nScipy: integration, statistical distributions, special functions, …\nMatplotlib: scientific plotting\nNumba: JIT Compiler for Python (very fast)\nPandas: Convenient processing of tabular data\n\nScikit-HEP Project\n\nuproot: fast reading and writing of ROOT Trees\nparticle: get particle properties from PDG IDs\nboost-histogram: multi-dimensional generalised histograms\niminuit: fitting and error computation package\nresample: Easy bootstrapping to compute uncertainty and bias\npyhf: for histogram fitting/limit setting and preserving likelihoods\n\nOther packages\n\njacobi: Easy numerical error propagation\nnumba-stats: Fast statistical distributions for fitting\ntabulate: Convert data tables to various formats (LaTeX, HTML, Markdown, …)\nscikit-learn: Basic machine learning tools\nxgboost: Gradient-boosting machine library that won many Kaggle competitions\n\n\nInstall everything with pip install LIBRARY" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#working-with-jupyter-notebooks", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#working-with-jupyter-notebooks", + "title": "Statistics I", + "section": "Working with Jupyter notebooks", + "text": "Working with Jupyter notebooks\n\nI recommend to use vscode\n\nVery good Jupyter notebook support\nRemote explorer: edit and run code on remote machines" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#prepare-computing-environment", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#prepare-computing-environment", + "title": "Statistics I", + "section": "Prepare computing environment", + "text": "Prepare computing environment\n\nIn terminal, create and enter virtual environment (you need Python-3.8 or later)\npython3 -m venv .venv\nsource .venv/bin/activate\nInstall packages\npython -m pip install --upgrade numba matplotlib uproot boost-histogram iminuit scipy particle numba_stats ipywidgets\n\n\n\nuproot provides fast and convenient access to ROOT trees\n\n\n\nimport uproot\n\nprint(f\"uproot {uproot.__version__}\")\n\nuproot 5.2.2\n\n\n\nf = uproot.open(\"example.root\")\nevent = f[\"event\"]\n \nevent.show()\n\nname | typename | interpretation \n---------------------+--------------------------+-------------------------------\ntrk_len | int32_t | AsDtype('>i4')\nmc_trk_len | int32_t | AsDtype('>i4')\ntrk_imc | int32_t[] | AsJagged(AsDtype('>i4'))\ntrk_px | float[] | AsJagged(AsDtype('>f4'))\ntrk_py | float[] | AsJagged(AsDtype('>f4'))\ntrk_pz | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_px | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_py | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_pz | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_pid | int32_t[] | AsJagged(AsDtype('>i4'))\n\n\n\nTree contains fake simulated LHCb events\nTruth in simulation\n\nmc_trk_len: number of true particles in event\nmc_trk_px: x-component of true momentum of particle\nmc_trk_py: y-component of true momentum of particle\nmc_trk_pz: z-component of true momentum of particle\nmc_trk_pid: particle ID (PDG ID)\n\nReconstructed simulation\n\ntrk_len: number reconstructed tracks in event\ntrk_px: x-component of momentum of reconstructed track\ntrk_py: y-component of momentum of reconstructed track\ntrk_pz: z-component of momentum of reconstructed track\n\nSpecial\n\ntrk_imc: index of matched true particle or -1\n\n\n\n# get branches as arrays\narr = event.arrays([\"trk_len\", \"trk_px\", \"mc_trk_len\", \"mc_trk_px\"])\n\ntrk_len = arr[\"trk_len\"]\ntrk_px = arr[\"trk_px\"]\nmc_trk_len = arr[\"mc_trk_len\"]\nmc_trk_px = arr[\"mc_trk_px\"]\n\n\ntrk_len\n\n[6,\n 7,\n 2,\n 6,\n 7,\n 7,\n 6,\n 7,\n 4,\n 6,\n ...,\n 2,\n 4,\n 13,\n 3,\n 6,\n 3,\n 4,\n 6,\n 3]\n------------------\ntype: 1000 * int32\n\n\nExercise: Explore some other arrays from event\n\n# do exercise here\n\n\nSome branches are 2D (e.g. trk_px)\nFirst index iterates over events\nSecond index iterates over tracks per event\n\n\nfor ievent in range(5):\n print(ievent, trk_len[ievent], trk_px[ievent])\n\n0 6 [-0.979, 0.232, -0.464, 0.629, 0.0287, 0.156]\n1 7 [-0.59, 0.102, -0.282, -0.585, 0.0525, -0.249, -0.0836]\n2 2 [0.124, 0.309]\n3 6 [-0.287, 0.0082, -0.671, -0.933, -0.011, 0.0323]\n4 7 [-0.086, -0.101, 0.0523, 0.204, -0.0287, 0.115, 0.0954]\n\n\n\nMathematical operators and slicing works on these arrays similar to Numpy\n\n\ntrk_len[:10]\n\n[6,\n 7,\n 2,\n 6,\n 7,\n 7,\n 6,\n 7,\n 4,\n 6]\n----------------\ntype: 10 * int32\n\n\n\ntrk_len > 10\n\n[False,\n False,\n False,\n False,\n False,\n False,\n False,\n False,\n False,\n False,\n ...,\n False,\n False,\n True,\n False,\n False,\n False,\n False,\n False,\n False]\n-----------------\ntype: 1000 * bool\n\n\n\nMany numpy functions work on awkward arrays\n\n\nimport numpy as np\nprint(np.__version__)\n\n1.24.4\n\n\n\nnp.sum(trk_len)\n\n5032\n\n\nExercise\n\nUse numpy functions to count how many events have zero tracks\nHint: use np.sum and array mask\n\n\n# do exercise here\n\n\n# solution\n\nf\"{np.sum(trk_len == 0)} events with zero tracks\"\n\n'5 events with zero tracks'\n\n\n\n# let's plot some things\nimport matplotlib.pyplot as plt\n\nplt.hist(trk_len);\n\n\n\n\n\nFor some operations and awkward arrays, we need the awkward library\n\n\nimport awkward as ak\nprint(ak.__version__)\n\n2.6.1\n\n\nExercise * Try to plot trk_px * Can you figure out from the error message why it does not work? * Fix the issue with the function ak.flatten\n\n# do exercise here\n\n\n# solution\n\n# plt.hist(trk_px) fails with helpful error message\nplt.hist(ak.flatten(trk_px));" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#reading-data-from-a-large-tree", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#reading-data-from-a-large-tree", + "title": "Statistics I", + "section": "Reading data from a large tree", + "text": "Reading data from a large tree\n\nTree often do not fit into memory (tree size > 100 GB and more possible)\nIterate over chunks\n\n\nfor chunk in event.iterate([\"trk_px\", \"mc_trk_px\"]):\n\n plt.hist(ak.flatten(chunk[\"trk_px\"]), label=\"rec\")\n plt.hist(ak.flatten(chunk[\"mc_trk_px\"]), label=\"gen\")\n plt.xlabel(\"px / MeVc-1\")\n plt.legend()\n\n # stop after reading one chunk\n break" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#accelerate-python-code-with-numba", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#accelerate-python-code-with-numba", + "title": "Statistics I", + "section": "Accelerate Python code with Numba", + "text": "Accelerate Python code with Numba\n\nJust-In-Time compiler for scientific Python code\nVery powerful and very fast\nFast for loops\nNot all Python code supported, works best with numerical code\nOnly use when needed\n\n\nimport numba as nb\nprint(f\"numba {nb.__version__}\")\n\nnumba 0.57.1\n\n\n\nlet’s subtract reconstructed from true momentum to check momentum resolution\nneed to select tracks matched to MC particles\nthis is relatively slow in Python, even if we use array computations\n\n\n%%timeit -n 1 -r 3\n\ndelta_px = []\nfor chunk in event.iterate([\"trk_px\", \"trk_imc\", \"mc_trk_px\"]):\n for px, imc, mc_px in zip(chunk[\"trk_px\"], chunk[\"trk_imc\"], chunk[\"mc_trk_px\"]):\n # select only tracks with associated true particle\n mask = imc >= 0\n subset_px = px[mask]\n subset_mc_px = mc_px[imc[mask]]\n # conversion to numpy needed\n delta_px = np.append(delta_px, subset_px - subset_mc_px) \n\n963 ms ± 35.4 ms per loop (mean ± std. dev. of 3 runs, 1 loop each)\n\n\n\n# awkward arrays can be used in Numba\n\n@nb.njit\ndef calc(chunk):\n r = []\n # loop over events\n for px, imc, mc_px in zip(chunk[\"trk_px\"], chunk[\"trk_imc\"], chunk[\"mc_trk_px\"]):\n # loop over reconstructed tracks per event\n for px_i, imc_i in zip(px, imc):\n if imc_i < 0:\n continue\n mc_px_i = mc_px[imc_i]\n r.append(px_i - mc_px_i)\n return r\n\n\n# function is JIT-compiled on first call with new data types\nfor chunk in event.iterate([\"trk_px\", \"trk_imc\", \"mc_trk_px\"]):\n calc(chunk)\n break\n\n\n%%timeit -n 1 -r 3\ndelta_px = []\nfor data in event.iterate([\"trk_px\", \"trk_imc\", \"mc_trk_px\"]):\n d = calc(data)\n delta_px = np.append(delta_px, d)\n\n4.17 ms ± 1.17 ms per loop (mean ± std. dev. of 3 runs, 1 loop each)\n\n\n\nCreated array delta_px could still become very large in memory\nBetter: Create histogram object first and then fill iteratively with chunks\n\n\n\nPython frontend to Boost.Histogram library in C++ from the Boost project\nVery fast and flexible\nMulti-dimensional histograms and other binned statistics (profiles!)\nSupports weighted data\nAnd much much more, see docs\n\n\nimport boost_histogram as bh\nprint(f\"boost-histogram {bh.__version__}\")\n\nboost-histogram 1.4.0\n\n\n\n# make an axis\nxaxis = bh.axis.Regular(20, -2, 2)\n\n# easy to make several histograms with same binning by reusing axis\nh_px = bh.Histogram(xaxis)\nh_mc_px = bh.Histogram(xaxis)\n\n\n# let's use a finer binning for this one\nh_delta_px = bh.Histogram(bh.axis.Regular(100, -2, 2))\n\n\n# incremental filling\nfor data in event.iterate([\"trk_px\", \"trk_imc\", \"mc_trk_px\"]):\n h_px.fill(ak.flatten(trk_px))\n\n\nh_px\n\nHistogram(Regular(20, -2, 2), storage=Double()) # Sum: 5032.0\n\n\n\nprint(h_px)\n\n ┌───────────────────────────────────────────────────────────┐\n[-inf, -2) 0 │ │\n[ -2, -1.8) 1 │▏ │\n[-1.8, -1.6) 3 │▎ │\n[-1.6, -1.4) 17 │█▎ │\n[-1.4, -1.2) 25 │█▊ │\n[-1.2, -1) 78 │█████▌ │\n[ -1, -0.8) 181 │████████████▉ │\n[-0.8, -0.6) 308 │█████████████████████▊ │\n[-0.6, -0.4) 453 │████████████████████████████████▏ │\n[-0.4, -0.2) 647 │█████████████████████████████████████████████▉ │\n[-0.2, 0) 819 │██████████████████████████████████████████████████████████ │\n[ 0, 0.2) 782 │███████████████████████████████████████████████████████▍ │\n[ 0.2, 0.4) 629 │████████████████████████████████████████████▌ │\n[ 0.4, 0.6) 480 │██████████████████████████████████ │\n[ 0.6, 0.8) 323 │██████████████████████▉ │\n[ 0.8, 1) 159 │███████████▎ │\n[ 1, 1.2) 77 │█████▌ │\n[ 1.2, 1.4) 37 │██▋ │\n[ 1.4, 1.6) 7 │▌ │\n[ 1.6, 1.8) 3 │▎ │\n[ 1.8, 2) 3 │▎ │\n[ 2, inf) 0 │ │\n └───────────────────────────────────────────────────────────┘\n\n\n\n# plot histogram with plt.stairs\nplt.stairs(h_px.values(), h_px.axes[0].edges);\n\n\n\n\nExercise * Use iterative loop to fill h_mc_px and h_delta_px, use calc from before * Plot all three histograms overlaid into one figure with plt.stairs\n\n# do exercise here\n\n\n# solution\n\nfor chunk in event.iterate([\"trk_px\", \"trk_imc\", \"mc_trk_px\"]):\n h_px.fill(ak.flatten(chunk[\"trk_px\"]))\n h_mc_px.fill(ak.flatten(chunk[\"trk_px\"]))\n h_delta_px.fill(calc(chunk))\n \nfor h, label in ((h_px, \"px\"), (h_mc_px, \"mc_px\"), (h_delta_px, \"delta_px\")):\n plt.stairs(h.values(), h.axes[0].edges, label=label)\n \nplt.legend();\n\n\n\n\n\nHistograms can be reduced\nShrink axis range with slices and using loc\nMake binning coarser with rebin\n\n\nfrom boost_histogram.tag import loc, rebin\n\nh_delta_px_2 = h_delta_px[loc(-0.5):loc(0.5):rebin(2)]\n\nfor h in (h_delta_px_2, h_delta_px):\n plt.stairs(h.values(), h.axes[0].edges,\n label=f\"{len(h.values())} bins\")\nplt.legend();" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#beyond-1d-histograms", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#beyond-1d-histograms", + "title": "Statistics I", + "section": "Beyond 1D histograms", + "text": "Beyond 1D histograms\n\nSupports advanced axis types (e.g. category axis)\nSupports higher dimensional histograms\nSupports generalized histograms with binned statistics (aka “profile” in ROOT)\n\n\n# check detector momentum resolution in x for different particle species\n\np_axis = bh.axis.Regular(50, -2, 2)\npid_axis = bh.axis.IntCategory([], growth=True)\n\n\n# profile: compute mean of reconstructed px in bins of mc_px for different particle species\n\np_2d = bh.Histogram(p_axis, pid_axis, storage=bh.storage.Mean())\n\n\nbranches = [\"trk_px\", \"trk_imc\", \"mc_trk_px\", \"mc_trk_pid\"]\nfor data in event.iterate(branches):\n for px, imc, mc_px, mc_pid in zip(*(data[b] for b in branches)):\n # select only tracks with associated true particle\n mask = imc >= 0\n px = px[mask]\n associated = imc[mask]\n mc_px = mc_px[associated]\n mc_pid = mc_pid[associated]\n\n p_2d.fill(mc_px, mc_pid, sample=px)\n\np_2d\n\nHistogram(\n Regular(50, -2, 2),\n IntCategory([-321, 211, 2212, -2212, -211, 321], growth=True),\n storage=Mean()) # Sum: Mean(count=3240, value=0.000433322, variance=0.263444)\n\n\n\nWe use particle library to look-up properties of particles based on their particle ID\nGet name, mass, life-time, quark content, …\n\n\nimport particle\nprint(f\"particle {particle.__version__}\")\n\nparticle 0.23.1\n\n\n\npids = p_2d.axes[1]\nfor i, pid in enumerate(pids):\n part = particle.Particle.from_pdgid(pid)\n\n x = p_2d.axes[0].centers\n y = p_2d.values()[:, i]\n ye = p_2d.variances()[:, i] ** 0.5\n plt.errorbar(x, y, ye, fmt=\"o\", label=f\"${part.latex_name}$ pid={int(part.pdgid)}\")\n\nplt.xlabel(\"mc_px\")\nplt.ylabel(\"px\")\nplt.legend();\n\n\n\n\nExercise * Only show bins with sufficient entries, use p_2d.counts()\n\n# do exercise here\n\n\n# solution\n\npids = p_2d.axes[1]\nfor i, pid in enumerate(pids):\n part = particle.Particle.from_pdgid(pid)\n\n x = p_2d.axes[0].centers\n y = p_2d.values()[:, i]\n ye = p_2d.variances()[:, i] ** 0.5\n\n ma = (p_2d.counts() > 5)[:, i]\n \n plt.errorbar(x[ma], y[ma], ye[ma], fmt=\"o\",\n label=f\"${part.latex_name}$ pid={int(part.pdgid)}\")\n\nplt.xlabel(\"mc_px\")\nplt.ylabel(\"px\")\nplt.legend();\n\n\n\n\n\nWe find that the reconstruction resolution is the same for all particle species\nWhich means we can remove this axis by summing over its entries (called projection in ROOT)\nboost-histogram keeps track of events in overflow and underflow bins by default to make summing exact\n\n\np_1d = p_2d[:, sum] # remove PID axis by summing\n\n\nx = p_1d.axes[0].centers\ny = p_1d.values()\nye = p_1d.variances() ** 0.5\nm = p_1d.counts() >= 5\n\nplt.errorbar(x[m], y[m], ye[m], fmt=\"o\")\n\nplt.xlabel(\"mc_px\")\nplt.ylabel(\"px\");" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#fits", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#fits", + "title": "Statistics I", + "section": "Fits", + "text": "Fits\n\nTypical analysis work flow (often automated with Snakemake)\n\nPre-select data\nMake histograms or profiles from pre-selected data\nFit histograms or profiles to extract physical parameters\n\nMany specialized fitting tools for individual purposes, e.g.: scipy.optimize.curve_fit\nGeneric libraries\n\nRooFit\nzfit\npyhf\niminuit\n…\n\nFitting (“estimation” in statistics)\n\nAdjust parametric statistical model to dataset and find model parameters which match dataset best\nAI community calls this “learning”\nEstimate: result of fit\n\nDataset: samples \\(\\{ \\vec x_i \\}\\)\n\nSample can be one-dimensional or multidimensional\nValues can be continuous (e.g. candidate mass) or discrete (e.g. bin counts)\n\nModel: Probability density or probability mass function \\(f(\\vec x; \\vec p)\\) which depends on unknown parameters \\(\\vec p\\)\nConjecture: maximum-likelihood estimate (MLE) \\(\\hat{\\vec p} = \\text{argmax}_{\\vec p} L(\\vec p)\\) is optimal, with \\(L(\\vec p) = \\prod_i f(\\vec x_i; \\vec p)\\)\n\nProven: Asymptotically optimal, unbiased with minimum variance\n\nNo proof for finite data sets, but unchallenged\n\nUniversal: MLE can be computed for any problem\nEquivalent: minimize negative log-likelihood (NLL), \\(\\hat{\\vec p} = \\text{argmin}_{\\vec p}(-\\ln L(\\vec p))\\)\n\nLimiting case of maximum-likelihood fit: least-squares fit aka chi-square fit\n\n\n\nPython frontend to Minuit2 C++ library maintained by ROOT team at CERN\nMaximum-likelihood and least-squares fits with error estimation\nEasy to use for beginners, flexible for power users\nComes with builtin cost functions for common fits\nLarge number of tutorials\n\n\nimport iminuit\nfrom iminuit import Minuit\n\nprint(f\"iminuit {iminuit.__version__}\")\n\niminuit 2.25.2" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#iminuit-vs.-other-packages", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#iminuit-vs.-other-packages", + "title": "Statistics I", + "section": "iminuit vs. other packages", + "text": "iminuit vs. other packages\n\npyhf\n\nBetter choice when you need to compute limits and preserve the likelihood\n\nzfit and RooFit\n\nHelp you build statistical models: automatic normalization & convolutions\n\nThis has to be done “by hand” in iminuit\n\nGood for standard tasks, but restrict your freedom\n\niminuit\n\nHas good documentation\nWell-designed API\nIs very flexible and open\nIs very fast with numba\n\nTypical fit in particle physics: mass distribution of decay candidates with signal peak over smooth background\nTask: get the number of signal candidates = signal yield\nCan be solved with a fit\n\n\ndef make_data(size, seed=1):\n rng = np.random.default_rng(seed)\n s = rng.normal(0.5, 0.1, size=size // 2)\n b = rng.exponential(1, size=2 * size)\n b = b[b < 1]\n b = b[: size // 2]\n x = np.append(s, b)\n return x\n\nx1 = make_data(10_000)\nplt.hist(x1, bins=100);\n\n\n\n\n\nTo apply maximum-likelihood method, we need statistical model\nFull sample is additive mixture of signal sample and background sample\nAssumption 1: signal is normal-distributed; pdf is \\(\\mathcal{N}(\\mu, \\sigma)\\) with parameters \\(\\mu\\) and \\(\\sigma\\)\nAssumption 2: background is truncated exponential distribution\n\n\n# fast implementations of statistical distributions, API similar to scipy.stats\nfrom numba_stats import norm, truncexpon\n\n# model pdf\ndef model1(x, z, mu, sigma, slope):\n s = norm.pdf(x, mu, sigma)\n b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)\n return (1 - z) * b + z * s\n\n# negative log-likelihood\ndef nll1(z, mu, sigma, slope):\n logL = np.log(model1(x1, z, mu, sigma, slope))\n return -np.sum(logL)\n\n\nm = Minuit(nll1, z=0.5, mu=0.5, sigma=1, slope=1)\nm.errordef = Minuit.LIKELIHOOD\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -1790\nNfcn = 209\n\n\nEDM = 762 (Goal: 0.0001)\n\n\n\nINVALID Minimum\nABOVE EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.500\n0.012\n\n\n\n\n\n\n\n1\nmu\n0.500\n0.003\n\n\n\n\n\n\n\n2\nsigma\n0.155\n0.005\n\n\n\n\n\n\n\n3\nslope\n1.00\n0.07\n\n\n\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n0.000149\n-1e-6 (-0.037)\n0.033e-3 (0.508)\n-0.25e-3 (-0.304)\n\n\nmu\n-1e-6 (-0.037)\n9.19e-06\n-0e-6 (-0.025)\n-68e-6 (-0.335)\n\n\nsigma\n0.033e-3 (0.508)\n-0e-6 (-0.025)\n2.91e-05\n-0.053e-3 (-0.147)\n\n\nslope\n-0.25e-3 (-0.304)\n-68e-6 (-0.335)\n-0.053e-3 (-0.147)\n0.00442\n\n\n\n\n\n\nFit did not work\nPlotting 1D or 2D slices of likelihood may help\n\n\nm.draw_profile(\"z\");\n\n\n\n\n\nIncrease print level\n\n\nm.print_level = 1\nm.migrad()\nm.print_level = 0\n\nW FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW VariableMetricBuilder No improvement in line search\nW VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001\nW VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001\nW FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW VariableMetricBuilder No improvement in line search\nW VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001\nW VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001\nW FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW VariableMetricBuilder No improvement in line search\nW VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001\nW VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001\nW FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW VariableMetricBuilder No improvement in line search\nW VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001\nW VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001\nW FCN result is NaN for [ 0.406325 0.495731 -0.0437361 1.13819 ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW FCN result is NaN for [ nan nan nan nan ]\nW VariableMetricBuilder No improvement in line search\nW VariableMetricBuilder Iterations finish without convergence; Edm 761.614 Requested 0.0001\nW VariableMetricBuilder No convergence; Edm 761.614 is above tolerance 0.001\n\n\n\nfit failed because parameters need limits\n\nExercise\n\nRepeat fit, but with parameter limits.\nSet parameter limits by calling m.limits[\"parameter\"] = (lower, upper) before m.migrad()\nYou can set identical limits for multiple parameters, for example, m.limits[\"a\", \"b\"] = (0, 1)\n\n\n# do exercise here\n\n\n# solution\n\nm.limits[\"z\", \"mu\"] = (0, 1)\nm.limits[\"sigma\", \"slope\"] = (0, None)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -2095\nNfcn = 481\n\n\nEDM = 1.78e-06 (Goal: 0.0001)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.498\n0.009\n\n\n0\n1\n\n\n\n1\nmu\n0.4968\n0.0020\n\n\n0\n1\n\n\n\n2\nsigma\n0.0994\n0.0020\n\n\n0\n\n\n\n\n3\nslope\n1.03\n0.06\n\n\n0\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.94e-05\n-1e-6 (-0.063)\n10e-6 (0.547)\n-0.12e-3 (-0.213)\n\n\nmu\n-1e-6 (-0.063)\n4.21e-06\n-0e-6 (-0.061)\n-27e-6 (-0.225)\n\n\nsigma\n10e-6 (0.547)\n-0e-6 (-0.061)\n3.94e-06\n-18e-6 (-0.156)\n\n\nslope\n-0.12e-3 (-0.213)\n-27e-6 (-0.225)\n-18e-6 (-0.156)\n0.00345\n\n\n\n\n\n\ncompute asymmetric errors with Minos algorithm (profile likelihood method)\n\n\nm.minos()\n\n\n\n\nMigrad\n\n\nFCN = -2095\nNfcn = 692\n\n\nEDM = 1.78e-06 (Goal: 0.0001)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.498\n0.009\n-0.009\n0.009\n0\n1\n\n\n\n1\nmu\n0.4968\n0.0021\n-0.0020\n0.0020\n0\n1\n\n\n\n2\nsigma\n0.0994\n0.0020\n-0.0019\n0.0020\n0\n\n\n\n\n3\nslope\n1.03\n0.06\n-0.06\n0.06\n0\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nError\n-0.009\n0.009\n-0.002\n0.002\n-0.0019\n0.0020\n-0.06\n0.06\n\n\nValid\nTrue\nTrue\nTrue\nTrue\nTrue\nTrue\nTrue\nTrue\n\n\nAt Limit\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\n\n\nMax FCN\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\n\n\nNew Min\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\nFalse\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.94e-05\n-1e-6 (-0.063)\n10e-6 (0.547)\n-0.12e-3 (-0.213)\n\n\nmu\n-1e-6 (-0.063)\n4.21e-06\n-0e-6 (-0.061)\n-27e-6 (-0.225)\n\n\nsigma\n10e-6 (0.547)\n-0e-6 (-0.061)\n3.94e-06\n-18e-6 (-0.156)\n\n\nslope\n-0.12e-3 (-0.213)\n-27e-6 (-0.225)\n-18e-6 (-0.156)\n0.00345\n\n\n\n\n\n\nyou can compute 2D confidence regions with mncontour and draw them with draw_mncontour\n\n\nm.draw_mncontour(\"z\", \"sigma\");\n\n\n\n\nExercise\n\nDraw 1,2,3 sigma contours, use keyword cl\n\n\n# do exercise here\n\n\n# solution\n\nm.draw_mncontour(\"z\", \"sigma\", cl=(1, 2, 3));\n\n\n\n\n\nfor nice overview: compute matrix of likelihood profiles with draw_mnmatrix\n\n\nm.draw_mnmatrix(figsize=(9,7));\n\n\n\n\n\n\n\n\nWriting unbinned negative-log-likelihood is repetitive, only model ever changes\nUse built-in cost function from iminuit to get nice extras\n\n\nfrom iminuit.cost import UnbinnedNLL\n\nnll1 = UnbinnedNLL(x1, model1)\nm = Minuit(nll1, z=0.5, mu=0.5, sigma=1, slope=1)\nm.limits[\"z\", \"mu\"] = (0, 1)\nm.limits[\"sigma\", \"slope\"] = (0, None)\n\n\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -4189\nNfcn = 169\n\n\nEDM = 8.37e-05 (Goal: 0.0002)\ntime = 0.4 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.498\n0.009\n\n\n0\n1\n\n\n\n1\nmu\n0.4968\n0.0020\n\n\n0\n1\n\n\n\n2\nsigma\n0.0994\n0.0020\n\n\n0\n\n\n\n\n3\nslope\n1.03\n0.06\n\n\n0\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.62e-05\n-1e-6 (-0.037)\n10e-6 (0.529)\n-0.11e-3 (-0.211)\n\n\nmu\n-1e-6 (-0.037)\n4.17e-06\n-0e-6 (-0.039)\n-28e-6 (-0.230)\n\n\nsigma\n10e-6 (0.529)\n-0e-6 (-0.039)\n3.83e-06\n-17e-6 (-0.152)\n\n\nslope\n-0.11e-3 (-0.211)\n-28e-6 (-0.230)\n-17e-6 (-0.152)\n0.00345\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:12:58.377610\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nExercise\n\nUse interactive mode via m.interactive() to check how fit reacts to parameter changes\nUseful for debugging and to find starting values by hand\n\n\n# do exercise here" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#faster-fits", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#faster-fits", + "title": "Statistics I", + "section": "Faster fits?", + "text": "Faster fits?\n\nOption 1: accelerate NLL computation with Numba\nOption 2: use binned data\n\n\nx2 = make_data(1_000_000)\n\n\nm = Minuit(UnbinnedNLL(x2, model1), z=0.5, mu=0.5, sigma=1, slope=1)\nm.limits[\"z\", \"mu\"] = (0, 1)\nm.limits[\"sigma\", \"slope\"] = (0, None)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -4.216e+05\nNfcn = 169\n\n\nEDM = 0.000135 (Goal: 0.0002)\ntime = 5.9 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n499.7e-3\n0.9e-3\n\n\n0\n1\n\n\n\n1\nmu\n499.98e-3\n0.20e-3\n\n\n0\n1\n\n\n\n2\nsigma\n99.59e-3\n0.20e-3\n\n\n0\n\n\n\n\n3\nslope\n0.997\n0.006\n\n\n0\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.87e-07\n-0.01e-6 (-0.049)\n0.10e-6 (0.548)\n-1.2e-6 (-0.229)\n\n\nmu\n-0.01e-6 (-0.049)\n4.18e-08\n-0 (-0.060)\n-0.26e-6 (-0.226)\n\n\nsigma\n0.10e-6 (0.548)\n-0 (-0.060)\n4.09e-08\n-0.18e-6 (-0.164)\n\n\nslope\n-1.2e-6 (-0.229)\n-0.26e-6 (-0.226)\n-0.18e-6 (-0.164)\n3.06e-05\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:13:04.515566\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nGeneral insights * Time spend in minimizer is negligible * Bottleneck is evaluating NLL function * Bottleneck inside NLL function is usually evaluation of model * NLL computation is trivially parallelizable: log(pdf) computed independently for each data point * Numba allows us to exploit auto-vectorization (SIMD instructions) and parallel computing on multiple cors\n\n@nb.njit(parallel=True, fastmath=True, error_model=\"numpy\")\ndef nll2(z, mu, sigma, slope):\n def model(x, z, mu, sigma, slope):\n s = norm.pdf(x, mu, sigma)\n b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)\n return (1 - z) * b + z * s\n\n logL = np.log(model(x2, z, mu, sigma, slope))\n return -np.sum(logL)\n\n# compile function\nnll2(0.5, 0.5, 0.1, 0.1);\n\nOMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.\n\n\n\nm = Minuit(nll2, z=0.5, mu=0.5, sigma=1, slope=1)\nm.errordef = Minuit.LIKELIHOOD\nm.limits[\"z\", \"mu\"] = (0, 1)\nm.limits[\"sigma\", \"slope\"] = (0, None)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -2.108e+05\nNfcn = 242\n\n\nEDM = 2.48e-06 (Goal: 0.0001)\ntime = 1.4 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n499.7e-3\n0.9e-3\n\n\n0\n1\n\n\n\n1\nmu\n499.98e-3\n0.20e-3\n\n\n0\n1\n\n\n\n2\nsigma\n99.59e-3\n0.20e-3\n\n\n0\n\n\n\n\n3\nslope\n0.997\n0.006\n\n\n0\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.87e-07\n-0.01e-6 (-0.049)\n0.10e-6 (0.548)\n-1.2e-6 (-0.229)\n\n\nmu\n-0.01e-6 (-0.049)\n4.18e-08\n-0 (-0.060)\n-0.26e-6 (-0.226)\n\n\nsigma\n0.10e-6 (0.548)\n-0 (-0.060)\n4.09e-08\n-0.18e-6 (-0.164)\n\n\nslope\n-1.2e-6 (-0.229)\n-0.26e-6 (-0.226)\n-0.18e-6 (-0.164)\n3.06e-05\n\n\n\n\n\n\n\nOften simpler: fit histograms\nNo bias from fitting histograms if done correctly, only loss in precision\nNeed model cdf instead of model pdf: \\(F(x; \\vec p) = \\int_{-\\infty}^x f(x'; \\vec p) \\text{d}x'\\)\nRecommended to use iminuit.cost.BinnedNLL, comes with many useful features\n\n\nfrom iminuit.cost import BinnedNLL\n\n# model is now a cdf!\ndef model3(x, z, mu, sigma, slope):\n s = norm.cdf(x, mu, sigma)\n b = truncexpon.cdf(x, 0.0, 1.0, 0.0, slope)\n return (1 - z) * b + z * s\n\nw, xe = np.histogram(x2, bins=100, range=(0, 1))\nnll3 = BinnedNLL(w, xe, model3)\n\n\nm = Minuit(nll3, z=0.5, mu=0.5, sigma=1, slope=1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 84.34 (χ²/ndof = 0.9)\nNfcn = 162\n\n\nEDM = 1.36e-07 (Goal: 0.0002)\ntime = 0.4 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n499.7e-3\n0.9e-3\n\n\n\n\n\n\n\n1\nmu\n499.99e-3\n0.20e-3\n\n\n\n\n\n\n\n2\nsigma\n99.58e-3\n0.20e-3\n\n\n\n\n\n\n\n3\nslope\n0.997\n0.006\n\n\n\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.98e-07\n-0.01e-6 (-0.049)\n0.10e-6 (0.535)\n-1.2e-6 (-0.236)\n\n\nmu\n-0.01e-6 (-0.049)\n4.22e-08\n-0 (-0.052)\n-0.26e-6 (-0.228)\n\n\nsigma\n0.10e-6 (0.535)\n-0 (-0.052)\n4.15e-08\n-0.18e-6 (-0.161)\n\n\nslope\n-1.2e-6 (-0.236)\n-0.26e-6 (-0.228)\n-0.18e-6 (-0.161)\n3.09e-05\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:13:17.369404\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nSee study of binned fits vs. unbinned fit for detailed comparison\nWith sufficient many bins (>= 20 in this case), increase in parameter uncertainties remains negligible\nBinned fits are (100-200)x faster than unbinned fits without using numba" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#how-well-is-model-matching-data", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#how-well-is-model-matching-data", + "title": "Statistics I", + "section": "How well is model matching data?", + "text": "How well is model matching data?\n\nJudgement requires binning of data\nLook at pull distribution\n\nCompute (data - model) / error for each bin\n\nCompare \\(\\chi^2\\) value (goodness-of-fit test statistic) against the degrees of freedom\n\nSimple check: reduced chi-square \\(\\chi^2/n_\\text{dof}\\) should be about 1\nBetter check: p-value \\(= \\int_{\\chi^2_\\text{observed}}^{\\infty} f(\\chi^2; n_\\text{dof}) \\, \\text{d}\\chi^2\\) (survival function)\nIf p-value very small (< 1%), wrong model or underestimated errors\n\n\n\nfrom scipy.stats import chi2\n\npulls = nll3.pulls(m.values)\ncx = xe[:-1] + np.diff(xe)\nplt.errorbar(cx, pulls, np.ones_like(pulls), fmt=\"o\");\n\n\n\n\n\n# m.fval is chi-square distributed for all built-in cost functions\n# except UnbinnedNLL and ExtendedUnbinnedNLL\npvalue = chi2(m.ndof).sf(m.fval)\nf\"chi2/ndof = {m.fmin.reduced_chi2:.2f} p-value = {pvalue:.2f}\"\n\n'chi2/ndof = 0.88 p-value = 0.80'\n\n\nExercise\n\nPlot histogram of pulls, should look like standard normal distribution\nCompute mean and standard deviation (use np.mean and np.std)\n\n\n# do exercise here\n\n\n# solution\n\nxe = plt.hist(pulls, bins=20)[1]\nplt.title(f\"mean = {np.mean(pulls):.2f} std = {np.std(pulls):.2f}\");" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#other-fits", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#other-fits", + "title": "Statistics I", + "section": "Other fits", + "text": "Other fits\n\nTo fit yields (counts) instead of fractions, use ExtendedUnbinnedNLL or ExtendedBinnedNLL\nTo fit (x, y) pairs, use LeastSquares\nTo fit simulated templates, use Template\nSee documentation how models should be constructed then" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#template-fits", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#template-fits", + "title": "Statistics I", + "section": "Template fits", + "text": "Template fits\n\nSometimes PDF/CDF not known in parametric form, but can be Monte-Carlo simulated\nIdea: fit a histogram of simulation output (template) to data histogram\nAdditional source of uncertainty from finite size of simulation\nCorrect error propagation with Barlow-Beeston method, Barlow and Beeston, Comput.Phys.Commun. 77 (1993) 219-228\niminuit.cost.Template\n\nExtension to weighted data and simuation, H. Dembinski, A. Abdelmotteleb, Eur.Phys.J.C 82 (2022) 11, 1043\nApproximation to Barlow-Beeston method for unweighted data\nFaster computation\nAllows you to mix parametric models and templates\n\n\n\ndef make_data2(rng, nmc, truth, bins):\n xe = np.linspace(0, 1, bins + 1)\n b = np.diff(truncexpon.cdf(xe, 0, 1, 0, 1))\n s = np.diff(norm.cdf(xe, 0.5, 0.05))\n n = rng.poisson(b * truth[0]) + rng.poisson(s * truth[1])\n t = np.array([rng.poisson(b * nmc), rng.poisson(s * nmc)])\n return xe, n, t\n\nrng = np.random.default_rng(1)\ntruth = 750, 250\nxe, n, t = make_data2(rng, 200, truth, 15)\n\n_, ax = plt.subplots(1, 2, figsize=(10, 4))\nax[0].stairs(n, xe, color=\"k\", label=\"data\")\nax[0].legend();\nax[1].stairs(t[0], xe, label=\"background\")\nax[1].stairs(t[1], xe, label=\"signal\")\nax[1].legend(title=\"templates\");\n\n\n\n\n\nfrom iminuit.cost import Template\n\nc = Template(n, xe, t)\nm = Minuit(c, 1, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 8.352 (χ²/ndof = 0.6)\nNfcn = 132\n\n\nEDM = 2.75e-07 (Goal: 0.0002)\ntime = 0.5 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n770\n70\n\n\n0\n\n\n\n\n1\nx1\n220\n40\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\n\n\nx0\n4.3e+03\n-0.8e3 (-0.354)\n\n\nx1\n-0.8e3 (-0.354)\n1.32e+03\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:13:24.390185\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nExercise\n\nWith make_data2 generate templates with 1 000 000 simulated points\nFit this data with the Template cost function\nCompare parameter uncertainties\n\n\n# do exercise here\n\n\n# solution\n\nxe, n, t = make_data2(rng, 1_000_000, truth, 15)\nc = Template(n, xe, t)\nm = Minuit(c, 1, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 4.329 (χ²/ndof = 0.3)\nNfcn = 110\n\n\nEDM = 2.66e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n820\n32\n\n\n0\n\n\n\n\n1\nx1\n201\n20\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\n\n\nx0\n1.03e+03\n-0.2e3 (-0.326)\n\n\nx1\n-0.2e3 (-0.326)\n415\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:13:24.547539\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nWith Template you can also fit a mix of template and parametric model\nOften template only needed for background(s)\n\n\n# density model for signal component, as for ExtendedBinnedNLL\ndef model5(x, s, mu, sigma):\n return s * norm.cdf(x, mu, sigma)\n\nc = Template(n, xe, (t[0], model5))\nm = Minuit(c, x0=700, x1_s=300, x1_mu=0.5, x1_sigma=0.1)\nm.limits[\"x1_mu\"] = (0, 1)\nm.limits[\"x1_sigma\"] = (0, None)\n\n\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 2.526 (χ²/ndof = 0.2)\nNfcn = 106\n\n\nEDM = 1.52e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n819\n34\n\n\n0\n\n\n\n\n1\nx1_s\n202\n23\n\n\n\n\n\n\n\n2\nx1_mu\n0.492\n0.006\n\n\n0\n1\n\n\n\n3\nx1_sigma\n0.049\n0.007\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1_s\nx1_mu\nx1_sigma\n\n\nx0\n1.16e+03\n-0.3e3 (-0.428)\n8.13e-3 (0.038)\n-79.31e-3 (-0.330)\n\n\nx1_s\n-0.3e3 (-0.428)\n541\n-8.17e-3 (-0.056)\n79.27e-3 (0.483)\n\n\nx1_mu\n8.13e-3 (0.038)\n-8.17e-3 (-0.056)\n3.99e-05\n-0 (-0.069)\n\n\nx1_sigma\n-79.31e-3 (-0.330)\n79.27e-3 (0.483)\n-0 (-0.069)\n4.98e-05\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:13:24.717132\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1_full.html#overfitting", + "href": "2024_LHCb_Starterkit/lecture_1_full.html#overfitting", + "title": "Statistics I", + "section": "Overfitting", + "text": "Overfitting\n\nDo not use models with degenerate parameters\nAvoid overfitting\n\n\nfrom iminuit.cost import LeastSquares\nfrom numba_stats import bernstein # Bernstein polynomials > Chebychev polynomials\n\nx = [-0.37, -0.25, -0.15, -0.05, 0.047, 0.147, 0.243, 0.374]\ny = [0.96, 0.90, 0.85, 0.81, 0.75, 0.66, 0.42, 0.14]\ney = [0.01, 0.01, 0.01, 0.02, 0.02, 0.03, 0.04, 0.05]\n\ndef model(x, p):\n return bernstein.density(x, p, -0.4, 0.4)\n\n\nstart = np.ones(3)\nm = Minuit(LeastSquares(x, y, ey, model), start)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 23.81 (χ²/ndof = 4.8)\nNfcn = 51\n\n\nEDM = 4.25e-21 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n0.950\n0.011\n\n\n\n\n\n\n\n1\nx1\n0.920\n0.027\n\n\n\n\n\n\n\n2\nx2\n0.21\n0.04\n\n\n\n\n\n\n\n\n\n\n\n\nx0\nx1\nx2\n\n\nx0\n0.000127\n-0.21e-3 (-0.680)\n0.14e-3 (0.321)\n\n\nx1\n-0.21e-3 (-0.680)\n0.000737\n-0.7e-3 (-0.666)\n\n\nx2\n0.14e-3 (0.321)\n-0.7e-3 (-0.666)\n0.00147\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:19:07.003025\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nExercise\n\nFit with increasing number of parameters (increase the length of the starting parameters)\n\nWhat do you observe? Look out for changes in chi2/ndof and the parameter correlations\n\nTry fitting this model\ndef model(x, p):\n return p[0] + bernstein.density(x, p[1:], -0.4, 0.4)\n\nWhat do you observe?\n\n\n\n# do exercise here\n\n\n# solution\n\n# def model(x, p):\n# return p[0] + bernstein.density(x, p[1:], -0.4, 0.4)\n\nstart = np.ones(6)\nm = Minuit(LeastSquares(x, y, ey, model), start)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 1.069 (χ²/ndof = 0.5)\nNfcn = 152\n\n\nEDM = 2.41e-15 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n0.956\n0.026\n\n\n\n\n\n\n\n1\nx1\n1.02\n0.13\n\n\n\n\n\n\n\n2\nx2\n0.47\n0.27\n\n\n\n\n\n\n\n3\nx3\n1.30\n0.34\n\n\n\n\n\n\n\n4\nx4\n0.29\n0.25\n\n\n\n\n\n\n\n5\nx5\n0.10\n0.08\n\n\n\n\n\n\n\n\n\n\n\n\nx0\nx1\nx2\nx3\nx4\nx5\n\n\nx0\n0.000693\n-3.0e-3 (-0.885)\n5.7e-3 (0.800)\n-6.2e-3 (-0.687)\n3.6e-3 (0.551)\n-0.7e-3 (-0.307)\n\n\nx1\n-3.0e-3 (-0.885)\n0.0161\n-0.033 (-0.961)\n0.037 (0.852)\n-0.022 (-0.698)\n0.004 (0.394)\n\n\nx2\n5.7e-3 (0.800)\n-0.033 (-0.961)\n0.0721\n-0.09 (-0.953)\n0.05 (0.822)\n-0.010 (-0.480)\n\n\nx3\n-6.2e-3 (-0.687)\n0.037 (0.852)\n-0.09 (-0.953)\n0.116\n-0.08 (-0.938)\n0.016 (0.583)\n\n\nx4\n3.6e-3 (0.551)\n-0.022 (-0.698)\n0.05 (0.822)\n-0.08 (-0.938)\n0.062\n-0.014 (-0.721)\n\n\nx5\n-0.7e-3 (-0.307)\n0.004 (0.394)\n-0.010 (-0.480)\n0.016 (0.583)\n-0.014 (-0.721)\n0.00649\n\n\n\n\n\n\n \n \n \n \n 2024-02-09T11:14:16.615819\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html", + "href": "2024_LHCb_Starterkit/lecture_2_full.html", + "title": "Statistics II", + "section": "", + "text": "Hans Dembinski, TU Dortmund" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#what-is-an-uncertainty-really", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#what-is-an-uncertainty-really", + "title": "Statistics II", + "section": "What is an uncertainty, really?", + "text": "What is an uncertainty, really?\n\nExample: \\(a = 3 \\pm 0.5\\), where 0.5 is the standard error (jargon: “error” = uncertainty).\nWhat we would like it to mean: “The true value is between 2.5 and 3.5 with a probability of 68 %”.\nWhat it technically means (in most cases): “The interval 2.5 to 3.5, if always constructed with the same procedure, contains the true value in 68 % of identically repeated random experiments.”\nBoth statements coincide only in the asymptotic limit of infinitely large samples." + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#statistical-and-systematic-uncertainties", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#statistical-and-systematic-uncertainties", + "title": "Statistics II", + "section": "Statistical and systematic uncertainties", + "text": "Statistical and systematic uncertainties\n\nStatistical uncertainty\n\nOrigin: We use finite sample instead of infinite distribution\nGoes down as simple size increases\nExample\n\nArithmetic mean: \\(\\bar x = \\frac 1N\\sum_i x_i\\)\nVariance of \\(\\bar x\\): \\(\\hat V_{\\bar x} = \\frac{1}{N - 1} \\frac 1 N\\sum_i (x_i - \\bar x)^2\\)\n\nWe have reliable standard recipes to calculate these\n\n\nVariance or standard deviation?\n\nStandard deviation \\(\\sigma = \\sqrt{V}\\)\nVariance is more fundamental\n\nSecond central moment of distribution\n\\(V = V_1 + V_2\\), but \\(\\sigma = \\sqrt{\\sigma_1^2 + \\sigma_2^2}\\)\n\nCommunicate standard deviations, compute with variances\n\n\n\n\nSystematic uncertainty\n\nOrigin: Imperfect appartus or technique, deviations of reality from simplified models\nQuantifies potential mistakes in analysis\nDoes not go down as sample size increases\nOften correlated\n\nReal-life: Measured the length of ten shoes with ruler than has factory tolerance of 0.1 mm\nLHCb: Measured cross-sections with value of luminosity that has uncertainty of 1-2%\n\nNot well covered in statistics books, but some theoretical results available" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#estimators", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#estimators", + "title": "Statistics II", + "section": "Estimators", + "text": "Estimators\n\nEstimator maps dataset to value \\(\\{x_i \\} \\to \\hat p\\)\n\nNotation: \\(\\hat p\\) is estimate of true value \\(p\\)\nEstimator can have functional form or be an algorithm\nArithmetic mean \\(\\hat \\mu = \\frac 1N \\sum_i x_i\\)\nSample median: Sort \\(x_i\\), pick value in the middle\n\nMany kinds of estimators, some better than others\nOptimal estimators\n\nAs close to true value as possible\nEasy to compute and/or easy to apply to any problem\nRobustness against outliers\n\nIn HEP, we mainly use\n\nPlug-in estimator\nMaximum-likelihood estimator (MLE)" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#bias-and-variance", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#bias-and-variance", + "title": "Statistics II", + "section": "Bias and variance", + "text": "Bias and variance\n\nIn one experiment, we obtain sample \\(\\{ x_i \\}\\) with size \\(N\\) and compute estimate \\(\\hat a\\)\nSample \\(\\{ x_i \\}\\) is random, estimate \\(\\hat a\\) also random\nIf experiment is identically repeated \\(K\\) times, we get sample \\(\\{ x_i \\}_k\\) and estimate \\(\\hat a_k\\) each time\nDerive accuracy of estimator from estimates \\(\\{ \\hat a_1, \\hat a_2, \\dots, \\hat a_K \\}\\)\n\nBias of \\(\\hat a\\) = \\(\\frac1K \\sum_i (\\hat a_i - a)\\)\nVariance of \\(\\hat a\\) = \\(\\frac1{K-1}\\sum_i (\\hat a_i - \\frac 1 K \\sum_k \\hat a_k)^2\\)\n\nExample for a biased estimator: \\(\\hat V = \\frac1N\\sum_i (x_i - \\hat\\mu)^2\\) is biased, if \\(\\hat \\mu\\) is estimated from same sample\n\n\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\nmu = 1\nsigma = 1\n\nN = 10\nK = 10000\nvar_1, var_2 = [], []\nfor k in range(K):\n rng = np.random.default_rng(seed=k)\n x_k = rng.normal(mu, sigma, size=N)\n v1 = np.var(x_k)\n v2 = np.mean((x_k - mu) ** 2)\n var_1.append(v1)\n var_2.append(v2)\n\n\nfig, ax = plt.subplots(1, 2, sharex=True, sharey=True, figsize=(10, 4))\nax[0].hist(var_1, bins=50, label=\"$\\hat V_1$\");\nax[1].hist(var_2, bins=50, label=\"$\\hat V_2$\");\nfor axi in ax:\n axi.axvline(sigma ** 2, color=\"C3\", label=\"true V\")\nax[0].axvline(np.mean(var_1), color=\"C1\", ls=\"--\", label=\"mean of $\\hat V_1$\")\nax[1].axvline(np.mean(var_2), color=\"C1\", ls=\"--\", label=\"mean of $\\hat V_2$\")\nfor axi in ax:\n axi.legend()\n\n\n\n\n\nWell-known bias correction for this case: \\[\n\\hat V_{1,\\text{corr}} = \\frac N{N-1}\\hat V_1\n\\] For derivation see e.g. F. James book (references at end)\n\nExercise * Compute \\(\\hat V_{1, \\text{corr}}\\) and compare its mean with true \\(V\\)\n\n# do exercise here\n\n\n# solution\n\nvar_1_corr = np.multiply(var_1, N / (N - 1))\n\nprint(f\"<hat V_1> = {np.mean(var_1):.2f}\")\nprint(f\"<hat V_1(corrected)> = {np.mean(var_1_corr):.2f}\")\nprint(f\"truth = {sigma**2:.2f}\")\n\n<hat V_1> = 0.91\n<hat V_1(corrected)> = 1.01\ntruth = 1.00\n\n\n\nPlug-in estimator\n\nPlug-in principle: replace true values in some formula with estimates, e.g.\n\n\\[\nc = g(a, b, \\dots) \\to \\hat c = g(\\hat a, \\hat b, \\dots)\n\\]\n\nJustification\n\nFor \\(N\\to \\infty\\), \\(\\Delta a = \\hat a - a \\to 0\\) for any reasonable estimator\n\\(g(\\hat a) = g(a + \\Delta a) = g(a) + O(\\Delta a) \\to g(a)\\) for smooth \\(g\\)\n\nExample: \\(\\sqrt{N}\\) uncertainty estimator for Poisson-distributed count \\(N\\)\n\n\\(P(N; \\lambda) = e^{-\\lambda} \\lambda^N / N!\\)\nVariance of counts: \\(V_N = \\lambda\\)\nEstimator \\(\\hat \\lambda = \\frac 1 K \\sum_k N_k\\); for \\(K = 1\\), \\(\\hat \\lambda = N\\)\nPlug-in principle: \\(\\hat V_N = \\hat \\lambda = N\\)\nStandard deviation: \\(\\hat \\sigma_N = \\sqrt{\\hat V_N} = \\sqrt{N}\\)\n\nPlug-in estimators can behave poorly in small samples\nPoisson example: \\(\\lambda = 0.01\\), sample \\(N = 0 \\to \\hat \\sigma_N = 0\\)" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#bootstrapping-bias-and-variance-of-estimators", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#bootstrapping-bias-and-variance-of-estimators", + "title": "Statistics II", + "section": "Bootstrapping bias and variance of estimators", + "text": "Bootstrapping bias and variance of estimators\n\nBootstrap method: generic way to compute bias and variance of any estimator\n\n\nParametric bootstrap\n\nAvailable:\n\nSample \\(\\{ x_i \\}\\) distributed along \\(f(x; a)\\), \\(a\\) is unknown\nEstimator for \\(\\hat a\\)\n\nWant to know bias and variance of estimator\nSimulate experiment \\(K\\) times\n\nDraw samples \\(\\{ x_i \\}_k\\) of equal size from \\(\\hat f(x; \\hat a)\\) (plugin-estimate)\nCompute \\(\\hat a_k\\)\nBias \\(\\frac 1 K \\sum_k (\\hat a_k - \\hat a)\\)\nVariance \\(\\frac 1 {K-1} \\sum_k (\\hat a_k - \\frac 1K \\sum_k \\hat a_k)^2\\)\n\n\n\nfrom numba_stats import norm, truncexpon, poisson\n\n\ndef make_data(s, mu, sigma, b, slope, seed):\n sr = poisson.rvs(s, size=1, random_state=seed)[0]\n br = poisson.rvs(b, size=1, random_state=seed)[0]\n s = norm.rvs(mu, sigma, size=sr, random_state=seed)\n b = truncexpon.rvs(0, 1, 0, slope, size=br, random_state=seed)\n return np.append(s, b)\n\ntruth = {\n \"s\": 1000,\n \"mu\": 0.5,\n \"sigma\": 0.05,\n \"b\": 500,\n \"slope\": 1,\n}\n\n\nx = make_data(**truth, seed=0)\n\nw, xe = plt.hist(x, bins=100)[:2];\n\n\n\n\n\nfrom iminuit import Minuit\nfrom iminuit.cost import ExtendedBinnedNLL\nfrom numba_stats import norm, truncexpon\n\ndef model(x, s, mu, sigma, b, slope):\n return s * norm.cdf(x, mu, sigma) + b * truncexpon.cdf(x, 0, 1, 0, slope)\n\nnll = ExtendedBinnedNLL(w, xe, model)\nm = Minuit(nll, **truth)\nm.limits[\"mu\"] = (0, 1)\nm.limits[\"s\", \"b\", \"sigma\", \"slope\"] = (0, None)\n\n\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 106.9 (χ²/ndof = 1.1)\nNfcn = 101\n\n\nEDM = 9.95e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\ns\n1000\n35\n\n\n0\n\n\n\n\n1\nmu\n0.4983\n0.0018\n\n\n0\n1\n\n\n\n2\nsigma\n0.0482\n0.0014\n\n\n0\n\n\n\n\n3\nb\n510\n27\n\n\n0\n\n\n\n\n4\nslope\n0.95\n0.15\n\n\n0\n\n\n\n\n\n\n\n\n\ns\nmu\nsigma\nb\nslope\n\n\ns\n1.2e+03\n-48.4e-6\n8.9237e-3 (0.177)\n-0.2e3 (-0.213)\n-0.368 (-0.073)\n\n\nmu\n-48.4e-6\n3.09e-06\n-0 (-0.005)\n61.5e-6 (0.001)\n-17.0e-6 (-0.066)\n\n\nsigma\n8.9237e-3 (0.177)\n-0 (-0.005)\n2.12e-06\n-8.9641e-3 (-0.231)\n-16.5e-6 (-0.078)\n\n\nb\n-0.2e3 (-0.213)\n61.5e-6 (0.001)\n-8.9641e-3 (-0.231)\n711\n0.354 (0.091)\n\n\nslope\n-0.368 (-0.073)\n-17.0e-6 (-0.066)\n-16.5e-6 (-0.078)\n0.354 (0.091)\n0.0213\n\n\n\n\n\n\n \n \n \n \n 2024-02-12T18:09:38.188990\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nFit can be written as estimator\n\n\ndef estimator(x):\n w = np.histogram(x, bins=xe)[0]\n nll = ExtendedBinnedNLL(w, xe, model)\n m = Minuit(nll, **truth)\n m.limits[\"mu\"] = (0, 1)\n m.limits[\"s\", \"b\", \"sigma\", \"slope\"] = (0, None)\n m.migrad()\n assert m.valid\n return m.values\n\n\nreplicates = []\nfor seed in range(100):\n x_k = make_data(*m.values, seed=seed+1)\n p_k = estimator(x_k)\n replicates.append(p_k)\n\nreplicates = np.transpose(replicates)\n\nExercise\n\nCompute bias and standard deviation of replicates for each parameter\nCompare standard deviation of replicates with Minuit’s uncertainty estimate, accessible via m.errors\n\n\n# do exercise here\n\n\n# solution\n\nfor i, name in enumerate(m.parameters):\n p = m.values[i]\n r = replicates[i]\n bias = np.mean(r - p)\n std = np.std(r, ddof=1)\n print(f\"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g} \"\n f\"bias/𝜎 = {bias / std:5.2f}\")\n\ns 𝜎(Minuit) = 35 𝜎(bootstrap) = 34 bias/𝜎 = -0.06\nmu 𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017 bias/𝜎 = -0.07\nsigma 𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0016 bias/𝜎 = -0.03\nb 𝜎(Minuit) = 27 𝜎(bootstrap) = 24 bias/𝜎 = -0.12\nslope 𝜎(Minuit) = 0.15 𝜎(bootstrap) = 0.16 bias/𝜎 = -0.09" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#nonparametric-bootstrap", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#nonparametric-bootstrap", + "title": "Statistics II", + "section": "Nonparametric bootstrap", + "text": "Nonparametric bootstrap\n\nCan do the same by sampling directly from \\(\\{ x_i \\}\\)\nStandard method for samples of fixed size \\(N\\):\n\nPick \\(N\\) random samples from \\(\\{ x_i \\}\\) with replacement to obtain \\(\\{ x_i \\}_k\\)\nCompute \\(\\hat a_k\\) from \\(\\{ x_i \\}_k\\) and proceed as before\n\n“Extended method” for samples of variable size (Poisson):\n\nRandomly draw \\(M\\) from Poisson distribution with \\(\\lambda = N\\)\nPick \\(M\\) random samples from \\(\\{ x_i \\}\\) with replacement to obtain \\(\\{ x_i \\}_k\\)\n…\n\nConvenient: use same algorithms for any distribution\nEfficient implementations for both algorithms in resample\n\n\n\nimport resample\n\nresample.__version__\n\n'1.7.1'\n\n\n\nfrom resample.bootstrap import resample\n\n# generate replicated samples of varying size (Poisson distributed)\nfor x_k in resample(x, size=100, method=\"extended\", random_state=1):\n print(x_k)\n break\n\n[0.58820262 0.58820262 0.52000786 ... 0.38532216 0.26892968 0.26892968]\n\n\nExercise\n\nAgain compute replicated fit results using resampe and store results in a list\nCopy previous code where possible\nAgain compute bias and standard deviation of replicates and compare with Minuit’s uncertainty estimate\n\n\n# do exercise here\n\n\n# solution\n\nreplicates = []\nfor x_k in resample(x, size=100, method=\"extended\", random_state=1):\n replicates.append(estimator(x_k))\n\nreplicates = np.transpose(replicates)\n\nfor i, name in enumerate(m.parameters):\n p = m.values[i]\n r = replicates[i]\n bias = np.mean(r - p)\n std = np.std(r, ddof=1)\n print(f\"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g}\"\n f\" bias/𝜎 = {bias / std:5.2f}\")\n\ns 𝜎(Minuit) = 35 𝜎(bootstrap) = 33 bias/𝜎 = -0.18\nmu 𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017 bias/𝜎 = 0.10\nsigma 𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0014 bias/𝜎 = -0.05\nb 𝜎(Minuit) = 27 𝜎(bootstrap) = 27 bias/𝜎 = -0.04\nslope 𝜎(Minuit) = 0.15 𝜎(bootstrap) = 0.17 bias/𝜎 = 0.18\n\n\n\nresample provides shortcut for computing variance\n\n\nfrom resample.bootstrap import variance\n\nvar = variance(estimator, x, method=\"extended\", size=100, random_state=1)\n\nfor i, name in enumerate(m.parameters):\n std = var[i] ** 0.5\n print(f\"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g}\")\n\ns 𝜎(Minuit) = 35 𝜎(bootstrap) = 33\nmu 𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017\nsigma 𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0014\nb 𝜎(Minuit) = 27 𝜎(bootstrap) = 27\nslope 𝜎(Minuit) = 0.15 𝜎(bootstrap) = 0.17\n\n\nExercise\n\nRepeat the variance calculation without using method=\"extended\"\nCompare the uncertainties of \\(s\\) and \\(b\\)\n\n\n# do exercise here\n\n\n# solution\n\nvar2 = variance(estimator, x, size=100, random_state=1)\n\n# uncertainties of s and b are too small, if normal bootstrap is used,\n# since size variation is not simulated\nfor i, name in enumerate(m.parameters):\n std = var[i] ** 0.5\n std2 = var2[i] ** 0.5\n print(f\"{name:8} 𝜎(extended bootstrap) = {std:6.2g} \"\n f\"𝜎(normal bootstrap) = {std2:6.2g}\")\n\ns 𝜎(extended bootstrap) = 33 𝜎(normal bootstrap) = 24\nmu 𝜎(extended bootstrap) = 0.0017 𝜎(normal bootstrap) = 0.0018\nsigma 𝜎(extended bootstrap) = 0.0014 𝜎(normal bootstrap) = 0.0016\nb 𝜎(extended bootstrap) = 27 𝜎(normal bootstrap) = 24\nslope 𝜎(extended bootstrap) = 0.17 𝜎(normal bootstrap) = 0.17\n\n\n\nBootstrap can be applied to most statistical problems in HEP\nRestriction: samples must be identically and independently distributed\nMay work well even with small samples (< 10)" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#error-propagation", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#error-propagation", + "title": "Statistics II", + "section": "Error propagation", + "text": "Error propagation\n\nFull analysis typically combines many intermediate results with uncertainties into final result\nConsider \\(\\vec b = g(\\vec a)\\) with uncertainties in \\(\\vec a\\)\nTwo ways of computing uncertainties of \\(\\vec b\\)\n\nMonte-Carlo simulation\nError propagation\n\nCan be equally used for statistical and systematic uncertainties\n\n\nMonte-Carlo simulation\n\nDraw variations of fit result from multivariate normal distribution\nCompute variable of interest from variations\nCompute standard deviation or covariance of\nExample: compute uncertainty of \\(\\hat N = \\hat s + \\hat b\\)\n\n\nrng = np.random.default_rng(1)\na_k = rng.multivariate_normal(m.values, m.covariance, size=1000)\n\ndef g(a):\n return a[0] + a[3]\n\nb_k = np.apply_along_axis(g, -1, a_k)\n\nprint(f\"𝜎(N, simulation) = {np.std(b_k):.3g}\")\nprint(f\"𝜎(N, naive) = {(m.errors[0]**2 + m.errors[3]**2)**0.5:.3g}\")\nprint(f\"𝜎(N, expected) = {np.sqrt(len(x)):.3g}\")\n\n𝜎(N, simulation) = 37.7\n𝜎(N, naive) = 43.7\n𝜎(N, expected) = 38.8\n\n\n\n\\(\\sigma_{N,\\text{simulation}}^2 < \\sigma_s^2 + \\sigma_b^2\\), because \\(s_k\\) and \\(b_k\\) are anti-correlated\n\n\n\nError propagation\n\nGiven: \\(\\vec a\\) with covariance matrix \\(C_a\\) and differentiable function \\(\\vec b = g(\\vec a)\\)\n\\(\\vec b\\) can have different dimensionality from \\(\\vec a\\)\nOne can derive via first-order Taylor expansion: \\[\nC_b = J \\, C_a \\, J^T \\text{ with } J_{ik} = \\frac{\\partial g_i}{\\partial a_k}\n\\]\nUseful special cases\n\nSum of scaled independent random values \\(c = \\alpha a + \\beta b\\) (\\(\\alpha, \\beta\\) are constant) \\[\n \\sigma^2_c = \\alpha^2 \\sigma^2_a + \\beta^2 \\sigma^2_b\n \\]\nProduct of powers of independent random values \\(c = a^\\alpha \\, b^\\beta\\) \\[\n \\left(\\frac{\\sigma_c}{c}\\right)^2 = \\alpha^2 \\left(\\frac{\\sigma_a}{a}\\right)^2 + \\beta^2 \\left(\\frac{\\sigma_b}{b}\\right)^2\n \\]\n\nFinal uncertainty often dominated by one term\nWhen estimating systematic uncertainties: put your effort in estimating largest term and be lax on others\nComputation of \\(J\\) by hand error-prone and tedious\njacobi library offers automatic error propagation based on a robust numerical derivative\n\n\n\nimport jacobi\n\njacobi.__version__\n\n'0.9.2'\n\n\n\nb, cov_b = jacobi.propagate(g, m.values, m.covariance)\n\nf\"𝜎(N, error propagation) = {cov_b ** 0.5:.3g} 𝜎(N, expected) = {np.sqrt(len(x)):.3g}\"\n\n'𝜎(N, error propagation) = 38.9 𝜎(N, expected) = 38.8'\n\n\n\njacobi.propagate works for any differentiable mapping, which even includes most fits!\n\n\nfrom iminuit.cost import LeastSquares\n\ndef h(pvec):\n rng = np.random.default_rng(1)\n a, b = pvec\n x = np.array([1, 2, 3])\n y = rng.normal(a + b * x, 0.1)\n m = Minuit(LeastSquares(x, y, 0.1, lambda x, a, b: a + b * x), a=0, b=0)\n m.strategy = 0\n m.migrad()\n assert m.valid\n return m.values\n\njacobi.propagate(h, (1, 2), (1, 1))\n\n(array([1.05143603, 1.99924264]),\n array([[1.00000000e+00, 1.44351198e-12],\n [1.44351198e-12, 1.00000000e+00]]))" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#systematic-uncertainties-known-unknowns", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#systematic-uncertainties-known-unknowns", + "title": "Statistics II", + "section": "Systematic uncertainties: Known unknowns", + "text": "Systematic uncertainties: Known unknowns\n“There are known knowns, things we know that we know; and there are known unknowns, things that we know we don’t know. But there are also unknown unknowns, things we do not know we don’t know.” Donald Rumsfeld\n\nWhy data analyses take so long: study data, detector, and methods to turn unknown unknowns into known knowns or known unknowns\nknown unknowns = systematic uncertainties\nNo formal way, but generally…\n\nWorry * What could go wrong? * What are my assumptions?\nExample * Efficiency or correction factor taken from simulation, but simulation differs from real experiment * Solution: perform control measurement to estimate how much simulation differs from real experiment\nMethods * Use methods that are known to be optimal for your problem or derive them from first principles * General purpose methods often safer to use than special-case methods * Maximum-likelihood fit > least-squares fit * Bootstrap estimate of uncertainty > other uncertainty estimates\nApply checks * Validate whole analysis on simulated data (sometimes you can use toy simulation) * Split data by magnet polarity, fill, etc., analyze splits separately, check for agreement\nGuidelines for handling systematic errors * Roger Barlow: Systematic Errors: Facts and Fictions (2002) * Roger Barlow, “Practical Statistics for Particle Physics” (2019), Section 6.5" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2_full.html#further-resources-for-learning", + "href": "2024_LHCb_Starterkit/lecture_2_full.html#further-resources-for-learning", + "title": "Statistics II", + "section": "Further resources for learning", + "text": "Further resources for learning\n\nLHCb Statistics and ML WG\n\nTWiki page\nStatistics Guidelines to collect information about best practices\n\n\n\nIntroductory statistics books\n\nList of books on TWiki pages of Statistis and Machine Learning WG\nBooks I personally like a lot\n\nGlen Cowan, “Statistical Data Analysis”, Oxford University Press, 1998\nF. James, “Statistical Methods in Experimental Physics”, 2nd edition, World Scientific, 2006\nR. Barlow, “Statistics: a Guide to the Use of Statistical Methods in the Physical Sciences”, Wiley, 1989\nB. Efron, R.J. Tibshirani, “An Introduction to the Bootstrap”, CRC Press, 1994\n\n\n\n\nLectures, papers, notebooks\n\nRoger Barlow\n\nIntroductory lecture (2019)\nSystematic Errors: Facts and Fictions (2002)\n“Practical Statistics for Particle Physics” (2019), Section 6.5\n\nMore Jupyter notebooks\n\nMatt Kenzie, PDG averaging\nMatt Kenzie, MLE uncertainties\niminuit tutorials on various fitting topics\n\nAll about sWeights and COWs with gentle derivation\n\nHD, M. Kenzie, C. Langenbruch, M. Schmelling, Nucl.Instrum.Meth.A 1040 (2022) 167270" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html", + "href": "2024_LHCb_Starterkit/lecture_1.html", + "title": "Statistics I", + "section": "", + "text": "Hans Dembinski, TU Dortmund" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#about-me", + "href": "2024_LHCb_Starterkit/lecture_1.html#about-me", + "title": "Statistics I", + "section": "About me", + "text": "About me\n\nWorking with the LHCb experiment since 2016\nBefore astroparticle physicist working with Pierre Auger Observatory and IceCube\nStatistics convener in LHCb (2019-2021, 2024-now)\nInterests\n\nStatistics\nHigh-performance computing\nAPI design: “Make Interfaces Easy to Use Correctly and Hard to Use Incorrectly” Scott Meyers" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#scikit-hep-project", + "href": "2024_LHCb_Starterkit/lecture_1.html#scikit-hep-project", + "title": "Statistics I", + "section": "Scikit-HEP project", + "text": "Scikit-HEP project\n\nProvides high-performance data analysis tools in Python\nPartially funded by NSF grants\nSeveral LHCb developers: Eduardo Rodrigues, Chris Burr, myself, …\nCore values\n\nSpecialized tools that integrate well with existing Python ecosystem\nEasy to use for beginners\nFlexible for power users\nGood documentation\nEasy installation" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#the-power-of-combinatorics", + "href": "2024_LHCb_Starterkit/lecture_1.html#the-power-of-combinatorics", + "title": "Statistics I", + "section": "The power of combinatorics", + "text": "The power of combinatorics\nBuild everything from basic components that fit together\n\nImage credit: Toni Zaat on Unsplash" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#important-packages", + "href": "2024_LHCb_Starterkit/lecture_1.html#important-packages", + "title": "Statistics I", + "section": "Important packages", + "text": "Important packages\n\nPython scientific stack\n\nNumpy: fast computation with arrays\nScipy: integration, statistical distributions, special functions, …\nMatplotlib: scientific plotting\nNumba: JIT Compiler for Python (very fast)\nPandas: Convenient processing of tabular data\n\nScikit-HEP Project\n\nuproot: fast reading and writing of ROOT Trees\nparticle: get particle properties from PDG IDs\nboost-histogram: multi-dimensional generalised histograms\niminuit: fitting and error computation package\nresample: Easy bootstrapping to compute uncertainty and bias\npyhf: for histogram fitting/limit setting and preserving likelihoods\n\nOther packages\n\njacobi: Easy numerical error propagation\nnumba-stats: Fast statistical distributions for fitting\ntabulate: Convert data tables to various formats (LaTeX, HTML, Markdown, …)\nscikit-learn: Basic machine learning tools\nxgboost: Gradient-boosting machine library that won many Kaggle competitions\n\n\nInstall everything with pip install LIBRARY" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#working-with-jupyter-notebooks", + "href": "2024_LHCb_Starterkit/lecture_1.html#working-with-jupyter-notebooks", + "title": "Statistics I", + "section": "Working with Jupyter notebooks", + "text": "Working with Jupyter notebooks\n\nI recommend to use vscode\n\nVery good Jupyter notebook support\nRemote explorer: edit and run code on remote machines" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#prepare-computing-environment", + "href": "2024_LHCb_Starterkit/lecture_1.html#prepare-computing-environment", + "title": "Statistics I", + "section": "Prepare computing environment", + "text": "Prepare computing environment\n\nIn terminal, create and enter virtual environment (you need Python-3.8 or later)\npython3 -m venv .venv\nsource .venv/bin/activate\nInstall packages\npython -m pip install --upgrade numba matplotlib uproot boost-histogram iminuit scipy particle numba_stats ipywidgets\nDownload this notebook from indico and example.root, put latter into same folder\nRun vscode on the current folder, then open the notebook inside the editor OR\nRun\npython -m pip install ipykernel\npython -m ipykernel install --name \"starterkit\"\njupyter notebook lecture_1.ipynb\nThen select starterkit as kernel\n\n\n\nuproot provides fast and convenient access to ROOT trees\n\n\n\nimport uproot\n\nprint(f\"uproot {uproot.__version__}\")\n\nuproot 5.2.2\n\n\n\nf = uproot.open(\"example.root\")\nevent = f[\"event\"]\n \nevent.show()\n\nname | typename | interpretation \n---------------------+--------------------------+-------------------------------\ntrk_len | int32_t | AsDtype('>i4')\nmc_trk_len | int32_t | AsDtype('>i4')\ntrk_imc | int32_t[] | AsJagged(AsDtype('>i4'))\ntrk_px | float[] | AsJagged(AsDtype('>f4'))\ntrk_py | float[] | AsJagged(AsDtype('>f4'))\ntrk_pz | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_px | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_py | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_pz | float[] | AsJagged(AsDtype('>f4'))\nmc_trk_pid | int32_t[] | AsJagged(AsDtype('>i4'))\n\n\n\nTree contains fake simulated LHCb events\nReconstructed tracks in simulation, branches with trk_*\nTruth in simulation, branches with mc_trk_*\nSome branches are 2D (e.g. trk_px)\n\nFirst index iterates over events\nSecond index iterates over tracks per event\n\nMathematical operators and slicing works on these arrays similar to Numpy\nMany numpy functions work on awkward arrays\n\n\nimport numpy as np\nnp.__version__\n\n'1.24.4'\n\n\n\ndata = event.arrays()\n\nnp.sum(data[\"trk_len\"])\n\n5032\n\n\nExercise\n\nUse np.sum and an array mask to count how many events have zero tracks\n\n\n# do exercise here\n\n\nFor some operations and awkward arrays, we need the awkward library\n\n\nimport awkward as ak\nak.__version__\n\n'2.6.1'\n\n\n\nFor plotting, we use matplotlib\n\n\nimport matplotlib\nimport matplotlib.pyplot as plt\nmatplotlib.__version__\n\n'3.8.2'\n\n\nExercise * Plot the branch trk_px with matplotlib.pyplot.hist * Can you figure out from the error message why it does not work? * Fix the issue with the function ak.flatten\n\n# do exercise here" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#reading-data-from-a-large-tree", + "href": "2024_LHCb_Starterkit/lecture_1.html#reading-data-from-a-large-tree", + "title": "Statistics I", + "section": "Reading data from a large tree", + "text": "Reading data from a large tree\n\nTree often do not fit into memory (tree size > 100 GB and more possible)\nIterate over chunks\nChunk-wise processing works very well with histograms\n\n\n\nPython frontend to Boost.Histogram library in C++ from the Boost project\nVery fast and flexible\nMulti-dimensional histograms and other binned statistics (profiles!)\nSupports weighted data\nAnd much much more, see docs\n\n\nimport boost_histogram as bh\nbh.__version__\n\n'1.4.0'\n\n\n\n# make an axis\nxaxis = bh.axis.Regular(20, -2, 2)\n\n# easy to make several histograms with same binning by reusing axis\nh_px = bh.Histogram(xaxis)\nh_mc_px = bh.Histogram(xaxis)\n\n\n# incremental filling, only read branch \"trk_px\" from tree\nfor data in event.iterate([\"trk_px\"]):\n h_px.fill(ak.flatten(data[\"trk_px\"]))\n\n\nh_px\n\nHistogram(Regular(20, -2, 2), storage=Double()) # Sum: 5032.0\n\n\n\nprint(h_px)\n\n ┌───────────────────────────────────────────────────────────┐\n[-inf, -2) 0 │ │\n[ -2, -1.8) 1 │▏ │\n[-1.8, -1.6) 3 │▎ │\n[-1.6, -1.4) 17 │█▎ │\n[-1.4, -1.2) 25 │█▊ │\n[-1.2, -1) 78 │█████▌ │\n[ -1, -0.8) 181 │████████████▉ │\n[-0.8, -0.6) 308 │█████████████████████▊ │\n[-0.6, -0.4) 453 │████████████████████████████████▏ │\n[-0.4, -0.2) 647 │█████████████████████████████████████████████▉ │\n[-0.2, 0) 819 │██████████████████████████████████████████████████████████ │\n[ 0, 0.2) 782 │███████████████████████████████████████████████████████▍ │\n[ 0.2, 0.4) 629 │████████████████████████████████████████████▌ │\n[ 0.4, 0.6) 480 │██████████████████████████████████ │\n[ 0.6, 0.8) 323 │██████████████████████▉ │\n[ 0.8, 1) 159 │███████████▎ │\n[ 1, 1.2) 77 │█████▌ │\n[ 1.2, 1.4) 37 │██▋ │\n[ 1.4, 1.6) 7 │▌ │\n[ 1.6, 1.8) 3 │▎ │\n[ 1.8, 2) 3 │▎ │\n[ 2, inf) 0 │ │\n └───────────────────────────────────────────────────────────┘\n\n\n\n# plot histogram with plt.stairs\nplt.stairs(h_px.values(), h_px.axes[0].edges);\n\n\n\n\n\nHistograms can be cut down to size after filling\n\nShrink axis range with slices and using loc\nMake binning coarser with rebin\nSee docs of boost-histogram or extended lecture\n\nBeyond 1D histograms\n\nSupports advanced axis types (e.g. category axis)\nSupports higher dimensional histograms\nSupports generalized histograms with binned statistics (aka “profile” in ROOT)\nSee docs of boost-histogram or extended lecture" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#fits", + "href": "2024_LHCb_Starterkit/lecture_1.html#fits", + "title": "Statistics I", + "section": "Fits", + "text": "Fits\n\nTypical analysis work flow (often automated with Snakemake)\n\nPre-select data\nMake histograms or profiles from pre-selected data\nFit histograms or profiles to extract physical parameters\n\nMany specialized fitting tools for individual purposes, e.g.: scipy.optimize.curve_fit\nGeneric libraries\n\nRooFit\nzfit\npyhf\niminuit\n…\n\nFitting (“estimation” in statistics)\n\nAdjust parametric statistical model to dataset and find model parameters which match dataset best\nAI community calls this “learning”\nEstimate: result of fit\n\nDataset: samples \\(\\{ \\vec x_i \\}\\)\nModel: Probability density or probability mass function \\(f(\\vec x; \\vec p)\\) which depends on unknown parameters \\(\\vec p\\)\nConjecture: maximum-likelihood estimate (MLE) \\(\\hat{\\vec p} = \\text{argmax}_{\\vec p} L(\\vec p)\\) is optimal, with \\(L(\\vec p) = \\prod_i f(\\vec x_i; \\vec p)\\)\n\nEquivalent: minimize negative log-likelihood (NLL), \\(\\hat{\\vec p} = \\text{argmin}_{\\vec p}(-\\ln L(\\vec p))\\)\n\nLimiting case of maximum-likelihood fit: least-squares fit aka chi-square fit\n\n\n\nPython frontend to Minuit2 C++ library maintained by ROOT team at CERN\nMaximum-likelihood and least-squares fits with error estimation\n\n\nimport iminuit\nfrom iminuit import Minuit\niminuit.__version__\n\n'2.25.2'\n\n\n\niminuit vs. other packages\n\npyhf\n\nBetter choice when you need to compute limits and preserve the likelihood\n\nzfit and RooFit\n\nHelp you build statistical models: automatic normalization & convolutions\n\nThis has to be done “by hand” in iminuit\n\nGood for standard tasks, but restrict your freedom\n\niminuit\n\nHas good documentation\nWell-designed API\nIs very flexible and open\nIs very fast with numba\n\n\nTypical fit in particle physics: mass distribution of decay candidates with signal peak over smooth background\nTask: fit signal fraction to calculate signal yield (fraction * number of candidates)\n\n\n# fast implementations of statistical distributions, API similar to scipy.stats\nimport numba_stats\nfrom numba_stats import norm, truncexpon\nnumba_stats.__version__\n\n'1.7.0'\n\n\n\ndef make_data(size, seed=0):\n z=0.5\n mu=0.5\n sigma=0.1\n slope=0.5\n s = norm.rvs(mu, sigma, size=int(z * size), random_state=seed)\n b = truncexpon.rvs(0, 1, 0, slope, size=int((1-z) * size), random_state=seed)\n return np.append(s, b)\n\nx1 = make_data(10_000)\nplt.hist(x1, bins=100);\n\n\n\n\n\nStatistical model assumptions\n\nFull sample is additive mixture of signal sample and background sample\nSignal is normal-distributed; pdf is \\(\\mathcal{N}(\\mu, \\sigma)\\) with parameters \\(\\mu\\) and \\(\\sigma\\)\nBackground is truncated exponential distribution\n\n\n\n# model pdf\ndef model1(x, z, mu, sigma, slope):\n s = norm.pdf(x, mu, sigma)\n b = truncexpon.pdf(x, 0.0, 1.0, 0.0, slope)\n return (1 - z) * b + z * s\n\n# we use NLL implementation provided by iminuit for convenience\nfrom iminuit.cost import UnbinnedNLL\n\nnll1 = UnbinnedNLL(x1, model1)\nm = Minuit(nll1, z=0.5, mu=0.5, sigma=0.1, slope=1)\n\n\n# need to set parameter limits or fit might fail\nm.limits[\"z\", \"mu\"] = (0, 1)\nm.limits[\"sigma\", \"slope\"] = (0, None)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = -4741\nNfcn = 99\n\n\nEDM = 2.03e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n0.489\n0.009\n\n\n0\n1\n\n\n\n1\nmu\n0.4983\n0.0020\n\n\n0\n1\n\n\n\n2\nsigma\n0.0977\n0.0019\n\n\n0\n\n\n\n\n3\nslope\n0.505\n0.016\n\n\n0\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.48e-05\n-2e-6 (-0.102)\n9e-6 (0.535)\n-0.06e-3 (-0.395)\n\n\nmu\n-2e-6 (-0.102)\n3.94e-06\n-0e-6 (-0.124)\n-5e-6 (-0.153)\n\n\nsigma\n9e-6 (0.535)\n-0e-6 (-0.124)\n3.7e-06\n-9e-6 (-0.280)\n\n\nslope\n-0.06e-3 (-0.395)\n-5e-6 (-0.153)\n-9e-6 (-0.280)\n0.000268\n\n\n\n\n\n\n \n \n \n \n 2024-02-13T09:27:46.977103\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nExercise\n\nUse interactive mode via m.interactive() to check how fit reacts to parameter changes\nUseful for debugging and to find starting values by hand\n\n\n# do exercise here\n\n\nyou can compute 2D confidence regions with Minuit.mncontour\n\nExercise * Draw the likelihood profile contour of parameters z and sigma with Minuit.draw_mncontour * Compute matrix of likelihood profile contours with Minuit.draw_mnmatrix\n\n# do exercise here" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#faster-fits", + "href": "2024_LHCb_Starterkit/lecture_1.html#faster-fits", + "title": "Statistics I", + "section": "Faster fits?", + "text": "Faster fits?\n\nOption 1: accelerate NLL computation with numba\nOption 2: use binned data\n\n\nx2 = make_data(1_000_000)\n\n\nOption 1\n\nNLL computation is trivially parallelizable: log(pdf) computed independently for each data point\nNumba allows us to exploit auto-vectorization (SIMD instructions) and parallel computing on multiple cores\nSee full lecture for an example to numba-JIT compile the NLL function\nSee benchmark of iminuit + numba vs. RooFit\n\nOption 2\n\nOften simpler: fit histograms\nNo bias from fitting histograms if done correctly, only loss in precision\nNeed model cdf instead of model pdf: \\(F(x; \\vec p) = \\int_{-\\infty}^x f(x'; \\vec p) \\text{d}x'\\)\n\n\n\nfrom iminuit.cost import BinnedNLL\n\n# model is now a cdf!\ndef model3(x, z, mu, sigma, slope):\n s = norm.cdf(x, mu, sigma)\n b = truncexpon.cdf(x, 0.0, 1.0, 0.0, slope)\n return z * s + (1 - z) * b\n\nw, xe = np.histogram(x2, bins=100, range=(0, 1))\nnll3 = BinnedNLL(w, xe, model3)\n\n\nm = Minuit(nll3, z=0.5, mu=0.5, sigma=1, slope=1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 94.55 (χ²/ndof = 1.0)\nNfcn = 192\n\n\nEDM = 1.76e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nz\n501.0e-3\n0.9e-3\n\n\n\n\n\n\n\n1\nmu\n500.3e-3\n0.2e-3\n\n\n\n\n\n\n\n2\nsigma\n100.06e-3\n0.20e-3\n\n\n\n\n\n\n\n3\nslope\n0.4982\n0.0017\n\n\n\n\n\n\n\n\n\n\n\n\nz\nmu\nsigma\nslope\n\n\nz\n8.91e-07\n-0.02e-6 (-0.103)\n0.10e-6 (0.557)\n-0.7e-6 (-0.422)\n\n\nmu\n-0.02e-6 (-0.103)\n4.03e-08\n-0 (-0.125)\n-0.05e-6 (-0.155)\n\n\nsigma\n0.10e-6 (0.557)\n-0 (-0.125)\n3.98e-08\n-0.10e-6 (-0.307)\n\n\nslope\n-0.7e-6 (-0.422)\n-0.05e-6 (-0.155)\n-0.10e-6 (-0.307)\n2.72e-06\n\n\n\n\n\n\n \n \n \n \n 2024-02-13T09:27:47.235298\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nSee study of binned fits vs. unbinned fit for detailed comparison\nWith sufficient many bins (>= 20 in this case), increase in parameter uncertainties remains negligible\nBinned fits are (100-200)x faster than unbinned fits without using numba" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#other-fits", + "href": "2024_LHCb_Starterkit/lecture_1.html#other-fits", + "title": "Statistics I", + "section": "Other fits", + "text": "Other fits\n\nTo fit yields (counts) instead of fractions, use ExtendedUnbinnedNLL or ExtendedBinnedNLL\nTo fit (x, y) pairs, use LeastSquares\nTo fit simulated templates, use Template\nSee documentation how models should be constructed then" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_1.html#template-fits", + "href": "2024_LHCb_Starterkit/lecture_1.html#template-fits", + "title": "Statistics I", + "section": "Template fits", + "text": "Template fits\n\nSometimes PDF/CDF not known in parametric form, but can be Monte-Carlo simulated\nAnsatz: Fit a histogram of simulation output (template) to data histogram\nAdditional source of uncertainty from finite size of simulation\nCorrect error propagation with Barlow-Beeston method, Barlow and Beeston, Comput.Phys.Commun. 77 (1993) 219-228\niminuit.cost.Template\n\nExtension to (s)weighted data and simuation, H. Dembinski, A. Abdelmotteleb, Eur.Phys.J.C 82 (2022) 11, 1043\nFast approximation to Barlow-Beeston method for unweighted data\nAllows you to mix parametric models and templates\n\n\n\ndef make_data2(rng, nmc, truth, bins):\n xe = np.linspace(0, 1, bins + 1)\n b = np.diff(truncexpon.cdf(xe, 0, 1, 0, 1))\n s = np.diff(norm.cdf(xe, 0.5, 0.05))\n n = rng.poisson(b * truth[0]) + rng.poisson(s * truth[1])\n t = np.array([rng.poisson(b * nmc), rng.poisson(s * nmc)])\n return xe, n, t\n\nrng = np.random.default_rng(1)\ntruth = 750, 250\nxe, n, t = make_data2(rng, 200, truth, 15)\n\n_, ax = plt.subplots(1, 2, figsize=(10, 4))\nax[0].stairs(n, xe, color=\"k\", label=\"data\")\nax[0].legend();\nax[1].stairs(t[0], xe, label=\"background\")\nax[1].stairs(t[1], xe, label=\"signal\")\nax[1].legend(title=\"templates\");\n\n\n\n\n\nfrom iminuit.cost import Template\n\nc = Template(n, xe, t)\nm = Minuit(c, 1, 1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 8.352 (χ²/ndof = 0.6)\nNfcn = 132\n\n\nEDM = 2.75e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n770\n70\n\n\n0\n\n\n\n\n1\nx1\n220\n40\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\n\n\nx0\n4.3e+03\n-0.8e3 (-0.354)\n\n\nx1\n-0.8e3 (-0.354)\n1.32e+03\n\n\n\n\n\n\n \n \n \n \n 2024-02-13T09:27:47.726719\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nWith Template you can also fit a mix of template and parametric model\nOften template only needed for background(s)\n\nExercise\n\nWith make_data2 generate templates with 1 000 000 simulated points\nFit this data with the Template cost function\nCompare parameter uncertainties\n\n\n# do exercise here" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html", + "href": "2024_LHCb_Starterkit/lecture_2.html", + "title": "Statistics II", + "section": "", + "text": "Hans Dembinski, TU Dortmund" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#what-is-an-uncertainty-really", + "href": "2024_LHCb_Starterkit/lecture_2.html#what-is-an-uncertainty-really", + "title": "Statistics II", + "section": "What is an uncertainty, really?", + "text": "What is an uncertainty, really?\n\nExample: \\(a = 3 \\pm 0.5\\), where 0.5 is the standard error (jargon: “error” = uncertainty)\nWhat we would like it to mean: “The true value is between 2.5 and 3.5 with a probability of 68 %”\nWhat it technically means (in most cases): “The interval 2.5 to 3.5, if always constructed with the same procedure, contains the true value in 68 % of identically repeated random experiments.”\nBoth statements coincide only in the asymptotic limit of infinitely large samples" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#statistical-and-systematic-uncertainties", + "href": "2024_LHCb_Starterkit/lecture_2.html#statistical-and-systematic-uncertainties", + "title": "Statistics II", + "section": "Statistical and systematic uncertainties", + "text": "Statistical and systematic uncertainties\n\nStatistical uncertainty\n\nOrigin: We use finite sample instead of infinite distribution\nGoes down as simple size increases\nExample\n\nArithmetic mean: \\(\\bar x = \\frac 1N\\sum_i x_i\\)\nVariance of \\(\\bar x\\): \\(\\hat V_{\\bar x} = \\frac{1}{N - 1} \\frac 1 N\\sum_i (x_i - \\bar x)^2\\)\n\nWe have reliable standard recipes to calculate these\n\n\nVariance or standard deviation?\n\nStandard deviation \\(\\sigma = \\sqrt{V}\\)\nVariance is more fundamental\n\nSecond central moment of distribution\n\\(V = V_1 + V_2\\), but \\(\\sigma = \\sqrt{\\sigma_1^2 + \\sigma_2^2}\\)\n\nCommunicate standard deviations, compute with variances\n\n\n\n\nSystematic uncertainty\n\nOrigin: Imperfect appartus or technique, deviations of reality from simplified models\nQuantifies potential mistakes in analysis\nDoes not go down as sample size increases\nOften correlated\n\nReal-life: Measured the length of ten shoes with ruler than has factory tolerance of 0.1 mm\nLHCb: Measured cross-sections with value of luminosity that has uncertainty of 1-2%\n\nNot well covered in statistics books, but some theoretical results available" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#estimators", + "href": "2024_LHCb_Starterkit/lecture_2.html#estimators", + "title": "Statistics II", + "section": "Estimators", + "text": "Estimators\n\nEstimator maps dataset to value \\(\\{x_i \\} \\to \\hat p\\)\n\nNotation: \\(\\hat p\\) is estimate of true value \\(p\\)\nEstimator can have functional form or be an algorithm\nArithmetic mean \\(\\hat \\mu = \\frac 1N \\sum_i x_i\\)\nSample median: Sort \\(\\{ x_i \\}\\), pick value in the middle\n\nMany kinds of estimators, some better than others\nOptimal estimators\n\nAs close to true value as possible\nEasy to compute and/or easy to apply to any problem\nRobustness against outliers\n\nIn HEP, we mainly use\n\nPlug-in estimator\nMaximum-likelihood estimator (MLE)" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#bias-and-variance", + "href": "2024_LHCb_Starterkit/lecture_2.html#bias-and-variance", + "title": "Statistics II", + "section": "Bias and variance", + "text": "Bias and variance\n\nIn one experiment, we obtain sample \\(\\{ x_i \\}\\) with size \\(N\\) and compute estimate \\(\\hat a\\)\nSample \\(\\{ x_i \\}\\) is random, estimate \\(\\hat a\\) also random\nIf experiment is identically repeated \\(K\\) times, we get sample \\(\\{ x_i \\}_k\\) and estimate \\(\\hat a_k\\) each time\nDerive accuracy of estimator from estimates \\(\\{ \\hat a_1, \\hat a_2, \\dots, \\hat a_K \\}\\)\n\nBias of \\(\\hat a\\) = \\(\\frac1K \\sum_i (\\hat a_i - a)\\)\nVariance of \\(\\hat a\\) = \\(\\frac 1 {K -1}\\sum_i (\\hat a_i - \\frac 1K \\sum_k \\hat a_k)^2\\)\n\nExample for biased estimator: \\(\\hat V = \\frac1N\\sum_i (x_i - \\hat\\mu)^2\\) is biased if \\(\\hat \\mu\\) is estimated from same sample\n\n\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\nmu = 1\nsigma = 1\n\nN = 10\nK = 10000\nvar_1, var_2 = [], []\nfor k in range(K):\n rng = np.random.default_rng(seed=k)\n x_k = rng.normal(mu, sigma, size=N)\n v1 = np.var(x_k)\n v2 = np.mean((x_k - mu) ** 2)\n var_1.append(v1)\n var_2.append(v2)\n\n\nfig, ax = plt.subplots(1, 2, sharex=True, sharey=True, figsize=(10, 4))\nax[0].hist(var_1, bins=50, label=\"$\\hat V_1$\");\nax[1].hist(var_2, bins=50, label=\"$\\hat V_2$\");\nfor axi in ax:\n axi.axvline(sigma ** 2, color=\"C3\", label=\"true V\")\nax[0].axvline(np.mean(var_1), color=\"C1\", ls=\"--\", label=\"mean of $\\hat V_1$\")\nax[1].axvline(np.mean(var_2), color=\"C1\", ls=\"--\", label=\"mean of $\\hat V_2$\")\nfor axi in ax:\n axi.legend()\n\n\n\n\n\nWell-known bias correction for this case: \\[\n\\hat V_{1,\\text{corr}} = \\frac N{N-1}\\hat V_1\n\\] For derivation see e.g. F. James book (references at end)\n\nExercise * Compute \\(\\hat V_{1, \\text{corr}}\\) and compare its mean with true \\(V\\)\n\n# do exercise here\n\n\nPlug-in estimator\n\nPlug-in principle: replace true values in some formula with estimates, e.g.\n\n\\[\nc = g(a, b, \\dots) \\to \\hat c = g(\\hat a, \\hat b, \\dots)\n\\]\n\nJustification\n\nFor \\(N\\to \\infty\\), \\(\\Delta a = \\hat a - a \\to 0\\) for any reasonable estimator\n\\(g(\\hat a) = g(a + \\Delta a) = g(a) + O(\\Delta a) \\to g(a)\\) for smooth \\(g\\)\n\nExample: \\(\\sqrt{N}\\) uncertainty estimator for Poisson-distributed count \\(N\\)\n\n\\(P(N; \\lambda) = e^{-\\lambda} \\lambda^N / N!\\)\nVariance of counts: \\(V_N = \\lambda\\)\nEstimator \\(\\hat \\lambda = \\frac 1 K \\sum_k N_k\\); for \\(K = 1\\), \\(\\hat \\lambda = N\\)\nPlug-in principle: \\(\\hat V_N = \\hat \\lambda = N\\)\nStandard deviation: \\(\\hat \\sigma_N = \\sqrt{\\hat V_N} = \\sqrt{N}\\)\n\nPlug-in estimators can behave poorly in small samples\nPoisson example: \\(\\lambda = 0.01\\), sample \\(N = 0 \\to \\hat \\sigma_N = 0\\)" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#bootstrapping-bias-and-variance-of-estimators", + "href": "2024_LHCb_Starterkit/lecture_2.html#bootstrapping-bias-and-variance-of-estimators", + "title": "Statistics II", + "section": "Bootstrapping bias and variance of estimators", + "text": "Bootstrapping bias and variance of estimators\n\nBootstrap method: generic way to compute bias and variance of any estimator\n\n\nParametric bootstrap\n\nAvailable:\n\nSample \\(\\{ x_i \\}\\) distributed along \\(f(x; a)\\), \\(a\\) is unknown\nEstimator for \\(\\hat a\\)\n\nWant to know bias and variance of estimator\nSimulate experiment \\(K\\) times\n\nDraw samples \\(\\{ x_i \\}_k\\) of equal size from \\(\\hat f(x; \\hat a)\\) (plugin-estimate)\nCompute \\(\\hat a_k\\)\nBias \\(\\frac 1 K \\sum_k (\\hat a_k - \\hat a)\\)\nVariance \\(\\frac 1 {K-1} \\sum_k (\\hat a_k - \\frac 1K \\sum_k \\hat a_k)^2\\)\n\n\n\nfrom numba_stats import norm, truncexpon, poisson\n\n\ndef make_data(s, mu, sigma, b, slope, seed):\n sr = poisson.rvs(s, size=1, random_state=seed)[0]\n br = poisson.rvs(b, size=1, random_state=seed)[0]\n s = norm.rvs(mu, sigma, size=sr, random_state=seed)\n b = truncexpon.rvs(0, 1, 0, slope, size=br, random_state=seed)\n return np.append(s, b)\n\ntruth = {\n \"s\": 1000,\n \"mu\": 0.5,\n \"sigma\": 0.05,\n \"b\": 500,\n \"slope\": 1,\n}\n\n\nx = make_data(**truth, seed=0)\n\nw, xe = plt.hist(x, bins=100)[:2];\n\n\n\n\n\nfrom iminuit import Minuit\nfrom iminuit.cost import ExtendedBinnedNLL\nfrom numba_stats import norm, truncexpon\n\ndef model(x, s, mu, sigma, b, slope):\n return s * norm.cdf(x, mu, sigma) + b * truncexpon.cdf(x, 0, 1, 0, slope)\n\nnll = ExtendedBinnedNLL(w, xe, model)\nm = Minuit(nll, **truth)\nm.limits[\"mu\"] = (0, 1)\nm.limits[\"s\", \"b\", \"sigma\", \"slope\"] = (0, None)\n\n\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 106.9 (χ²/ndof = 1.1)\nNfcn = 101\n\n\nEDM = 9.95e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\ns\n1000\n35\n\n\n0\n\n\n\n\n1\nmu\n0.4983\n0.0018\n\n\n0\n1\n\n\n\n2\nsigma\n0.0482\n0.0014\n\n\n0\n\n\n\n\n3\nb\n510\n27\n\n\n0\n\n\n\n\n4\nslope\n0.95\n0.15\n\n\n0\n\n\n\n\n\n\n\n\n\ns\nmu\nsigma\nb\nslope\n\n\ns\n1.2e+03\n-48.4e-6\n8.9237e-3 (0.177)\n-0.2e3 (-0.213)\n-0.368 (-0.073)\n\n\nmu\n-48.4e-6\n3.09e-06\n-0 (-0.005)\n61.5e-6 (0.001)\n-17.0e-6 (-0.066)\n\n\nsigma\n8.9237e-3 (0.177)\n-0 (-0.005)\n2.12e-06\n-8.9641e-3 (-0.231)\n-16.5e-6 (-0.078)\n\n\nb\n-0.2e3 (-0.213)\n61.5e-6 (0.001)\n-8.9641e-3 (-0.231)\n711\n0.354 (0.091)\n\n\nslope\n-0.368 (-0.073)\n-17.0e-6 (-0.066)\n-16.5e-6 (-0.078)\n0.354 (0.091)\n0.0213\n\n\n\n\n\n\n \n \n \n \n 2024-02-13T09:47:06.936461\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\nFit can be written as estimator\n\n\ndef estimator(x):\n w = np.histogram(x, bins=xe)[0]\n nll = ExtendedBinnedNLL(w, xe, model)\n m = Minuit(nll, **truth)\n m.limits[\"mu\"] = (0, 1)\n m.limits[\"s\", \"b\", \"sigma\", \"slope\"] = (0, None)\n m.migrad()\n assert m.valid\n return m.values\n\n\nreplicates = []\nfor seed in range(100):\n x_k = make_data(*m.values, seed=seed+1)\n p_k = estimator(x_k)\n replicates.append(p_k)\n\nreplicates = np.transpose(replicates)\n\nExercise\n\nCompute bias and standard deviation of replicates for each parameter\nCompare standard deviation of replicates with Minuit’s uncertainty estimate, accessible via m.errors\n\n\n# do exercise here" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#nonparametric-bootstrap", + "href": "2024_LHCb_Starterkit/lecture_2.html#nonparametric-bootstrap", + "title": "Statistics II", + "section": "Nonparametric bootstrap", + "text": "Nonparametric bootstrap\n\nCan do the same by sampling directly from \\(\\{ x_i \\}\\)\nStandard method for samples of fixed size \\(N\\):\n\nPick \\(N\\) random samples from \\(\\{ x_i \\}\\) with replacement to obtain \\(\\{ x_i \\}_k\\)\nCompute \\(\\hat a_k\\) from \\(\\{ x_i \\}_k\\) and proceed as before\n\n“Extended method” for samples of variable size (Poisson):\n\nRandomly draw \\(M\\) from Poisson distribution with \\(\\lambda = N\\)\nPick \\(M\\) random samples from \\(\\{ x_i \\}\\) with replacement to obtain \\(\\{ x_i \\}_k\\)\n…\n\nConvenient: use same algorithms for any distribution\nEfficient implementations for both algorithms in resample\n\n\n\nimport resample\n\nresample.__version__\n\n'1.7.1'\n\n\n\nfrom resample.bootstrap import resample\n\n# generate replicated samples of varying size (Poisson distributed)\nfor x_k in resample(x, size=100, method=\"extended\", random_state=1):\n print(x_k)\n break\n\n[0.58820262 0.58820262 0.52000786 ... 0.38532216 0.26892968 0.26892968]\n\n\nExercise\n\nAgain compute replicated fit results using resampe and store results in a list\nCopy previous code where possible\nAgain compute bias and standard deviation of replicates and compare with Minuit’s uncertainty estimate\n\n\n# do exercise here\n\n\nresample provides shortcut for computing variance\n\n\nfrom resample.bootstrap import variance\n\nvar = variance(estimator, x, method=\"extended\", size=100, random_state=1)\n\nfor i, name in enumerate(m.parameters):\n std = var[i] ** 0.5\n print(f\"{name:8} 𝜎(Minuit) = {m.errors[i]:6.2g} 𝜎(bootstrap) = {std:6.2g}\")\n\ns 𝜎(Minuit) = 35 𝜎(bootstrap) = 33\nmu 𝜎(Minuit) = 0.0018 𝜎(bootstrap) = 0.0017\nsigma 𝜎(Minuit) = 0.0015 𝜎(bootstrap) = 0.0014\nb 𝜎(Minuit) = 27 𝜎(bootstrap) = 27\nslope 𝜎(Minuit) = 0.15 𝜎(bootstrap) = 0.17\n\n\nExercise\n\nRepeat the variance calculation without using method=\"extended\"\nCompare the uncertainties of \\(s\\) and \\(b\\)\n\n\n# do exercise here\n\n\nBootstrap can be applied to most statistical problems in HEP\nRestriction: samples must be identically and independently distributed\nMay work well even with small samples (< 10)" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#error-propagation", + "href": "2024_LHCb_Starterkit/lecture_2.html#error-propagation", + "title": "Statistics II", + "section": "Error propagation", + "text": "Error propagation\n\nFull analysis typically combines many intermediate results with uncertainties into final result\nConsider \\(\\vec b = g(\\vec a)\\) with uncertainties in \\(\\vec a\\)\nTwo ways of computing uncertainties of \\(\\vec b\\)\n\nMonte-Carlo simulation\nError propagation\n\nCan be equally used for statistical and systematic uncertainties\n\n\nMonte-Carlo simulation\n\nDraw variations of fit result from multivariate normal distribution\nCompute variable of interest from variations\nCompute standard deviation or covariance of\nExample: compute uncertainty of \\(\\hat N = \\hat s + \\hat b\\)\n\n\nrng = np.random.default_rng(1)\na_k = rng.multivariate_normal(m.values, m.covariance, size=1000)\n\ndef g(a):\n return a[0] + a[3]\n\nb_k = np.apply_along_axis(g, -1, a_k)\n\nprint(f\"𝜎(N, simulation) = {np.std(b_k):.3g}\")\nprint(f\"𝜎(N, naive) = {(m.errors[0]**2 + m.errors[3]**2)**0.5:.3g}\")\nprint(f\"𝜎(N, expected) = {np.sqrt(len(x)):.3g}\")\n\n𝜎(N, simulation) = 37.7\n𝜎(N, naive) = 43.7\n𝜎(N, expected) = 38.8\n\n\n\n\\(\\sigma_{N,\\text{simulation}}^2 < \\sigma_s^2 + \\sigma_b^2\\), because \\(s_k\\) and \\(b_k\\) are anti-correlated\n\n\n\nError propagation\n\nGiven: \\(\\vec a\\) with covariance matrix \\(C_a\\) and differentiable function \\(\\vec b = g(\\vec a)\\)\n\\(\\vec b\\) can have different dimensionality from \\(\\vec a\\)\nOne can derive via first-order Taylor expansion: \\[\nC_b = J \\, C_a \\, J^T \\text{ with } J_{ik} = \\frac{\\partial g_i}{\\partial a_k}\n\\]\nUseful special cases\n\nSum of scaled independent random values \\(c = \\alpha a + \\beta b\\) (\\(\\alpha, \\beta\\) are constant) \\[\n \\sigma^2_c = \\alpha^2 \\sigma^2_a + \\beta^2 \\sigma^2_b\n \\]\nProduct of powers of independent random values \\(c = a^\\alpha \\, b^\\beta\\) \\[\n \\left(\\frac{\\sigma_c}{c}\\right)^2 = \\alpha^2 \\left(\\frac{\\sigma_a}{a}\\right)^2 + \\beta^2 \\left(\\frac{\\sigma_b}{b}\\right)^2\n \\]\n\nFinal uncertainty often dominated by one term\nWhen estimating systematic uncertainties: put your effort in estimating largest term and be lax on others\nComputation of \\(J\\) by hand error-prone and tedious\njacobi library offers automatic error propagation based on a robust numerical derivative\n\n\n\nimport jacobi\n\njacobi.__version__\n\n'0.9.2'\n\n\n\nb, cov_b = jacobi.propagate(g, m.values, m.covariance)\n\nf\"𝜎(N, error propagation) = {cov_b ** 0.5:.3g} 𝜎(N, expected) = {np.sqrt(len(x)):.3g}\"\n\n'𝜎(N, error propagation) = 38.9 𝜎(N, expected) = 38.8'\n\n\n\njacobi.propagate works for any differentiable mapping, which even includes most fits!\n\n\nfrom iminuit.cost import LeastSquares\n\ndef h(pvec):\n a, b = pvec\n x = np.array([1, 2, 3])\n y = a + b * x\n m = Minuit(LeastSquares(x, y, 1, lambda x, a, b: a + b * x), a=0, b=0)\n m.strategy = 0\n m.migrad()\n assert m.valid\n return m.values\n\njacobi.propagate(h, (1, 2), (1, 1))\n\n(array([1., 2.]),\n array([[ 1.00000000e+00, -1.84267416e-12],\n [-1.84267416e-12, 1.00000000e+00]]))\n\n\n\njacobi.propagate makes complex error propagation feasible\nExtensively used in the pp @ 13 TeV analysis of prompt charged particle production" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#systematic-uncertainties-known-unknowns", + "href": "2024_LHCb_Starterkit/lecture_2.html#systematic-uncertainties-known-unknowns", + "title": "Statistics II", + "section": "Systematic uncertainties: Known unknowns", + "text": "Systematic uncertainties: Known unknowns\n“There are known knowns, things we know that we know; and there are known unknowns, things that we know we don’t know. But there are also unknown unknowns, things we do not know we don’t know.” Donald Rumsfeld\n\nWhy data analyses take so long: study data, detector, and methods to turn unknown unknowns into known knowns or known unknowns\nknown unknowns = systematic uncertainties\nNo formal way, but generally…\n\nWorry * What could go wrong? * What are my assumptions?\nExample * You take an efficiency or correction factor from simulation, but simulation differs from real experiment. * Solution: Perform control measurement to estimate how much simulation differs from real experiment.\nMethods * Use methods that are known to be optimal for your problem or derive them from first principles * General purpose methods often safer to use than special-case methods * Maximum-likelihood fit > least-squares fit * Bootstrap estimate of uncertainty > other uncertainty estimates\nApply checks * Validate whole analysis on simulated data (sometimes you can use toy simulation) * Split data by magnet polarity, fill, etc., analyze splits separately, check for agreement\nGuidelines for handling systematic errors * Roger Barlow: Systematic Errors: Facts and Fictions (2002) * Roger Barlow, “Practical Statistics for Particle Physics” (2019), Section 6.5" + }, + { + "objectID": "2024_LHCb_Starterkit/lecture_2.html#further-resources-for-learning", + "href": "2024_LHCb_Starterkit/lecture_2.html#further-resources-for-learning", + "title": "Statistics II", + "section": "Further resources for learning", + "text": "Further resources for learning\n\nLHCb Statistics and ML WG\n\nTWiki page\nStatistics Guidelines to collect information about best practices\n\n\n\nIntroductory statistics books\n\nList of books on TWiki pages of Statistis and Machine Learning WG\nBooks I personally like a lot\n\nGlen Cowan, “Statistical Data Analysis”, Oxford University Press, 1998\nF. James, “Statistical Methods in Experimental Physics”, 2nd edition, World Scientific, 2006\nR. Barlow, “Statistics: a Guide to the Use of Statistical Methods in the Physical Sciences”, Wiley, 1989\nB. Efron, R.J. Tibshirani, “An Introduction to the Bootstrap”, CRC Press, 1994\n\n\n\n\nLectures, papers, notebooks\n\nRoger Barlow\n\nIntroductory lecture (2019)\nSystematic Errors: Facts and Fictions (2002)\n“Practical Statistics for Particle Physics” (2019), Section 6.5\n\nMore Jupyter notebooks\n\nMatt Kenzie, PDG averaging\nMatt Kenzie, MLE uncertainties\niminuit tutorials on various fitting topics\n\nAll about sWeights and COWs with gentle derivation\n\nHD, M. Kenzie, C. Langenbruch, M. Schmelling, Nucl.Instrum.Meth.A 1040 (2022) 167270\n\nMachine learning\n\nScikit-Learn\nUvA Deep Learning tutorials (JAX/FLAX or Torch)" + }, + { + "objectID": "posts/Wilson Score Interval with Weighted Histograms.html", + "href": "posts/Wilson Score Interval with Weighted Histograms.html", + "title": "The Wilson score interval and weighted histograms", + "section": "", + "text": "The Wilson Score interval (WSI) was won the LHCb internal challenge for the most suitable interval for efficiencies. The question what to do when events are weighted was left open at the time. It is addressed here, the WSI can be used also with weighted histograms if ordinary counts in unweighted histograms are replaced with effective counts in weighted histograms.\n\nimport numpy as np\nimport boost_histogram as bh\nfrom matplotlib import pyplot as plt\n\nhttps://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval\nWilson score interval: \\[\np_\\text{min} = \\frac{n_S + \\frac 12 z^2}{n_S + n_F + z^2} - \\frac{z}{n_S + n_F + z^2} \\sqrt{\\frac{n_S n_F}{n_S + n_F} + \\frac{z^2}{4}}\n\\] \\[\np_\\text{max} = \\frac{n_S + \\frac 12 z^2}{n_S + n_F + z^2} + \\frac{z}{n_S + n_F + z^2} \\sqrt{\\frac{n_S n_F}{n_S + n_F} + \\frac{z^2}{4}}\n\\] with \\(n_S\\) counts of selected events, \\(n_F\\) counts of rejected events, \\(z\\) number of sigmas (\\(z = 1\\) for standard intervals).\nIn case of weighted events, the \\(n_S,n_F\\) are the effective counts, computed as \\[\nn_\\text{effective} = \\frac{\\big(\\sum_i w_i\\big)^2}{\\sum_i w^2_i}.\n\\] In boost-histogram, this is computed as n_eff = h.values() ** 2 / h.variances() for a histogram h with WeightedSum storage.\n\ndef wilson_score_interval(n_s, n_f, z=1):\n n = n_s + n_f\n z2 = z * z\n center = (n_s + 0.5 * z2) / (n + z2)\n delta = z / (n + z2) * np.sqrt(n_s * n_f / n + 0.25 * z2)\n return center - delta, center + delta\n\n\np_truth = 0.2\n\nrng = np.random.default_rng(1)\n\nw_s = bh.accumulators.WeightedSum()\nw_f = bh.accumulators.WeightedSum()\n\nw = rng.exponential(10, size=100)\nm = rng.uniform(size=len(w)) < p_truth\n\nw_s.fill(w[m])\nw_f.fill(w[~m])\n\nWeightedSum(value=944.245, variance=24939.8)\n\n\n\nn_s = w_s.value ** 2 / w_s.variance if w_s.variance > 0 else 0\nn_f = w_f.value ** 2 / w_f.variance if w_f.variance > 0 else 0\n\np = n_s / (n_s + n_f)\np_min, p_max = wilson_score_interval(n_s, n_f)\n\nplt.errorbar([1], [p], [[p-p_min], [p_max-p]], fmt=\"ok\")\n\nprint(f\"efficiency = {p:.2f} - {p-p_min:.2f} + {p_max-p:.2f}\")\n\nefficiency = 0.16 - 0.05 + 0.06\n\n\n\n\n\nNote that the error bar is asymmetric in general. This is especially noticable when p is close to 0 or 1.\nLet’s check with a parametric bootstrap simulation whether this uncertainty is correct. It is parametric because I put in we know that the weights are exponentially distributed.\n\nrng = np.random.default_rng(1)\n\nw_mean = np.mean(w)\n\np_boot = []\nfor iboot in range(1000):\n wi = rng.exponential(w_mean, size=len(w))\n mi = rng.uniform(size=len(w)) < p\n t_s = bh.accumulators.WeightedSum()\n t_f = bh.accumulators.WeightedSum()\n t_s.fill(wi[mi])\n t_f.fill(wi[~mi])\n p_boot.append(t_s.value / (t_s.value + t_f.value))\n \ncenter = np.mean(p_boot)\ndelta = np.std(p_boot)\np_min_boot = center - delta\np_max_boot = center + delta\n\nplt.errorbar([1], [p], [[p-p_min], [p_max-p]], fmt=\"ok\")\nplt.errorbar([2], [p], [[p-p_min_boot], [p_max_boot-p]], fmt=\"sr\")\nplt.axhline(p_truth, ls=\"--\", color=\"0.5\")\n\nprint(f\"efficiency = {p:.2f} - {p-p_min:.2f} + {p_max-p:.2f}\")\nprint(f\"efficiency(boot) = {p:.2f} - {p-p_min_boot:.2f} + {p_max_boot-p:.2f}\")\n\nefficiency = 0.16 - 0.05 + 0.06\nefficiency(boot) = 0.16 - 0.05 + 0.05\n\n\n\n\n\nOk, check passed." + }, + { + "objectID": "posts/Sleep mode power consumption.html", + "href": "posts/Sleep mode power consumption.html", + "title": "Power consumption in sleep mode", + "section": "", + "text": "Some people make a big fuss about machines consuming power in standby. While it is very important to conserve energy, it makes no sense to turn off all devices but then run the air conditioner during the full night or run the car for a few minutes before you leave to heat it up. It is important to save energy where it really matters.\nHere I compute that the CO2 emissions produced by a MacBook in sleep mode over a year are equal to burning 900 g of wood. So if you are worried about this, put one log less into the fireplace on Christmas, and it is compensated.\n\nkg_of_co2_for_burning_kg_of_wood = 1.9\nkg_of_co2_per_wh = 0.94e-3 \nmacbook_power_consumption_off_in_watt = 0.07\nmacbook_power_consumption_sleep_mode_in_watt = 0.34\n\n\nsleep_hours_per_year = 52 * (5 * 10 + 2 * 24)\n\n\nmacbook_power_consumption_sleep_mode_in_wh_per_year = sleep_hours_per_year * macbook_power_consumption_sleep_mode_in_watt\n\n\nmacbook_power_consumption_sleep_mode_in_wh_per_year * kg_of_co2_per_wh / kg_of_co2_for_burning_kg_of_wood\n\n0.8572008421052631" + }, + { + "objectID": "posts/parse_latex.html", + "href": "posts/parse_latex.html", + "title": "Transform LaTeX to Unicode", + "section": "", + "text": "It is possible to transform a subset of LaTeX to Unicode, as demonstrated by unicodeit website. Unfortunately, unicodeit only works on short LaTeX strings.\nHere, I write a simple parser with Lark to process more complex LaTeX strings. A more complete version of this prototype is available as unicodeitplus.\n\n# Wikipedia: https://en.wikipedia.org/wiki/Unicode_subscripts_and_superscripts\nimport string\n\nCOMMANDS = {}\n\nsuperscript_numbers = \"⁰¹²³⁴⁵⁶⁷⁸⁹\"\nfor i, ch in enumerate(superscript_numbers):\n COMMANDS[f\"^{{{i}}}\"] = ch\n\nsubscript_numbers = \"₀₁₂₃₄₅₆₇₈₉\"\nfor i, ch in enumerate(subscript_numbers):\n COMMANDS[f\"_{{{i}}}\"] = ch\n\nsuperscript_lowercase = \"ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖ𐞥ʳˢᵗᵘᵛʷˣʸᶻ\"\nsuperscript_uppercase = \"ᴬᴮꟲᴰᴱꟳᴳᴴᴵᴶᴷᴸᴹᴺᴼᴾꟴᴿ ᵀᵁⱽᵂ \"\nfor latex, ch in zip(\n string.ascii_letters, superscript_lowercase + superscript_uppercase\n):\n if ch != \" \":\n COMMANDS[f\"^{{{latex}}}\"] = ch\n\nsubscript_lowercase = \"ₐ ₑ ₕᵢⱼₖₗₘₙₒₚ ᵣₛₜᵤᵥ ₓ \"\nfor latex, ch in zip(string.ascii_letters, subscript_lowercase * 2):\n if ch != \" \":\n COMMANDS[f\"_{{{latex}}}\"] = ch\n\ngreek_lowercase = (\n r\"\\alpha\",\n r\"\\beta\",\n r\"\\gamma\",\n r\"\\delta\",\n r\"\\epsilon\",\n r\"\\zeta\",\n r\"\\eta\",\n r\"\\theta\",\n r\"\\iota\",\n r\"\\kappa\",\n r\"\\lambda\",\n r\"\\mu\",\n r\"\\nu\",\n r\"\\xi\",\n \"o\",\n r\"\\pi\",\n r\"\\rho\",\n r\"\\sigma\",\n r\"\\tau\",\n r\"\\upsilon\",\n r\"\\phi\",\n r\"\\chi\",\n r\"\\psi\",\n r\"\\omega\",\n)\n\nsuperscript_lowercase_greek = \" ᵝᵞᵟᵋ ᶿᶥ ᶹᵠᵡ \"\nsubscript_lowercase_greek = \" ᵦᵧ ᵨ ᵩᵪ \"\nfor latex, sup, sub in zip(\n greek_lowercase, superscript_lowercase_greek, subscript_lowercase_greek\n):\n if sup != \" \":\n COMMANDS[f\"^{{{latex}}}\"] = sup\n if sub != \" \":\n COMMANDS[f\"_{{{latex}}}\"] = sub\n\n\n# Symbols extracted from http://milde.users.sourceforge.net/LUCR/Math/data/unimathsymbols.txt, which is under Copyright 2011 by Günter Milde and licensed under the LaTeX Project Public License (LPPL)\n\nfrom pathlib import Path\nfrom urllib.request import urlopen\n\n\ndef match(comments):\n matches = [\n (\"PLUS\", \"+\"),\n (\"MINUS\", \"-\"),\n (\"EQUALS\", \"=\"),\n (\"LEFT PARENTHESIS\", \"(\"),\n (\"RIGHT PARENTHESIS\", \")\"),\n ]\n for match, latex in matches:\n if match in comments:\n return latex\n assert False, f\"unmatched: {comments}\" # never arrive here\n\n\nwith urlopen(\"http://milde.users.sourceforge.net/LUCR/Math/data/unimathsymbols.txt\") as response:\n body = response.read().decode()\n for line in body.split(\"\\n\"):\n if not line or line.startswith(\"#\"):\n continue\n items = line.split(\"^\")\n _, ch, latex, latex2, clas, category, requirements, comments = items\n comments = comments[:-1]\n if latex:\n if len(ch) > 1:\n COMMANDS[latex] = ch[1]\n else:\n COMMANDS[latex] = ch\n elif latex2:\n COMMANDS[latex2] = ch\n elif comments.startswith(\"SUPERSCRIPT\"):\n latex = f\"^{{{match(comments)}}}\"\n COMMANDS[latex] = ch\n elif comments.startswith(\"SUBSCRIPT\"):\n latex = f\"_{{{match(comments)}}}\"\n COMMANDS[latex] = ch\n else:\n pass\n\n\n# enhancements\nCOMMANDS[r\"\\to\"] = COMMANDS[r\"\\rightarrow\"]\nCOMMANDS[r\"^{\\ast}\"] = \"*\"\nCOMMANDS[r\"\\hbar\"] = COMMANDS[r\"\\hslash\"]\nCOMMANDS[\"h\"] = \"ℎ\"\n\nLark is awesome, because it generates the parser from an EMBL string, is fast and lightweight.\n\nfrom lark import Lark\nfrom lark import Tree\nfrom lark.visitors import Transformer, Visitor, Discard, v_args, Interpreter\nfrom copy import deepcopy\n\nparser = Lark(r\"\"\"\nstart: (item | math)*\n\n?atom: CHARACTER\n | COMMAND\n\n?item: atom\n | WS+\n | group\n\nCHARACTER: /[^%#&\\{\\}^_]/ | ESCAPED\nESCAPED: \"\\\\\\\\\" | \"\\\\#\" | \"\\\\%\" | \"\\\\&\" | \"\\\\{\" | \"\\\\}\" | \"\\\\_\"\ngroup: \"{\" item* \"}\"\nmath: \"$\" item* \"$\"\nSUBSCRIPT: \"_\"\nSUPERSCRIPT: \"^\"\nCOMMAND: ((\"\\\\\" WORD WS*) | SUBSCRIPT | SUPERSCRIPT)\n\n%import common.WS\n%import common.WORD\n\"\"\", parser=\"lalr\")\n\n\nHAS_ARG = {\n r\"_\",\n r\"^\",\n r\"\\grave\",\n r\"\\acute\",\n r\"\\hat\",\n r\"\\tilde\",\n r\"\\bar\",\n r\"\\overline\",\n r\"\\breve\",\n r\"\\dot\",\n r\"\\ddot\",\n r\"\\mathring\",\n r\"\\check\",\n r\"\\utilde\",\n r\"\\underbar\",\n r\"\\underline\",\n r\"\\not\",\n r\"\\lvec\",\n r\"\\vec\",\n r\"\\LVec\",\n r\"\\vec\",\n r\"\\dddot\",\n r\"\\ddddot\",\n r\"\\overleftrightarrow\",\n r\"\\underleftarrow\",\n r\"\\underrightarrow\",\n r\"\\mathbf\",\n r\"\\text\",\n r\"\\mathrm\",\n r\"\\left\",\n r\"\\right\",\n r\"\\big\",\n r\"\\Big\",\n r\"\\Bigg\",\n r\"\\sqrt\",\n}\n\nIGNORE_AS_FALLBACK = {\n r\"\\text\",\n r\"\\mathbf\",\n r\"\\mathrm\",\n r\"\\left\",\n r\"\\right\",\n r\"\\big\",\n r\"\\Big\",\n r\"\\Bigg\",\n}\n\nESCAPED = {\n r\"\\}\": \"}\",\n r\"\\{\": \"{\",\n \"\\\\\\\\\": \"\\\\\",\n}\n\n\ndef handle_cmd(state, x):\n # - x can be a character or a command, like \\alpha\n # - state[\"command\"] contains stack with commands, may be empty\n # - to transform ^{\\alpha} or \\text{x} correctly, we first try to \n # convert innermost command and x as a unit\n # - they are treated independently only if previous step fails\n cmd_stack = state[\"command\"].copy()\n if state[\"math\"]:\n cmd = cmd_stack[-1] if cmd_stack else \"\"\n latex = f\"{cmd}{{{x}}}\"\n if cmd and latex in COMMANDS:\n x = COMMANDS[latex]\n cmd_stack.pop()\n elif x.startswith(r\"\\\\\"):\n x = COMMANDS.get(x, x)\n elif cmd in (r\"\\text\", r\"\\mathrm\"):\n cmd_stack.pop()\n else:\n x = COMMANDS.get(x, x)\n for cmd in reversed(cmd_stack):\n if cmd in COMMANDS:\n # must be some unicode modifier, e.g. \\dot, \\vec\n assert cmd in HAS_ARG\n x += COMMANDS[cmd]\n else:\n latex = f\"{cmd}{{{x}}}\"\n if latex in COMMANDS:\n x = COMMANDS[latex]\n elif cmd not in IGNORE_AS_FALLBACK:\n x = latex\n else:\n for cmd in reversed(state[\"command\"]):\n x = f\"{cmd}{{{x}}}\"\n if state[\"command\"] and not state[\"group\"]:\n state[\"command\"].pop()\n return x\n\n\ndef transform(ch, state=None):\n if state is None:\n state = {\n \"math\": False,\n \"command\": [],\n \"group\": False,\n }\n\n if isinstance(ch, Tree):\n r = []\n undo_math = False\n if ch.data == \"math\":\n state[\"math\"] = True\n if ch.data == \"group\":\n state[\"group\"] = True\n for x in ch.children:\n r.append(transform(x, state))\n if ch.data == \"math\":\n state[\"math\"] = False\n if ch.data == \"group\":\n state[\"group\"] = False\n if state[\"command\"]:\n state[\"command\"].clear()\n return \"\".join(r)\n\n if ch.type == \"CHARACTER\":\n x = ESCAPED.get(ch.value, ch.value)\n return handle_cmd(state, x)\n if ch.type == \"WS\":\n return \"\" if state[\"math\"] else \" \"\n if ch.type == \"COMMAND\":\n x = ch.value.strip()\n if x in HAS_ARG:\n if x == r\"\\sqrt\":\n state[\"command\"].append(r\"\\overline\")\n return COMMANDS[r\"\\sqrt\"]\n state[\"command\"].append(x)\n return \"\"\n return handle_cmd(state, x)\n # never arrive here\n assert False, f\"unknown token {ch}\" \n\n\ndef parse(s):\n tree = parser.parse(s)\n return transform(tree)\n\nLet’s see how well (or not) this works on a few examples.\n\ns = r\"foo?!-1+2. \\} \\\\ $\\left(\\mathbf{\\alpha + 1}^2_x y\\right)$ bar $\\beta^{12}$ $\\bar p {}^foo$ $\\bar \\mathrm{t}$ \"\nprint(s)\nparse(s)\n\nfoo?!-1+2. \\} \\\\ $\\left(\\mathbf{\\alpha + 1}^2_x y\\right)$ bar $\\beta^{12}$ $\\bar p {}^foo$ $\\bar \\mathrm{t}$ \n\n\n'foo?!-1+2. } \\\\ (𝛂+𝟏²ₓ𝑦) bar 𝛽¹² 𝑝̄ᶠ𝑜𝑜 t̄ '\n\n\n\nparse(r\"$D^{\\ast\\ast} \\to hhee$\")\n\n'𝐷**→ℎℎ𝑒𝑒'\n\n\n\nparse(r\"$\\mathbf{xyz + 1}$\")\n\n'𝐱𝐲𝐳+𝟏'\n\n\n\nparse(r\"$\\sqrt {1Aas\\alpha}$\")\n\n'√1̅𝐴̅𝑎̅𝑠̅𝛼̅'\n\n\n\nparse(r\"$\\vec{x} b^2 \\vec\\alpha\\overline\\alpha K^0_S p_\\text{T} \\text T$\")\n\n'𝑥⃗𝑏²𝛼⃗𝛼̅𝐾⁰ₛ𝑝ₜT'\n\n\n\nparse(r\"$\\sqrt{abcd}$\")\n\n'√𝑎̅𝑏̅𝑐̅𝑑̅'\n\n\n\nparse(r\"$p_T / \\text{GeV}c^{-1}$\")\n\n'𝑝ₜ/GeV𝑐⁻¹'\n\n\n\nparse(r\"Search for resonant $ \\mathrm{t}\\overline{\\mathrm{t}} $ production in proton-proton collisions at $ \\sqrt{s}=13 $ TeV\")\n\n'Search for resonant tt̅ production in proton-proton collisions at √𝑠̅=13 TeV'" + }, + { + "objectID": "posts/cows.html", + "href": "posts/cows.html", + "title": "Fitting mixture models with COWs", + "section": "", + "text": "This is a demo of how to use custom orthogonal weight functions (COWs) to generate weights which extract one component of a mixture. Find out more in our paper.\n\nimport numpy as np\nfrom numba_stats import bernstein, norm, expon\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import quad\nfrom typing import Callable\n\n\nrng = np.random.default_rng(1)\n\ns_x = rng.normal(1, 0.1, size=10000)\nb_x = rng.exponential(1, size=100000)\ns_y = rng.exponential(1, size=len(s_x))\nb_y = rng.normal(0, 1, size=len(b_x))\n\nx = np.append(s_x, b_x)\ny = np.append(s_y, b_y)\nid = np.append(np.ones_like(s_x, dtype=bool), np.zeros_like(b_x, dtype=bool))\n\nxrange = 0, 2\nyrange = 0, 4\nm = (x > xrange[0]) & (x < xrange[1]) & (y > yrange[0]) & (y < yrange[1])\nx = x[m]\ny = y[m]\nid = id[m]\n\ns_x = x[id]\ns_y = y[id]\nb_x = x[~id]\nb_y = y[~id]\n\nplt.figure()\nplt.hist(x, bins=100, range=xrange)\nplt.hist(s_x, bins=100, range=xrange)\n\nplt.figure()\nplt.hist(y, bins=100, range=yrange)\nplt.hist(s_y, bins=100, range=yrange);\n\n\n\n\n\n\n\n\ndef gs(m):\n return norm.pdf(m, 1, 0.1)\n\ndef gb(m):\n return expon.pdf(m, 0, 1)\n\ndef g(m):\n return gs(m) + 10 * gb(m) / 11.\n\ndef Gs(m):\n return norm.pdf(m, 1.1, 0.2)\n\ndef flat(m):\n return np.ones_like(m)\n\n\ndef _A(I, S, B, xrange=(-np.inf, np.inf)):\n nB = len(B)\n W = np.empty((1 + nB, 1 + nB))\n F = [S] + B\n for i in range(1 + nB):\n for j in range(i + 1):\n W[i, j] = quad(lambda m: F[i](m) * F[j](m) / I(m), *xrange)[0]\n if i != j:\n W[j, i] = W[i, j]\n return np.linalg.inv(W)\n\ndef cow(m, I, S, B, xrange=(-np.inf, np.inf)):\n if isinstance(B, Callable):\n B = [B]\n a = _A(I, S, B, xrange=xrange)\n F = [S(m)] + [Bi(m) for Bi in B]\n return (a[0] @ F) / I(m)\n\n\nw0 = cow(x, g, gs, gb, xrange=xrange)\nplt.plot(x, w0, \".\", ms=1, mew=0);\n\n\n\n\n\nplt.hist(y, weights=w0, density=True, label=\"estimate\")\nplt.hist(s_y, density=True, edgecolor=\"r\", facecolor=\"None\", label=\"truth\")\nplt.legend(title=f\"var(w0) = {np.var(w0):.2f}\");\n\n\n\n\n\nw0 = cow(x, g, Gs, gb, xrange=xrange)\nplt.plot(x, w0, \".\", ms=1, mew=0);\n\n\n\n\n\nplt.hist(y, weights=w0, density=True, label=\"estimate\")\nplt.hist(s_y, density=True, edgecolor=\"r\", facecolor=\"None\", label=\"truth\")\nplt.legend(title=f\"var(w0) = {np.var(w0):.2f}\");\n\n\n\n\n\nw0 = cow(x, flat, Gs, gb, xrange=xrange)\nplt.plot(x, w0, \".\", ms=1, mew=0);\n\n\n\n\n\nplt.hist(y, weights=w0, density=True, label=\"estimate\")\nplt.hist(s_y, density=True, edgecolor=\"r\", facecolor=\"None\", label=\"truth\")\nplt.legend(title=f\"var(w0) = {np.var(w0):.2f}\");\n\n\n\n\n\nimport numba as nb\n\nclass Base:\n def __init__(self, n, i, xrange):\n self.beta = np.zeros(n)\n self.beta[i] = 1\n self.xrange = xrange\n\n def __call__(self, m):\n return bernstein.density(m, self.beta, *self.xrange)\n\ndef make_basis(n, xrange):\n return [Base(n, i, xrange) for i in range(n)]\n\nN = 7\nB = make_basis(N, xrange)\nfor Bi in B:\n plt.plot(x, Bi(x), \".\", ms=1, mew=0)\n\nw0 = cow(x, flat, gs, B, xrange)\nplt.figure()\nplt.plot(x, w0, \".\", ms=1, mew=0);\n\n\n\n\n\n\n\n\nplt.hist(y, weights=w0, density=True, label=\"estimate\")\nplt.hist(s_y, density=True, edgecolor=\"r\", facecolor=\"None\", label=\"truth\")\nplt.legend(title=f\"var(w0) = {np.var(w0):.2f}\");" + }, + { + "objectID": "posts/Interactive plotting in Jupyter with matplotlib.html", + "href": "posts/Interactive plotting in Jupyter with matplotlib.html", + "title": "Interactive plotting in Jupyter with matplotlib", + "section": "", + "text": "This is a little demo on how to make interactive plots with matplotlib and ipympl.\n\n#!pip install ipywidgets matplotlib scipy numpy ipympl\n%matplotlib widget\nfrom matplotlib import pyplot as plt\nfrom scipy.stats import johnsonsu, norm, t\nfrom ipywidgets import interact, widgets\nimport numpy as np\n\n\nx = np.linspace(-5, 5, 1000)\n\nfig = plt.figure()\nplt.plot(x, norm.pdf(x), label=\"normal\")\nline, = plt.plot(x, x, label=\"other\")\nplt.legend()\nplt.ylim(0, 1)\n\ndef update(a, b, which):\n if which == \"Johnson SU\":\n y = johnsonsu.pdf(x, a, np.exp(b))\n elif which == \"Student-t\":\n y = t.pdf(x, np.exp(b))\n line.set_ydata(y)\n fig.canvas.draw_idle()\n \ninteract(update,\n a=(-10.0, 10.0),\n b=widgets.FloatSlider(0.0, min=-10.0, max=5.0),\n which=[\"Johnson SU\", \"Student-t\"]);" + }, + { + "objectID": "posts/Cross-section.html", + "href": "posts/Cross-section.html", + "title": "Inelastic cross-sections of proton-proton and pion-proton collisions", + "section": "", + "text": "According to a simple parton model, the cross-section for an inelastic proton-proton collision should be 3/2 of a pion-proton collision. In this picture, only the valence quarks matter. We know that at high energies most of the collisions involve sea quarks and gluons, so another theory says that the cross-sections might become equal at high energies.\nI check this here by downloading cross-section data from the Particle Data Group, interpolating the data points and then plot the ratio. We get indeed a ratio of about 3/2=1.5. In reality, it is a bit larger, about 1.6. There is no visible trend that it does down to 1 with energy.\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport requests\n\n\ndef read(url):\n r = requests.get(url)\n tab = r.text\n xy = []\n skip = 11\n for line in tab.strip().split(\"\\n\"):\n if skip:\n skip -= 1\n continue\n items = line.split()\n xi = float(items[3])\n yi = float(items[4])\n yei = (0.5 * sum(float(i) ** 2 for i in items[5:9])) ** 0.5\n xy.append((xi, yi, yei))\n xy.sort(key=lambda p: p[0])\n return np.transpose(xy)\n\n\npim_p = read(\"https://pdg.lbl.gov/2020/hadronic-xsections/rpp2020-pimp_total.dat\")\npip_p = read(\"https://pdg.lbl.gov/2020/hadronic-xsections/rpp2020-pipp_total.dat\")\np_p = read(\"https://pdg.lbl.gov/2020/hadronic-xsections/rpp2020-pp_total.dat\")\n\n\nplt.figure(figsize=(16, 6))\nplt.errorbar(*pim_p, fmt=\"o\", label=\"pi- p\")\nplt.errorbar(*pip_p, fmt=\"s\", label=\"pi+ p\")\nplt.errorbar(*p_p, fmt=\"D\", label=\"p p\")\nplt.loglog()\nplt.legend()\n\n<matplotlib.legend.Legend at 0x12b7185b0>\n\n\n\n\n\n\nfrom scipy.interpolate import interp1d\nl_pip_p = interp1d(*pip_p)\nl_p_p = interp1d(*p_p)\nx = np.geomspace(1, 1e4)\nplt.plot(x, l_p_p(x) / l_pip_p(x))\nplt.axhline(3/2, ls=\"--\", color=\"0.5\")\nplt.semilogx()\n\n[]" + }, + { + "objectID": "posts/comparison_chisquare_test_statistics.html", + "href": "posts/comparison_chisquare_test_statistics.html", + "title": "Comparison of asymptotically chisquare-distributed test statistics", + "section": "", + "text": "This notebook is released under the MIT license. See LICENSE.\nFor GoF tests, we often use a test statistic that is asymptotically \\(\\chi^2\\) distributed.\nWe compute the test statistic on binned data with \\(k\\) bins with Poisson-distributed counts \\(n_i\\) for which we have estimates of the expected counts \\(\\nu_i\\), typically obtained from a fitted model with \\(m\\) free parameters. If the data is truely sampled from the model (the \\(H_0\\) hypothesis), the test statistic \\(X\\) is asymptotically distributed as \\(\\chi^2(k - m)\\).\nSmall p-values \\(P = \\int_{X}^\\infty \\chi^2(X'; k - m) \\, \\text{d}X'\\) can be used as evidence against the hypothesis \\(H_0\\).\nNote: the asymptotic limit is assumed to be taken while keeping the binning constant. If the binning is adaptive to the sample size so that \\(\\nu_i\\) remains constant, \\(X\\) will not approach the \\(\\chi^2(k - m)\\) distribution.\nThere are candidates for an asymptotically \\(\\chi^2\\)-distributed test statistic.\n\nPearson’s test statistic \\[\nX_P = \\sum_i \\frac{(n_i - \\nu_i)^2}{\\nu_i}\n\\]\nPearson’s test statistic with \\(\\nu_i\\) replaced by bin-wise estimate \\(n_i\\) \\[\nX_N = \\sum_i \\frac{(n_i - \\nu_i)^2}{n_i}\n\\]\nLikelihood ratio to saturated model (see e.g. introduction of https://doi.org/10.1088/1748-0221/4/10/P10009) \\[\nX_{L} = 2\\sum_i \\Big( n_i \\ln\\frac{n_i}{\\nu_i} - n_i + \\nu_i \\Big)\n\\] The statistic is equal to \\(-2\\ln(L/L_\\text{saturated})\\), with \\(L = \\prod_i P_\\text{poisson}(n_i; \\nu_i)\\) and \\(L_\\text{saturated} = \\prod_i P_\\text{poisson}(n_i; n_i)\\), the likelihood for the saturated model with \\(\\nu_i = n_i\\). The saturated model has the largest possible likelihood given the data.\nTransform-based test statistic \\[\nX_T = \\sum_i z_i^2 \\quad\\text{with}\\quad z_i = q_\\text{norm}\\bigg(\\sum_{k=0}^{n_i} P_\\text{poisson}(k; v_i)\\bigg)\n\\] where \\(q_\\text{norm}\\) is the quantile of the standard normal distribution. The double-transform converts a Poisson distributed count \\(k\\) into a variable \\(z\\) that has a standard normal distribution.\n\n\nfrom sympy import *\n\nn = symbols(\"n\", integer=True, positive=True)\nv = symbols(\"v\", real=True, positive=True)\n\n\ndef poisson(n, v):\n return v ** n * exp(-v) / factorial(n)\n\npoisson(n, v)\n\n\\(\\displaystyle \\frac{v^{n} e^{- v}}{n!}\\)\n\n\n\nX_L = simplify(- 2 * (log(poisson(n, v)) - log(poisson(n, n)))); X_L\n\n\\(\\displaystyle 2 n \\log{\\left(n \\right)} - 2 n \\log{\\left(v \\right)} - 2 n + 2 v\\)\n\n\nGiven this choice, it is fair to ask which test statistic performs best. The test statistics may approach the \\(\\chi^2(k - m)\\) distribution with different speeds.\nWe check this empirically with toy simulations. For each toy experiment, we use a fixed expected value \\(\\nu_i = \\mu\\) and draw \\(k\\) Poisson-distributed numbers \\(n_i\\) from \\(\\mu\\), corresponding to \\(k\\) bins in the toy experiment. We then compute the test statistics \\(X\\) and its p-value, assuming it is \\(\\chi^2(k)\\) distributed (\\(m = 0\\) here). This is repeated many times to get a histogram of p-values. The distribution is uniform if the test statistic \\(X\\) is indeed \\(\\chi^2(k)\\) distributed. Deviations from uniformity indicate that the test statistic has not reached the asymptotic limit yet.\nWe then quantify the agreement of the p-value distribution with a flat distribution with the reduced \\(\\chi^2\\) value.\n\nimport numpy as np\nimport numba as nb\nfrom scipy.stats import chi2, poisson, norm\nimport matplotlib.pyplot as plt\nfrom pyik.mplext import plot_hist\n\n\n# numba currently does not support scipy, so we cannot access\n# scipy.stats.norm.ppf and scipy.stats.poisson.cdf in a JIT'ed\n# function. As a workaround, we wrap special functions from\n# scipy to implement the needed functions here.\n\nfrom numba.extending import get_cython_function_address\nimport ctypes\n\n\ndef get(name, narg):\n addr = get_cython_function_address(\"scipy.special.cython_special\", name)\n functype = ctypes.CFUNCTYPE(ctypes.c_double, *([ctypes.c_double] * narg))\n return functype(addr)\n\n\nerfinv = get(\"erfinv\", 1)\ngammaincc = get(\"gammaincc\", 2)\n\n\n@nb.vectorize('float64(float64)')\ndef norm_ppf(p):\n return np.sqrt(2) * erfinv(2 * p - 1)\n\n\n@nb.vectorize('float64(intp, float64)')\ndef poisson_cdf(k, m):\n return gammaincc(k + 1, m)\n\n\n# check implementations\nm = np.linspace(0.1, 3, 20)[:,np.newaxis]\nk = np.arange(10)\ngot = poisson_cdf(k, m)\nexpected = poisson.cdf(k, m)\nnp.testing.assert_allclose(got, expected)\n\np = np.linspace(0, 1, 10)\ngot = norm_ppf(p)\nexpected = norm.ppf(p)\nnp.testing.assert_allclose(got, expected)\n\n\n@nb.njit\ndef xn(n, v):\n r = 0.0\n k = 0\n for ni in n:\n if ni > 0:\n k += 1\n r += (ni - v) ** 2 / ni\n return r, k\n\n\n@nb.njit\ndef xp(n, v):\n return np.sum((n - v) ** 2 / v), len(n)\n\n\n@nb.njit\ndef xl(n, v):\n r = 0.0\n for ni in n:\n if ni > 0:\n r += ni * np.log(ni / v) + v - ni\n else:\n r += v\n return 2 * r, len(n)\n\n\n@nb.njit\ndef xt(n, v):\n p = poisson_cdf(n, v)\n z = norm_ppf(p)\n return np.sum(z ** 2), len(n)\n \n\n@nb.njit(parallel=True)\ndef run(n, mu, nmc):\n args = []\n for i in range(len(n)):\n for j in range(len(mu)):\n for imc in range(nmc):\n args.append((i, j, imc))\n results = np.empty((4, len(n), len(mu), nmc, 2))\n for m in nb.prange(len(args)):\n i, j, imc = args[m]\n ni = n[i]\n mui = mu[j]\n np.random.seed(imc)\n x = np.random.poisson(mui, size=ni)\n rp = xp(x, mui)\n rn = xn(x, mui)\n rl = xl(x, mui)\n rt = xt(x, mui)\n results[0, i, j, imc, 0] = rp[0]\n results[0, i, j, imc, 1] = rp[1]\n results[1, i, j, imc, 0] = rn[0]\n results[1, i, j, imc, 1] = rn[1]\n results[2, i, j, imc, 0] = rl[0]\n results[2, i, j, imc, 1] = rl[1]\n results[3, i, j, imc, 0] = rt[0]\n results[3, i, j, imc, 1] = rt[1]\n return results\n\n\nmu = np.geomspace(1e-1, 1e3, 17)\nn = 1, 3, 10, 30, 100, 1000\nnmc = 10000\n\nresult = run(n, mu, nmc)\n\n\ndef reduced_chi2(r):\n p = 1 - chi2.cdf(*np.transpose(r))\n bins = 20\n xe = np.linspace(0, 1, bins + 1)\n n = np.histogram(p, bins=xe)[0]\n v = len(p) / bins\n return np.sum((n - v) ** 2 / v) / bins\n\nmatrix = np.empty((len(result), len(n), len(mu)))\nfor k in range(len(result)):\n for i, ni in enumerate(n):\n for j, mui in enumerate(mu):\n matrix[k, i, j] = reduced_chi2(result[k, i, j])\n\n\ndef plot(r, **kwargs):\n p = 1 - chi2.cdf(*np.transpose(r))\n bins = 20\n xe = np.linspace(0, 1, bins + 1)\n n = np.histogram(p, bins=xe)[0]\n plot_hist(xe, n, **kwargs)\n\n\nfor i, ni in enumerate(n):\n for j, mui in enumerate(mu):\n if j % 4 == 0: # draw only every forth value of mu\n fig, ax = plt.subplots(1, 4, figsize=(15, 5))\n plt.suptitle(f\"k = {ni} mu = {mui:.2f}\")\n for k, t in enumerate(\"PNLT\"):\n plt.sca(ax[k])\n plot(result[k, i, j])\n plt.title(f\"$X_{t}$ ($\\chi^2/n_\\mathrm{{dof}} = ${matrix[k, i, j]:.2g})\")\n\nRuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`).\n fig, ax = plt.subplots(1, 4, figsize=(15, 5))\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe can by eye that \\(X_P\\) gives the best results. Its p-value distribution converges to a uniform distribution for lower values of \\(\\mu\\) and \\(n\\) than the other two test statistics. We summarize this by plotting the reduced \\(\\chi^2\\) for flatness as a function of \\(\\mu\\) and \\(n\\).\n\nfig, ax = plt.subplots(1, len(result), figsize=(15, 5), sharex=True, sharey=True)\nfor i, ni in enumerate(n):\n for axi, matrixi in zip(ax, matrix):\n axi.plot(mu, matrixi[i],\n label=f\"$k = {ni:.1f}$\")\n \nfor axi, t in zip(ax, \"PNLT\"):\n axi.set_title(f\"$X_{t}$\")\n axi.axhline(1, ls=\":\", color=\"0.5\", zorder=0)\n axi.set_xlabel(\"$\\mu$\")\nax[0].set_ylabel(\"$\\chi^2 / n_\\mathrm{dof}$\")\nax[1].legend()\nplt.loglog();\n\n\n\n\n\nfor i, ni in enumerate(n):\n plt.figure()\n plt.title(f\"k = {ni}\")\n for j, t in enumerate(\"PNLT\"):\n plt.plot(mu, matrix[j, i] / matrix[0, i],\n label=f\"$X_{t}$\")\n plt.loglog()\n plt.xlabel(\"$\\mu$\")\n plt.ylabel(\"red. $\\chi^2$ ratio\")\n plt.legend()\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe classic test statistics \\(X_P\\) perform best overall, it converges to the asymptotic \\(\\chi^2\\) distribution for smaller values of \\(\\mu\\) and \\(k\\) and is also the simplest to compute.\nThe statistic \\(X_L\\) shows a curious behavior, it converges well around \\(k=10\\), on par with \\(X_P\\), but the convergence gets worse again for \\(k > 10\\).\nThe study confirms that none of the test statistics is doing well if the expected count per bin is smaller than about 10. If the original binned data contains bins with small counts, it is recommended to compute the distribution of the test statistic via parametric bootstrapping from the fitted model instead of using the asymptotic \\(\\chi^2\\) distribution. Parametric bootstrapping in this case simplifies to drawing new samples for each data bin from a Poisson distribution whose expectation is equal to the expected count per bin predicted by the model. For many such samples one computes the test statistic to obtain a distribution. The p-value is then given by the fraction of sample values that are higher than the original value." + }, + { + "objectID": "posts/template_with_distortion.html", + "href": "posts/template_with_distortion.html", + "title": "Template fits with distortion correction", + "section": "", + "text": "I recently gave a review talk on template fits in CERN’s PHYSTAT seminar, where I presented results from our recent paper on this subject. In the summary, I discussed on of the downsides of template fits: if the simulation does not match the experiment perfectly, one cannot easily adjust for these discrepancies, as one would do in a parametric fit. For example, if I fit a peak over a smooth background, and the peak location is slightly off in the simulation, a parametric fit would simply fit a different peak location and give a good fit, while a template fit with templates taken from simulation would give poor results.\nWe explore this on a toy example. We want to fit a peak over smooth background using templates. The sample which provides a template for the peak is distorted. Its mean is shifted and it also too narrow. We want to fit the yield of this peak under these conditions.\nimport numpy as np\nfrom iminuit.cost import template_chi2_da\nfrom iminuit import Minuit\nfrom numba_stats import truncexpon, truncnorm\nimport matplotlib.pyplot as plt\nfrom resample.bootstrap import resample\nfrom scipy.optimize import differential_evolution\nimport warnings\nfrom iminuit.warnings import IMinuitWarning\n\ns_sample = truncnorm.rvs(0, 1, 0.5, 0.1, size=1000, random_state=1)\nb_sample = truncexpon.rvs(0, 1, 0, 1, size=1000, random_state=1)\nx_sample = np.append(s_sample, b_sample)\n\nst_sample = truncnorm.rvs(0, 1, 0.6, 0.05, size=1000, random_state=2)\nbt_sample = truncexpon.rvs(0, 1, 0, 1, size=1000, random_state=2)\n\nn_data, xe, _ = plt.hist(\n x_sample, bins=20, range=(0, 1), alpha=0.5, color=\"k\", label=\"data\"\n)\ntemplate_s = plt.hist(st_sample, bins=xe, alpha=0.5, label=\"signal template\")[0]\ntemplate_b = plt.hist(bt_sample, bins=xe, alpha=0.5, label=\"background template\")[0]\nplt.legend();\nIn practice, this situation is common. The simulation is a good, but not perfect approximation of reality. Template fits could work very well in these situations if we find a way to fit a slightly distorted template instead of the raw one. For this to work, the distortion has to be a simple function, for example, an affine transformation like in this case. Then, the transformation can be fitted together with the yields. The parameters of the transformation are nuisance parameters. Note that the uncertainty of the fitted yield will always increase when such a transformation has to be estimated from the data as well." + }, + { + "objectID": "posts/template_with_distortion.html#the-catch", + "href": "posts/template_with_distortion.html#the-catch", + "title": "Template fits with distortion correction", + "section": "The catch", + "text": "The catch\nThe idea to fit a transformed template is straight-forward, but there is a catch. Standard minimizers employed in HEP use a method based on gradient descent, and the gradient is computed numerically using a finite-difference formula. These minimizers only work correctly if the function to minimize is smooth: its first and second derivatives must be smoothly varying. But this is not the case the in the outlined approach, because the transformation will cause points from the samples that form the template to migrate from bin to bin. This causes jumps in the objective function and thus discontinuities in the derivatives. MINUIT does not like that at all. It can fail to converge, get stuck in a local minimum, or may report convergence to a point which is not a minimum. All of these outcomes are bad. The following cells demonstrate how MINUIT fails on such a fit. As objective function, we use the transformed approximate likelihood for template fits that we derived in our paper (implemented in template_chi2_da).\n\ndef model(s, b, offset, scale):\n mu_st_sample = np.mean(st_sample)\n st_sample_modified = scale * (st_sample - mu_st_sample) + mu_st_sample + offset\n template_s_modified = np.histogram(st_sample_modified, bins=xe)[0]\n # a small constant is added to avoid dividing by zero\n ts = template_s_modified / (np.sum(template_s_modified) + 1e-6)\n tb = template_b / np.sum(template_b)\n mu = s * ts + b * tb\n mu_var = s * ts**2 + b * tb**2\n return mu, mu_var\n\n\ndef objective(s, b, offset, scale):\n mu, mu_var = model(s, b, offset, scale)\n return template_chi2_da(n_data, mu, mu_var)\n\n\n# if errordef and ndata on the objective are set,\n# iminuit computes the chi2 gof test statistic for us\nobjective.errordef = Minuit.LEAST_SQUARES\nobjective.ndata = len(xe) - 1\n\nm = Minuit(objective, s=1000, b=1000, offset=0, scale=1)\nm.limits[\"s\", \"b\"] = (0, None)\nm.limits[\"scale\"] = (0.01, 2.5)\nm.limits[\"offset\"] = (-0.2, 0.2)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 499 (χ²/ndof = 31.2)\nNfcn = 296\n\n\nEDM = 2.72e-12 (Goal: 0.0002)\n\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\ns\n494\n33\n\n\n0\n\n\n\n\n1\nb\n1.51e3\n0.05e3\n\n\n0\n\n\n\n\n2\noffset\n-9.0380e-3\n0.0033e-3\n\n\n-0.2\n0.2\n\n\n\n3\nscale\n999.17e-3\n0.31e-3\n\n\n0.01\n2.5\n\n\n\n\n\n\n\n\ns\nb\noffset\nscale\n\n\ns\n1.08e+03\n-0.4e3 (-0.299)\n-2.648168e-6 (-0.025)\n-124.42e-6 (-0.012)\n\n\nb\n-0.4e3 (-0.299)\n2.02e+03\n3.171095e-6 (0.022)\n149.02e-6 (0.011)\n\n\noffset\n-2.648168e-6 (-0.025)\n3.171095e-6 (0.022)\n1.06e-11\n0.497e-9 (0.498)\n\n\nscale\n-124.42e-6 (-0.012)\n149.02e-6 (0.011)\n0.497e-9 (0.498)\n9.39e-08\n\n\n\n\n\nMINUIT reports successful convergence, but signal and background yields are far from the truth. Also, the chi2/ndof ratio is far from 1. We can see that this a poor fit by drawing the fitted model and by looking at the profile-likelihood for each parameter.\n\n# let's draw the fitted model\nplt.stairs(model(*m.values)[0], xe, label=\"fitted model\")\nplt.stairs(n_data, xe, label=\"data\")\nplt.legend();\n\n\n\n\n\nplt.figure()\nm.draw_profile(\"s\")\nplt.figure()\nm.draw_profile(\"b\")\n# ignore some iminuit warnings about failing to converge during profile computation\nwith warnings.catch_warnings():\n warnings.simplefilter(\"ignore\", IMinuitWarning)\n plt.figure()\n m.draw_mnprofile(\"scale\", bound=(0.5, 3.5))\n plt.figure()\n m.draw_mnprofile(\"offset\", bound=(-0.5, 0.2))\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe can see in the profile that we are not in the global minimum, which is near offset = -0.1 and scale = 2.2." + }, + { + "objectID": "posts/template_with_distortion.html#solution", + "href": "posts/template_with_distortion.html#solution", + "title": "Template fits with distortion correction", + "section": "Solution", + "text": "Solution\nTwo solutions come to mind, both have caveats. I explore one below, which I consider to be the simpler one. The second solution is to replace the transformed template by a kernel density estimate. This would lead to additional hyperparameters introduced by the kernel density estimation, which would need to be carefully chosen for each problem, and require the derivation of a new complicated likelihood function for this case.\n\nUse a global derivative-free minimizer and bootstrapping\nSince we cannot use a local minimizer which uses gradients for our objective function, we switch to a global derivative-free minimizer. Global derivative-free minimizers require many more function calls than standard local minimizers like MINUIT’s MIGRAD, and this number moreover scales quadratically with the number of fitted parameters. Fortunately, our objective function is relative fast to compute and does not have many parameters, so global minimization is feasible. In the following, I apply the differential_evolution algorithm from scipy to this problem.\n\nr = differential_evolution(\n lambda x: objective(*x),\n bounds=((0, 2000), (0, 2000), (-0.2, 0.2), (0.5, 3.5)),\n polish=False,\n seed=1,\n)\nprint(r.message)\n\nchi2 = objective(*r.x)\n\nprint(f\"chi2/ndof = {chi2/(objective.ndata - m.nfit):.1f}\")\nprint(f\"s = {r.x[0]:.0f}\")\nprint(f\"b = {r.x[1]:.0f}\")\nprint(f\"offset = {r.x[2]:.2f}\")\nprint(f\"scale = {r.x[3]:.1f}\")\n\n# let's draw the fitted model\nplt.stairs(model(*r.x)[0], xe, label=\"fitted model\")\nplt.stairs(n_data, xe, label=\"data\")\nplt.legend();\n\nOptimization terminated successfully.\nchi2/ndof = 1.0\ns = 1017\nb = 987\noffset = -0.09\nscale = 2.0\n\n\n\n\n\nThis looks great! We got an excellent chi2/ndof ratio and the fitted model agrees well with the data. Our idea works, and the global minimizer did not take too much time either.\nNow we need to address the next challenge: we also want to know the parameter uncertainties. We cannot use the usual technique of inverting the appropriately scaled Hesse matrix of the objective function at the minimum to get the covariance matrix, because this would again require a numerical computation of second derivatives.\nI will nevertheless attempt to compute the covariance matrix with Minuit’s HESSE algorithm to see what happens.\n\nm.values = r.x\nm.hesse()\n\n\n\n\nExternal\n\n\nFCN = 15.5 (χ²/ndof = 1.0)\nNfcn = 9332\n\n\nEDM = 0.0125 (Goal: 0.0002)\n\n\n\nINVALID Minimum\nABOVE EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\ns\n1.02e3\n0.04e3\n\n\n0\n\n\n\n\n1\nb\n990\n40\n\n\n0\n\n\n\n\n2\noffset\n-87.39e-3\n0.14e-3\n\n\n-0.2\n0.2\n\n\n\n3\nscale\n1.9848\n0.0015\n\n\n0.01\n2.5\n\n\n\n\n\n\n\n\ns\nb\noffset\nscale\n\n\ns\n1.85e+03\n-0.7e3 (-0.380)\n4.261e-6\n-185.2e-6 (-0.003)\n\n\nb\n-0.7e3 (-0.380)\n1.7e+03\n-28.018e-6 (-0.005)\n-217.3e-6 (-0.003)\n\n\noffset\n4.261e-6\n-28.018e-6 (-0.005)\n2.08e-08\n0.050e-6 (0.230)\n\n\nscale\n-185.2e-6 (-0.003)\n-217.3e-6 (-0.003)\n0.050e-6 (0.230)\n2.29e-06\n\n\n\n\n\nHESSE reports succeess, but the uncertainties on offset and scale are suspiciously small. We should not trust this result. This is one of the possible outcomes if the second derivatives are not well-defined bad. In other cases, HESSE may fail or report that the covariance matrix had to be forced to be positive definite.\nSince this does not work, we employ the bootstrap technique, using the resample library. This library provides a function which readily computes the covariance for a function that depends on a single dataset, but we have a more complicated case here. We need to resample three samples simultaneously, which have different sizes, the data sample and the two samples that generate the templates. We use the “extended” resampling method to also vary the sizes of these samples, which is usually the correct thing in high-energy physics, but not how the standard bootstrap in the literature works.\n\n# function which does the whole estimation; taking samples as input and\n# returning parameters of interest as output\ndef fit(x_sample, st_sample, bt_sample):\n n_data = np.histogram(x_sample, bins=xe)[0]\n template_b = np.histogram(bt_sample, bins=xe)[0]\n\n def model(s, b, offset, scale):\n mu_st_sample = np.mean(st_sample)\n st_sample_modified = scale * (st_sample - mu_st_sample) + mu_st_sample + offset\n template_s_modified = np.histogram(st_sample_modified, bins=xe)[0]\n # a small constant is added to avoid dividing by zero\n ts = template_s_modified / (np.sum(template_s_modified) + 1e-6)\n tb = template_b / np.sum(template_b)\n mu = s * ts + b * tb\n mu_var = s * ts**2 + b * tb**2\n return mu, mu_var\n\n def objective(s, b, offset, scale):\n mu, mu_var = model(s, b, offset, scale)\n return template_chi2_da(n_data, mu, mu_var)\n\n r = differential_evolution(\n lambda x: objective(*x),\n bounds=((0, 2000), (0, 2000), (-0.2, 0.2), (0.5, 3.5)),\n polish=False,\n seed=1,\n )\n assert r.success\n return r.x\n\n\nnumber_of_bootstrap_samples = 20\nkwargs = {\"size\": number_of_bootstrap_samples, \"method\": \"extended\", \"random_state\": 1}\n\nresults = []\nfor xs, sts, bts in zip(\n resample(x_sample, **kwargs),\n resample(st_sample, **kwargs),\n resample(bt_sample, **kwargs),\n):\n xb = fit(xs, sts, bts)\n results.append(xb)\n\ncov = np.cov(results, rowvar=False)\n\nfor name, val, err in zip(m.parameters, r.x, np.diag(cov) ** 0.5):\n print(f\"{name} = {val:.2f} +/- {err:.2f}\")\n\ns = 1016.88 +/- 71.60\nb = 987.33 +/- 51.06\noffset = -0.09 +/- 0.00\nscale = 1.98 +/- 0.16\n\n\nThe calculation takes a while but eventually completes. We now find larger uncertainties compared to the HESSE estimate above, which essentially treated the transform like it was fixed. We can see that the increase in the uncertainties indeed originates from the fitted transform, if we repeat the bootstrap with fixed the transformation parameters.\n\ndef fit(x_sample, st_sample, bt_sample):\n n_data = np.histogram(x_sample, bins=xe)[0]\n template_b = np.histogram(bt_sample, bins=xe)[0]\n\n def model(s, b, offset, scale):\n mu_st_sample = np.mean(st_sample)\n st_sample_modified = scale * (st_sample - mu_st_sample) + mu_st_sample + offset\n template_s_modified = np.histogram(st_sample_modified, bins=xe)[0]\n # a small constant is added to avoid dividing by zero\n ts = template_s_modified / (np.sum(template_s_modified) + 1e-6)\n tb = template_b / np.sum(template_b)\n mu = s * ts + b * tb\n mu_var = s * ts**2 + b * tb**2\n return mu, mu_var\n\n def objective(s, b, offset, scale):\n mu, mu_var = model(s, b, offset, scale)\n return template_chi2_da(n_data, mu, mu_var)\n\n r = differential_evolution(\n lambda x: objective(*x, -0.09, 1.99), bounds=((0, 2000), (0, 2000)), \n polish=False,\n seed=1\n )\n assert r.success\n return r.x\n\n\nkwargs = {\"size\": number_of_bootstrap_samples, \"method\": \"extended\", \"random_state\": 1}\n\nresults = []\nfor xs, sts, bts in zip(\n resample(x_sample, **kwargs),\n resample(st_sample, **kwargs),\n resample(bt_sample, **kwargs),\n):\n xb = fit(xs, sts, bts)\n results.append(xb)\n\ncov = np.cov(results, rowvar=False)\n\nfor name, val, err in zip(m.parameters, r.x, np.diag(cov) ** 0.5):\n print(f\"{name} = {val:.2f} +/- {err:.2f}\")\n\ns = 1016.88 +/- 50.19\nb = 987.33 +/- 39.07" + }, + { + "objectID": "posts/template_with_distortion.html#conclusion", + "href": "posts/template_with_distortion.html#conclusion", + "title": "Template fits with distortion correction", + "section": "Conclusion", + "text": "Conclusion\nAt least in some cases, it is possible to fit distorted templates and estimate the distortion parameters as nuisance parameters, and get good results with modest effort. The solution described here will work best if the components in the mixture are well separated and if the same distortion can be applied to all templates (for example, because it models a calibration mismatch).\nIn the following cases, this method will introduce large uncertainties on the fitted yields: - the components are strongly overlapping - different distortions are needed for each template - the distortion function is very flexible and has many parameters\nIn some of these difficult cases, reasonable results may be attainable if the distortion parameters are further constrained with penalty terms, that prevent the distortion function from diverging too much from the identity." + }, + { + "objectID": "posts/invariant mass combinatorial background.html", + "href": "posts/invariant mass combinatorial background.html", + "title": "Approximate PDF for the invariant mass distribution of combinatorial background", + "section": "", + "text": "We derive an approximate pdf for the invariant mass distribution from combinatorial background. Combinatorial background refers to random pairs of unrelated particles that are combined to compute the invariant mass of the hypothetical ancestor particle.\nIt is known empirically that an exponential distribution often provides a good description of this background. We show here why that is. We work in coordinates where \\(c=1\\).\nWe express the 3-momentum of a particle in \\(p_T, \\eta, \\phi\\) since the probability distributions approximately factorize:\n\\[\ng(p_T) \\, h_1(\\phi) \\, h_2(\\eta)\n\\]\nwhere \\(h_1(\\phi)\\) is a uniform distribution from \\(-\\pi, \\pi\\) and \\(h_2(\\eta)\\) is a somewhat bell-shaped pdf around 0 with a broadened center and with a dip at zero. We will see that the result does not depend on \\(h_2(\\eta)\\) within the limits of the approximation.\nThe distribution \\(g(p_T)\\) is approximately exponential for \\(p_T \\gg 1\\,\\text{GeV}c^{-1}\\). We will assume it is exponential, which is the first approximation.\nWe now express the invariant mass \\(m\\) of the hypothetical ancestor particle in these coordinates.\nfrom sympy import *\ndef ancestor_mass(pt1, eta1, phi1, pt2, eta2, phi2, m1, m2):\n # see e.g. Wikipedia https://en.wikipedia.org/wiki/Pseudorapidity\n \n px1 = cos(phi1) * pt1\n py1 = sin(phi1) * pt1\n pz1 = sinh(eta1) * pt1\n\n px2 = cos(phi2) * pt2\n py2 = sin(phi2) * pt2\n pz2 = sinh(eta2) * pt2\n\n # momentum squared of ancestor\n p_sq = (px1 + px2) ** 2 + (py1 + py2) ** 2 + (pz1 + pz2) ** 2\n \n # momentum squared of particles 1 and 2\n p1_sq = px1 ** 2 + py1 ** 2 + pz1 ** 2\n p2_sq = px2 ** 2 + py2 ** 2 + pz2 ** 2\n \n # energies of particles 1 and 2\n e1 = sqrt(p1_sq + m1 ** 2)\n e2 = sqrt(p2_sq + m2 ** 2)\n\n return sqrt((e1 + e2) ** 2 - p_sq)\nWe use SymPy to transform and simplify the formula for the ancestor mass. First we declare the variables.\npt1, pt2, m1, m2, lambd, m = symbols(\"p_T_1 p_T_2 m_1 m_2 lambda m\", positive=True, real=True)\n\neta1, eta2, phi1, phi2 = symbols(\"eta_1 eta_2 phi_1 phi_2\", real=True)\nancestor_mass(pt1, eta1, phi1, pt2, eta2, phi2, m1, m2)\n\n\\(\\displaystyle \\sqrt{- \\left(p_{T 1} \\sin{\\left(\\phi_{1} \\right)} + p_{T 2} \\sin{\\left(\\phi_{2} \\right)}\\right)^{2} - \\left(p_{T 1} \\cos{\\left(\\phi_{1} \\right)} + p_{T 2} \\cos{\\left(\\phi_{2} \\right)}\\right)^{2} - \\left(p_{T 1} \\sinh{\\left(\\eta_{1} \\right)} + p_{T 2} \\sinh{\\left(\\eta_{2} \\right)}\\right)^{2} + \\left(\\sqrt{m_{1}^{2} + p_{T 1}^{2} \\sin^{2}{\\left(\\phi_{1} \\right)} + p_{T 1}^{2} \\cos^{2}{\\left(\\phi_{1} \\right)} + p_{T 1}^{2} \\sinh^{2}{\\left(\\eta_{1} \\right)}} + \\sqrt{m_{2}^{2} + p_{T 2}^{2} \\sin^{2}{\\left(\\phi_{2} \\right)} + p_{T 2}^{2} \\cos^{2}{\\left(\\phi_{2} \\right)} + p_{T 2}^{2} \\sinh^{2}{\\left(\\eta_{2} \\right)}}\\right)^{2}}\\)\nThe raw expression for the ancestor mass is rather complex. We ask SymPy to simplify it.\nm_expr = ancestor_mass(pt1, eta1, phi1, pt2, eta2, phi2, m1, m2).simplify(); m_expr\n\n\\(\\displaystyle \\sqrt{m_{1}^{2} + m_{2}^{2} - 2 p_{T 1} p_{T 2} \\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} - 2 p_{T 1} p_{T 2} \\sinh{\\left(\\eta_{1} \\right)} \\sinh{\\left(\\eta_{2} \\right)} + 2 \\sqrt{m_{1}^{2} + p_{T 1}^{2} \\cosh^{2}{\\left(\\eta_{1} \\right)}} \\sqrt{m_{2}^{2} + p_{T 2}^{2} \\cosh^{2}{\\left(\\eta_{2} \\right)}}}\\)\nThis looks much better.\nWe now introduce the second approximation. We assume that \\(m_i \\ll p_{T,i}\\), which plays along nicely with the first approximation \\(p_{T,i} \\gg 1\\,\\text{GeV}c^{-1}\\) if the outgoing particles are light. In other words, this approximation can be expected to work well for decays into particles with masses \\(m_i \\ll 1\\,\\text{GeV}c^{-2}\\).\nWe expand the expression to first order in \\(m_{i}\\) in a Taylor series. The first order happens to be zero, so we simply get:\nm_approx_expr = m_expr.subs(m1, 0).subs(m2, 0).simplify(); Eq(m, m_approx_expr)\n\n\\(\\displaystyle m = \\sqrt{2} \\sqrt{p_{T 1}} \\sqrt{p_{T 2}} \\sqrt{- \\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} + \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}}\\)\nTo compute the pdf of the ancestor mass \\(m\\), we need to perform a change of variable and then integrate over the remaining coordinates.\nWe first make the change of variables.\nThe old variables are \\(p_{T,1}, p_{T,2}, \\eta_1, \\eta_2, \\phi_1, \\phi_2\\).\nThe new variables are \\(m, p_{T,2}, \\eta_1, \\eta_2, \\phi_1, \\phi_2\\).\nThe pdf in the new variables is\n\\[\nf'(m, p_{T,2}, \\eta_1, \\eta_2, \\phi_1, \\phi_2) = g(p_{T,1}) \\, g(p_{T,2}) \\, h_1(\\phi_1) \\, h_1(\\phi_2) \\, h_2(\\eta_1) \\, h_2(\\eta_2) \\,\n\\left|\\frac{\\partial(m, p_{T,2}, \\eta_1, \\eta_2, \\phi_1, \\phi_2)}{\\partial(p_{T,1}, p_{T,2}, \\eta_1, \\eta_2, \\phi_1, \\phi_2)} \\right|^{-1} \\\\\nf'(m, p_{T,2}, \\eta_1, \\eta_2, \\phi_1, \\phi_2) = g(p_{T,1}(m, p_{T,2}, \\dots)) \\, g(p_{T,2}) \\, h_1(\\phi_1) \\, h_1(\\phi_2) \\, h_2(\\eta_1) \\, h_2(\\eta_2) \\, \\left| \\frac{\\partial m}{\\partial p_{T, 1}} \\right|^{-1}\n\\]\nWe manually express \\(\\sqrt{p_{T, 1}}\\) as a function of \\(m\\), to replace it later. The SymPy solver has trouble doing this, as it tends to break down the \\(\\cosh\\) into exponentials, but we want to keep it intact.\nsqrt_pt1 = m / m_approx_expr * sqrt(pt1); Eq(sqrt(pt1), sqrt_pt1)\n\n\\(\\displaystyle \\sqrt{p_{T 1}} = \\frac{\\sqrt{2} m}{2 \\sqrt{p_{T 2}} \\sqrt{- \\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} + \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}}}\\)\npt1_m = sqrt_pt1 ** 2; Eq(pt1, pt1_m)\n\n\\(\\displaystyle p_{T 1} = \\frac{m^{2}}{2 p_{T 2} \\left(- \\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} + \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}\\right)}\\)\nWe compute \\(\\left| \\frac{\\partial m}{\\partial p_{T, 1}} \\right|^{-1}\\) and express it as a function of \\(m\\).\ndpt1_dm = 1/diff(m_approx_expr, pt1); dpt1_dm\n\n\\(\\displaystyle \\frac{\\sqrt{2} \\sqrt{p_{T 1}}}{\\sqrt{p_{T 2}} \\sqrt{- \\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} + \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}}}\\)\ndpt1_dm = dpt1_dm.subs(sqrt(pt1), sqrt_pt1); dpt1_dm\n\n\\(\\displaystyle \\frac{m}{p_{T 2} \\left(- \\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} + \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}\\right)}\\)\n# exponential pdf\ndef g(pt):\n return lambd * exp(-lambd * pt)\n# uniform pdf from -pi, pi\ndef h1(phi):\n return 1/(2 * pi)\n# unspecified pdf for eta\nh2 = Function(\"h2\")\nf_prime = simplify(dpt1_dm * g(pt1_m) * g(pt2) * h1(phi1) * h1(phi2) * h2(eta1) * h2(eta2)); f_prime\n\n\\(\\displaystyle - \\frac{\\lambda^{2} m \\operatorname{h_{2}}{\\left(\\eta_{1} \\right)} \\operatorname{h_{2}}{\\left(\\eta_{2} \\right)} e^{\\lambda \\left(\\frac{m^{2}}{2 p_{T 2} \\left(\\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} - \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}\\right)} - p_{T 2}\\right)}}{4 \\pi^{2} p_{T 2} \\left(\\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} - \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}\\right)}\\)\nTo obtain \\(f(m)\\), we need to integrate \\(f'\\) over \\(p_{T,2}, \\phi_1, \\phi_2, \\eta_1, \\eta_2\\).\nIntegral(f_prime, \n (pt2, 0, oo),\n (eta1, \"eta_a\", \"eta_b\"), (eta2, \"eta_a\", \"eta_b\"),\n (phi1, -pi, pi), (phi2, -pi, pi))\n\n\\(\\displaystyle \\int\\limits_{- \\pi}^{\\pi}\\int\\limits_{- \\pi}^{\\pi}\\int\\limits_{\\eta_{a}}^{\\eta_{b}}\\int\\limits_{\\eta_{a}}^{\\eta_{b}}\\int\\limits_{0}^{\\infty} \\left(- \\frac{\\lambda^{2} m \\operatorname{h_{2}}{\\left(\\eta_{1} \\right)} \\operatorname{h_{2}}{\\left(\\eta_{2} \\right)} e^{\\lambda \\left(\\frac{m^{2}}{2 p_{T 2} \\left(\\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} - \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}\\right)} - p_{T 2}\\right)}}{4 \\pi^{2} p_{T 2} \\left(\\cos{\\left(\\phi_{1} - \\phi_{2} \\right)} - \\cosh{\\left(\\eta_{1} - \\eta_{2} \\right)}\\right)}\\right)\\, dp_{T 2}\\, d\\eta_{1}\\, d\\eta_{2}\\, d\\phi_{1}\\, d\\phi_{2}\\)\nThis integral does not look analytically tractable. Since it contains an exponential function, Laplace’s method comes to mind as an approximation. To apply it, the function \\(g(\\vec x)\\) in the exponential \\(e^{M g(\\vec x)}\\) must have a global maximum inside the integration bounds. We investigate \\(g(\\vec x)\\) numerically to see whether that is the case.\ndef g(x, y, z):\n return 1/(x * (np.cos(z) - np.cosh(y) + 1e-100)) - x\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nxn = np.linspace(0.05, 3, 100)\nyn = np.linspace(2, 5)\nzn = np.linspace(-2*np.pi, 2* np.pi)\n\nfig, ax = plt.subplots(1, 3, figsize=(14, 4))\nplt.sca(ax[0])\nY, Z = np.meshgrid(yn, zn)\nplt.pcolormesh(Y, Z, g(0.5, Y, Z), shading=\"nearest\")\nplt.colorbar()\nplt.sca(ax[1])\nX, Y = np.meshgrid(xn, yn)\nplt.pcolormesh(X, Y, g(X, Y, 0), shading=\"nearest\")\nplt.colorbar()\nplt.sca(ax[2])\nfor yi in yn[::10]:\n plt.plot(xn, g(xn, yi, 0))\nThere is indeed a global maximum as a function of \\(p_{T,2}\\) between 0 and infinity, but not with respect to \\(\\Delta\\eta\\). The maximum for \\(\\Delta\\eta\\) is located at the upper bound of the \\(\\eta\\) interval, which means that Laplace’s method cannot be applied to the integral as a whole.\nWe cannot solve the integral completely, so we do the next-best thing. We partially solve it by approximately carrying out the integration over \\(p_{T,2}\\) for each value of \\(C(\\Delta\\eta, \\Delta\\phi) = \\cosh\\Delta\\eta - \\cos\\Delta\\phi\\) with Laplace’s method:\n\\[\n\\int h(x) e^{M g(x)} \\text{d}x \\approx \\sqrt{\\frac{2\\pi}{M |g''(x_0)|}} h(x_0) e^{M g(x_0)}\n\\]\nwith \\(M > 0, M \\rightarrow \\infty, h(x) > 0\\) and \\(x_0\\) being the maximum of \\(g(x)\\).\nTo obtain simpler formulas, we make a bunch of substitutions:\ndeta, dphi = symbols(r\"\\Delta\\eta \\Delta\\phi\")\n\nx = symbols(\"x\", real=True, positive=True)\nC = Function(\"C\", real=True, positive=True)(deta, dphi)\n\nf_prime = f_prime.subs((cosh(eta1 - eta2)-cos(phi1 - phi2)), C).subs(eta1, eta2 + deta).subs(phi1, phi2 + dphi).subs(pt2, x).simplify(); f_prime\n\n\\(\\displaystyle \\frac{\\lambda^{2} m \\operatorname{h_{2}}{\\left(\\eta_{2} \\right)} \\operatorname{h_{2}}{\\left(\\Delta\\eta + \\eta_{2} \\right)} e^{- \\lambda \\left(\\frac{m^{2}}{2 x C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}} + x\\right)}}{4 \\pi^{2} x C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}\\)\nM = lambd\ng = -m ** 2 / (2 * x * C) - x; g\n\n\\(\\displaystyle - \\frac{m^{2}}{2 x C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}} - x\\)\nh = f_prime / exp(M * g).simplify(); h\n\n\\(\\displaystyle \\frac{\\lambda^{2} m \\operatorname{h_{2}}{\\left(\\eta_{2} \\right)} \\operatorname{h_{2}}{\\left(\\Delta\\eta + \\eta_{2} \\right)}}{4 \\pi^{2} x C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}\\)\nassert simplify(h * exp(M * g)) == f_prime\ngx = diff(g, x); gx\n\n\\(\\displaystyle \\frac{m^{2}}{2 x^{2} C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}} - 1\\)\ngxx = diff(g, x, 2); gxx\n\n\\(\\displaystyle - \\frac{m^{2}}{x^{3} C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}\\)\nx0 = solve(Eq(gx, 0), x)[0]; x0 # there is only one solution, since x > 0\n\n\\(\\displaystyle \\frac{\\sqrt{2} m}{2 \\sqrt{C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}}\\)\ngxx0 = gxx.subs(x, x0); gxx0\n\n\\(\\displaystyle - \\frac{2 \\sqrt{2} \\sqrt{C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}}{m}\\)\nh0 = h.subs(x, x0); h0\n\n\\(\\displaystyle \\frac{\\sqrt{2} \\lambda^{2} \\operatorname{h_{2}}{\\left(\\eta_{2} \\right)} \\operatorname{h_{2}}{\\left(\\Delta\\eta + \\eta_{2} \\right)}}{4 \\pi^{2} \\sqrt{C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}}\\)\ng0 = g.subs(x, x0); g0\n\n\\(\\displaystyle - \\frac{\\sqrt{2} m}{\\sqrt{C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}}\\)\nNow we have everything to get Laplace’s approximate solution.\nsqrt(2 * pi / (M * -gxx0)) * h0 * exp(M * g0).simplify()\n\n\\(\\displaystyle \\frac{\\sqrt[4]{2} \\lambda^{\\frac{3}{2}} \\sqrt{m} \\operatorname{h_{2}}{\\left(\\eta_{2} \\right)} \\operatorname{h_{2}}{\\left(\\Delta\\eta + \\eta_{2} \\right)} e^{- \\frac{\\sqrt{2} \\lambda m}{\\sqrt{C{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}}}}{4 \\pi^{\\frac{3}{2}} C^{\\frac{3}{4}}{\\left(\\Delta\\eta,\\Delta\\phi \\right)}}\\)\nIn other words, the approximate solution for each value of \\(C(\\Delta\\eta, \\Delta\\phi)\\) is\n\\[\nf(m, C) \\propto m^{1/2} \\, e^{- m \\, \\lambda_C}\n\\]\nwhere \\(\\lambda_C\\) is proportional to \\(C^{-1/2}\\). Since the function has a broad maximum, the actual function \\(f(m)\\) will be a superposition of many \\(f(m, C)\\) with similar slopes.\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nyn = np.linspace(2, 5)\nplt.plot(yn, 1/np.sqrt(np.cosh(yn) - 1), label=r\"$C^{-1/2}(\\eta, 0)$\")\nplt.legend()\nplt.xlabel(r\"$\\eta$\");" + }, + { + "objectID": "posts/invariant mass combinatorial background.html#simulation-study", + "href": "posts/invariant mass combinatorial background.html#simulation-study", + "title": "Approximate PDF for the invariant mass distribution of combinatorial background", + "section": "Simulation study", + "text": "Simulation study\nWe confirm the analytical results in a toy simulation. We first generate some toy data. The \\(p_T\\) distribution is exponential. The distribution in \\(\\eta\\) and \\(\\phi\\) are uniform. We then form all unique pairs and compute the invariant mass for the case with candidate masses assumed to be zero.\n\nimport numpy as np\n\nrng = np.random.default_rng(1)\n\npt = rng.exponential(size=1000)\neta = rng.uniform(2, 5, size=1000)\nphi = rng.uniform(-np.pi, np.pi, size=1000)\n\nf = lambdify((pt1, eta1, phi1, pt2, eta2, phi2), m_approx_expr, \"math\")\n\n# this is slow in pure Python, but we don't care for a one-shot\nm = []\nfor i in range(1000):\n for j in range(i):\n m.append(f(pt[i], eta[i], phi[i], pt[j], eta[j], phi[j]))\nm = np.array(m)\n\nWe make a histogram of the resulting distribution.\n\nn, xe = np.histogram(m, bins=50)\n\nfrom pyik.mplext import plot_hist\n\nplot_hist(xe, n)\nplt.xlabel(\"m\")\nplt.ylabel(\"counts\")\nplt.semilogy();\n\n\n\n\nIt looks approximately like an exponential, but there are differences. The peak at zero is cut off and the slope varies, being steeper near zero and flatter in the tail. In a narrow interval, however, the exponential distribution is expected to give a good description.\nWe fit a simple exponential model and a more complex model that takes our analytical solution in account. It is a superposition of three distributions of the form \\(A_i \\sqrt{m} e^{-B_i m}\\) with approximate the integral that we cannot compute analytically.\n\nfrom iminuit import Minuit\nfrom iminuit.cost import LeastSquares\n\ndef model_A(x, a, b):\n return a * np.exp(- x * b)\n\ndef model_B(x, par):\n npar = len(par) // 2\n par = np.asarray(par)[:, None]\n return x ** 0.5 * np.sum(par[:npar] * np.exp(-x * par[npar:]), axis=0)\n\ncx = (xe[1:] + xe[:-1]) / 2\nnerr = (n + 0.5) ** 0.5\n\n\nmin1 = Minuit(LeastSquares(cx, n, nerr, model_A), 1, 1)\nmin1.limits = (0, None)\nmin1.migrad()\n\n\n\n\nFCN = 7712\nNfcn = 177\n\n\nEDM = 4.59e-07 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n153.21e3\n0.30e3\n\n\n0\n\n\n\n\n1\nb\n691.0e-3\n0.9e-3\n\n\n0\n\n\n\n\n\n\n\n\n\na\nb\n\n\na\n8.84e+04\n0.182 (0.678)\n\n\nb\n0.182 (0.678)\n8.14e-07\n\n\n\n\n\n\nmin2 = Minuit(LeastSquares(cx, n, nerr, model_B), np.ones(6))\nmin2.limits = (0, None)\nmin2.migrad(ncall=10000)\n\n\n\n\nFCN = 149\nNfcn = 1583\n\n\nEDM = 8.62e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n3.2e3\n1.5e3\n\n\n0\n\n\n\n\n1\nx1\n197e3\n6e3\n\n\n0\n\n\n\n\n2\nx2\n123e3\n5e3\n\n\n0\n\n\n\n\n3\nx3\n0.500\n0.033\n\n\n0\n\n\n\n\n4\nx4\n1.744\n0.033\n\n\n0\n\n\n\n\n5\nx5\n0.864\n0.017\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\nx2\nx3\nx4\nx5\n\n\nx0\n2.35e+06\n-6.8e+06 (-0.769)\n4.87e+06 (0.607)\n49.3 (0.982)\n29.4 (0.583)\n23.5 (0.900)\n\n\nx1\n-6.8e+06 (-0.769)\n3.32e+07\n-2.85e+07 (-0.946)\n-128 (-0.680)\n-163 (-0.859)\n-93.8 (-0.956)\n\n\nx2\n4.87e+06 (0.607)\n-2.85e+07 (-0.946)\n2.74e+07\n86.3 (0.503)\n165 (0.958)\n78.7 (0.884)\n\n\nx3\n49.3 (0.982)\n-128 (-0.680)\n86.3 (0.503)\n0.00107\n0.000538 (0.499)\n0.00046 (0.825)\n\n\nx4\n29.4 (0.583)\n-163 (-0.859)\n165 (0.958)\n0.000538 (0.499)\n0.00108\n0.000464 (0.830)\n\n\nx5\n23.5 (0.900)\n-93.8 (-0.956)\n78.7 (0.884)\n0.00046 (0.825)\n0.000464 (0.830)\n0.00029\n\n\n\n\n\n\nfig, ax = plt.subplots(1, 2, figsize=(10, 4), constrained_layout=True)\nplt.sca(ax[0])\nplot_hist(xe, n)\nxm = np.linspace(xe[0], xe[-1], 1000)\nplt.plot(xm, model_A(xm, *min1.values), label=f\"exponential: $\\\\chi^2/n_{{dof}} = {min1.fval:.1f}/{len(cx) - min1.nfit}$\")\nplt.plot(xm, model_B(xm, min2.values), label=f\"superposition: $\\\\chi^2/n_{{dof}} = {min2.fval:.1f}/{len(cx) - min2.nfit}$\")\nplt.semilogy()\nplt.legend()\nplt.xlabel(\"m\")\nplt.sca(ax[1])\nplot_hist(xe, n)\nplt.plot(xm, model_A(xm, *min1.values))\nplt.plot(xm, model_B(xm, min2.values))\nplt.xlabel(\"m\")\n\nText(0.5, 0, 'm')\n\n\n\n\n\nThe sophisticated superposition model globally describes the distribution much better than the plain exponential model. The cut-off near zero is modeled by the term \\(\\sqrt{m}\\), which suppresses the rise of the pure exponentials near zero. The superposition of three exponentials with different slopes is able to match the changing slope.\nEven for the sophisticated model, \\(\\chi^2\\) does not look so good, but that is because we use the approximate least-squares method here to keep it simple, in which we do not integrate the model over each bin. The implied approximation is bad near \\(m = 0\\), where the distribution has a sharp peak.\nFor an approximation over a local interval not too close to \\(m = 0\\), a simple exponential distribution will also perform well, since locally one of the exponentail distributions is dominant." + }, + { + "objectID": "posts/MachineLearning.html", + "href": "posts/MachineLearning.html", + "title": "Introduction into Machine Learning", + "section": "", + "text": "Hans Dembinski" + }, + { + "objectID": "posts/MachineLearning.html#overview", + "href": "posts/MachineLearning.html#overview", + "title": "Introduction into Machine Learning", + "section": "Overview", + "text": "Overview\n\n\nCaution: Field is riddled with buzzwords\n\nMachine learning, artificial intelligence, deep learning, belief networks, attention-based networks, long short-term memory, features, hyperparameters …\n\nAnthropomorphized and technical names hide what are usually basic mathematical algorithms\nMachine learning: umbrella term for multivariate statistical methods for classification and regression\n\nMultivariate: algorithms use more than one variable as input, e.g. (x, y, z)\nStatistical: algorithms can handle noisy input and are founded on statistical theory\nClassification: sort inputs based on input variables into categories\nRegression: find a mapping of input vectors to output vectors (fitting)\n\n\n\n\nRise of the machines: Why do we need them?\n\nHumans good at working with 1D or 2D data, because we can visualize it\nHumans bad at working with higer-dimensional data\nAlgorithms (“machines”) scale well to high-dimensional data\n\n\n\nNotable software\n\nMultiple algorithms and utilities\n\nScikit-Learn\nROOT TMVA\n\nGradient-boosted decision trees (aka gradient boosting machines)\n\nXGBoost\nLightGBM\nCatBoost\n…\n\nNeural networks\n\nTensorflow\nKeras (high-level interface to Tensorflow)\nPyTorch\n…\n\n\n\n\nLearning resources\n\nInternet provides excellent resources for autodidactic learning\n\nExample code\nTutorials\nDocumentation \n\nScikit-Learn\nKaggle\nAll methods can do classification and regression and work with any input\nWhat to use?\n\nStructured data (i.e. tables of numbers): Ensembles of decision trees\n\nGood results with default settings\nCan process continuous and categorical input (m/w, red/green/blue, …)\nNo data preprocessing required\nCompute feature importance\n\nUnstructured data (i.e. images, waveforms, text…): Neural networks\n\nDeep networks learn to extract features automatically\nRequires data preprocessing\nNetworks difficult to optimize for non-experts\nOften need very large data sets for training\n\n\n\nSome definitions * sample: one or more numbers which describe outcome of stochastic process * dataset: consists of samples * features: subset of numbers from each sample which carry interesting information, also numbers computed from each sample * signal: samples which originate from stochastic process of interest, typically particle decay * background: samples which do not originate from process of interest\nDiscuss how the typical decay \\(K^0_S \\to \\pi^+ \\pi^-\\) is reconstructed and why in general there is always some background.\nClarify how the previous terms would be used in the context of this example." + }, + { + "objectID": "posts/MachineLearning.html#classification", + "href": "posts/MachineLearning.html#classification", + "title": "Introduction into Machine Learning", + "section": "Classification", + "text": "Classification\n\nSetting: dataset is mix of two or more components, typically signal + one or more backgrounds\nWant only signal component, invert a cut to select signal candidates\nFeatures are variables which allow to discriminate between signal and background, for example, reconstructed mass of decay candidate\n\n\n#!pip install scipy matplotlib numpy scikit-learn xgboost\n\n\nimport scipy\nimport matplotlib\nimport numpy\nimport sklearn\nimport xgboost\n\nprint(f\"scipy {scipy.__version__}\")\nprint(f\"matplotlib {matplotlib.__version__}\")\nprint(f\"numpy {numpy.__version__}\")\nprint(f\"sklearn {sklearn.__version__}\")\nprint(f\"xgboost {xgboost.__version__}\")\n\nscipy 1.10.0\nmatplotlib 3.7.1\nnumpy 1.21.6\nsklearn 1.2.2\nxgboost 1.7.4\n\n\n\nfrom scipy.stats import norm\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\nbackground = norm(0, 1).rvs(1000, random_state=1)\nsignal = norm(1, 0.1).rvs(500, random_state=1)\n\nx = np.append(signal, background)\ny = np.append(np.ones(len(signal)), np.zeros(len(background)))\n\n\ndef plot(x, y):\n signal = x[y == 1]\n background = x[y == 0]\n t, xe = np.histogram(x, bins=100)\n s = np.histogram(signal, bins=xe)[0]\n b = np.histogram(background, bins=xe)[0]\n plt.stairs(b, xe, fill=True, label=\"background\")\n plt.stairs(t, xe, baseline=b, fill=True, label=\"signal\")\n plt.xlabel(\"x\")\n plt.legend();\n return xe\n\nplot(x, y);\n\n\n\n\nExercise: * Propose simple selection (“cut”) that keeps most of the signal and removes most of the background * Is it possible to eliminate the background completely?\n\nEfficiency and purity of a selection\n\nEfficiency: fraction of signal left after applying selection\n\nNumber of true signal samples after cut / Number of all true signal samples\n\nPurity: fraction of background still contained in selected sample\n\nNumber of true signal samples after cut / Number of selected samples\n\nWant to maximize efficiency and purity\n\n\nSingle-parameter cut\n\nxmin = 0.7\n\nselected_signal = np.sum(x[y==1] > xmin)\ntotal_signal = np.sum(y==1)\nselected = np.sum(x > xmin)\n\nefficiency = selected_signal / total_signal\npurity = selected_signal / selected\n\nplot(x, y)\nplt.axvspan(-3.5, xmin, facecolor=\"k\", alpha=0.5)\nplt.legend(title=f\"efficiency={efficiency:.2f}\\npurity={purity:.2f}\");\n\n\n\n\n\nxmin = np.linspace(-3, 3, 100)\nefficiency = []\npurity = []\nfor xm in xmin:\n selected_signal = np.sum(signal > xm)\n total_signal = len(signal)\n selected = np.sum(x > xm)\n e = selected_signal / total_signal\n p = selected_signal / selected\n efficiency.append(e)\n purity.append(p)\nplt.plot(xmin, efficiency, label=\"efficiency\")\nplt.plot(xmin, purity, label=\"purity\")\nplt.xlabel(\"xmin\")\nplt.legend();\n\n\n\n\n\nplt.scatter(efficiency, purity, c=xmin);\nplt.xlim(0, 1)\nplt.ylim(0, 1)\nplt.xlabel(\"efficiency\")\nplt.ylabel(\"purity\")\nplt.colorbar().set_label(\"xmin\")\n\n\n\n\n\narg = np.argmax(purity)\nprint(\"efficiency at max purity\", efficiency[arg])\n\nefficiency at max purity 0.904\n\n\n\nHow to choose cut?\n\nWant to maximize efficiency and purity, but what trade-off between the two?\n\nOften subjective choice is acceptable, because exact answer is difficult\nObjective answer given by figure-of-merit (FOM)\n\nPopular is Punzi FOM\nMost FOM (including Punzi FOM) only minimize statistical uncertainty\nCorrect FOM has to minimize total uncertainty = statistical ⊕ systematic\nSystematic uncertainty often difficult to quantify: fallback to Punzi or subjective choice\n\n\n\n\nBDT-based cut\n\nWe can do better with two cuts, on xmin and xmax\nSpace of cut parameters then already two-dimensional\nWith 2D data\n\nFour rectangular cuts possible (4 cut parameters)\nInfinitely many non-rectangular cuts possible\n\nUse machine instead, needs to be trained on labeled data\nUse BDT classifier from XGBoost\nGradient boosting machines regularly win Kaggle competitions\nWell documented, fast, parallizable, tunable\n\n\nfrom xgboost.sklearn import XGBClassifier\nfrom sklearn.model_selection import train_test_split\n\n# generate training data, ideally we have more than actual data\nbackground = norm(0, 1).rvs(100000, random_state=2)\nsignal = norm(1, 0.1).rvs(50000, random_state=2)\nx2 = np.append(signal, background)\n\n# features in general matrix [N, K] for N samples and K features\nX = x2.reshape((len(x2), 1))\n# labels: signal=1, background=0\ny = np.append(np.ones_like(signal), np.zeros_like(background))\n\n\n# ALWAYS do this: split into train and test datasets\nX_train, X_test, y_train, y_test = train_test_split(X, y, random_state=3)\n\n# actual magic happens here\nclf = XGBClassifier(random_state=4)\nclf.fit(X_train, y_train);\n\n\nfig, ax = plt.subplots(2, sharex=True)\n\nplt.sca(ax[0])\nxe = plot(X_test[:, 0], y_test)\nplt.xlabel(None) # unset label set by `plot`\n\nplt.sca(ax[1])\nscore = clf.score(X_test, y_test)\nxm = np.linspace(xe[0], xe[-1], 200)\nps, pb = clf.predict_proba(xm).T\n\nplt.xlabel(\"x\")\nplt.plot(xm, ps, \"C0-\", label=\"background probability\")\nplt.plot(xm, pb, \"C1-\", label=\"signal probability\")\nplt.legend(title=f\"score={score:.2f}\", frameon=False);\n\n\n\n\nExercise: * Vary some hyperparameters in XGBClassifier to see how score, runtime, and predicted probability changes * n_estimators try e.g. 2, 5, 10 * max_depth try e.g. 1, 5, 10 * learning_rate try e.g. 0.001, 0.05, 0.5, 1\n\nBDT predicts signal probability\nWe use this to implement our cut: select regions with high signal probability\n\n\nps = clf.predict_proba(X_test[y_test==1])[:, 1]\npb = clf.predict_proba(X_test[y_test==0])[:, 1]\nxe = plt.hist(ps, bins=20, alpha=0.5, label=\"true signal\")[1]\nplt.hist(pb, bins=xe, alpha=0.5, label=\"true background\")\nplt.legend()\nplt.xlabel(\"predicted signal probability\")\nplt.ylim(0, 5e3);\n\n\n\n\n\nxe = plot(X_test[:, 0], y_test)\n\n# chosen cut: signal probability > 0.6\n# visualize accepted region\nxm = np.linspace(xe[0], xe[-1], 200)\nmask = clf.predict_proba(xm)[:, 1] > 0.6\nplt.plot(xm, mask * 5000, \"k-\");\n\n\n\n\n\npmin = np.linspace(0, 1, 100)\nefficiency = []\npurity = []\nfor pm in pmin:\n mask = clf.predict_proba(X_test)[:, 1] > pm\n selected = np.sum(mask)\n selected_signal = np.sum(y_test[mask])\n total_signal = np.sum(y_test)\n efficiency.append(selected_signal / total_signal)\n with np.errstate(divide=\"ignore\", invalid=\"ignore\"):\n purity.append(selected_signal / selected)\nplt.plot(pmin, efficiency, label=\"efficiency\")\nplt.plot(pmin, purity, label=\"purity\")\nplt.xlabel(\"pmin\")\nplt.legend();\n\n\n\n\n\nplt.scatter(efficiency, purity, c=pmin);\nplt.xlim(0, 1)\nplt.ylim(0, 1)\nplt.xlabel(\"efficiency\")\nplt.ylabel(\"purity\")\nplt.colorbar().set_label(\"pmin\")\n\n\n\n\n\nPhysicists like efficiency and purity, but ROC curve is standard\nReceiver-operator characteristic\nROC plots true positive rate (fraction of signal that passes cut) over false positive rate (fraction of background that passes cut)\nScikit-Learn provides a quick way to generate this plot\n\n\nfrom sklearn.metrics import RocCurveDisplay\n\nRocCurveDisplay.from_estimator(clf, X_test, y_test);\n\n\n\n\n\narea under curve (AUC) is a common metric to compare the performance of two different classifiers\ncan also use Scikit-Learn to compute ROC points and AUC and plot these ourselves\n\n\nfrom sklearn.metrics import roc_curve, auc\n\n# plot XGBoost curve again\nfpr, tpr, thresholds = roc_curve(y_test, clf.predict_proba(X_test)[:, 1])\nplt.plot(fpr, tpr, label=f\"XGBoost AUC={auc(fpr, tpr):.2f}\")\n\n# plot ROC curve for our simple x > 0.7 cut for comparison\nfpr2, tpr2, _ = roc_curve(y_test, X_test[:, 0] > 0.7)\nplt.plot(fpr2, tpr2, label=f\"x > 0.7 AUC={auc(fpr2, tpr2):.2f}\")\n\nplt.xlabel(\"False positive rate\")\nplt.ylabel(\"True positive rate\")\nplt.legend();\n\n\n\n\n\nExercise: run scikit-learn example\nTry adding XGBClassifier into the comparison\nPlay around with the hyperparameters of the classifiers" + }, + { + "objectID": "posts/MachineLearning.html#regression", + "href": "posts/MachineLearning.html#regression", + "title": "Introduction into Machine Learning", + "section": "Regression", + "text": "Regression\n\nRegression (aka fitting): “learn” function y = f(x) from points(x, y), where both x and y can be vectors\nMachines can do this, but we rarely need this in HEP\nWe usually prefer to use theoretically motivated models or hand-crafted low-order polynoms\nLet’s try regression on one-dimensional function\n\n\ndef model(x, a, b, c, d):\n return a + b * np.sin(c * x) * np.exp(- d * x)\n\ntruth = 0.2, 1, 2, 0.5\n\nx = np.linspace(0, 10, 1000)\nplt.plot(x, model(x, *truth))\nplt.xlabel(\"x\")\nplt.ylabel(\"y\");\n\n\n\n\n\ngenerate random points on this curve and add some noise to make it more realistic\n\n\nrng = np.random.default_rng(1)\n\nx = rng.uniform(0, 10, 1000)\ny = model(x, 0.2, 1, 2, 0.5)\ny += rng.normal(0, 0.05, len(y))\n\nX = np.reshape(x, (-1, 1))\n\n\n(X, y) are inputs for regression similar to before, but y is a real number now, not a label (integer)\n\n\nNeural network\n\nWe use MLPRegressor from Scikit-Learn, a basic neural network\nNeural networks always need preprocessing\nScikit-Learn offers pipelines to automate this, use them\n\n\nfrom sklearn.neural_network import MLPRegressor\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.pipeline import make_pipeline\n\n# network with 5 hidden layers with 100 nodes each\nregressor = MLPRegressor(\n hidden_layer_sizes=(100,)*5,\n random_state=1)\n\n# we use basic StandardScaler for preprocessing\npipeline = make_pipeline(StandardScaler(), regressor)\npipeline.fit(X, y);\n\n\n# compute fitted curve\nX2 = np.reshape(np.linspace(0, 10, 1000), (-1, 1))\ny2 = pipeline.predict(X2)\n\nplt.plot(x, y, \".\")\nplt.plot(X2[:, 0], y2, color=\"r\", label=\"Neural net\")\nplt.plot(X2[:, 0], model(X2[:, 0], *truth), color=\"k\", label=\"truth\")\nplt.legend()\nplt.xlabel(\"x\")\nplt.ylabel(\"y\");\n\n\n\n\n\nExercise: play around with hyperparameters of MLPRegressor\nWhat happens if you remove the StandardScaler?\nWhat performs better, deep or wide networks?\n\n\n\nGaussian Process\n\nFor low-dimensional data, Gaussian Process Regression can work well\nNot easy to set up: need to select suitable kernels and configure them\nAdvantage: Predicts uncertainty and automatically provides smooth solution\n\n\nfrom sklearn.gaussian_process import GaussianProcessRegressor\nfrom sklearn.gaussian_process.kernels import RBF, WhiteKernel\n\nkernel = RBF() + WhiteKernel()\n\ngpr = GaussianProcessRegressor(kernel=kernel)\ngpr.fit(X, y)\ny3, y3err = gpr.predict(X2, return_std=True)\n\n\nplt.plot(x, y, \".\")\nplt.plot(X2[:, 0], model(X2[:, 0], *truth), color=\"k\", label=\"truth\")\nplt.plot(X2[:, 0], y2, color=\"r\", label=\"Neural net\")\nl = plt.plot(X2[:, 0], y3, label=\"Gaussian Process\")[0]\nplt.fill_between(X2[:, 0], y3 - y3err, y3 + y3err, alpha=0.8, color=l.get_color())\nplt.legend();" + }, + { + "objectID": "posts/tracking_efficiency.html", + "href": "posts/tracking_efficiency.html", + "title": "Tracking efficiency for two-body decay", + "section": "", + "text": "Consider decay \\(A \\to B C\\)\nWant to know corrected efficiency \\(\\epsilon'_A\\)\n\nGiven: \\(\\epsilon_A(x)\\), \\(\\epsilon_B(x)\\), \\(\\epsilon_C(x)\\), correction factor \\(\\gamma(x)\\) for B, C\nEfficiency and correction factors depend on variable \\(x\\) (can be multidimensional)\n\\(f(x_B, x_C)\\) is probability density to find decay products \\(B\\) and \\(C\\) at locations \\(x_B\\) and \\(x_C\\), respectively\n\\[\n\\epsilon'_A = \\int \\text{d}x_B\\! \\int \\text{d}x_C\\, \\epsilon'_B(x_B) \\, \\epsilon'_C(x_C) \\; f(x_B, x_C)\n\\]\n\\[\n\\epsilon'_A = \\int \\text{d}x_B\\! \\int \\text{d}x_C \\, \\gamma(x_B) \\, \\gamma(x_C) \\, \\epsilon_B(x_B) \\, \\epsilon_C(x_C) \\; f(x_B, x_C)\n\\]\n\\[\n\\epsilon'_A = \\text{E}[\\gamma(x_B) \\, \\gamma(x_C) \\, \\epsilon_B(x_B) \\, \\epsilon_C(x_C)]\n\\]" + }, + { + "objectID": "posts/tracking_efficiency.html#compute-sample-estimator", + "href": "posts/tracking_efficiency.html#compute-sample-estimator", + "title": "Tracking efficiency for two-body decay", + "section": "Compute sample estimator", + "text": "Compute sample estimator\n\nApply plugin estimate \\(\\text{E}[f(x)] \\to \\frac 1N \\sum_k f(x_k)\\), \\(N\\) is generated number of A decays\n\n\\[\n\\epsilon'_A = \\frac 1 N \\sum_k \\gamma(x_{B,k}) \\, \\gamma(x_{C,k}) \\, \\epsilon_B(x_{B,k}) \\, \\epsilon_C(x_{C,k})\n\\]\n\nApply second plugin estimate: \\(\\sum_k \\epsilon(x_k) \\to \\sum_k \\beta_k\\), with \\(\\beta_k \\in \\{0, 1\\}\\), \\(\\beta_k\\) is outcome of the Bernoulli process which determines whether child is reconstructed or not\n\n\\[\n\\epsilon'_A = \\frac 1 N \\sum_k \\gamma(x_{B,k}) \\, \\gamma(x_{C,k}) \\, \\beta_{B,k} \\, \\beta_{C,k}\n\\]\n\nFor \\(\\gamma(x)\\) that is piece-wise constant over \\(x\\)-bins \\(i\\), this is equal to \\[\n\\epsilon'_A = \\frac 1 N \\sum_{i,j} n_{ij} \\, \\gamma_{i} \\, \\gamma_{j} \\quad\n\\text{with} \\quad n_{ij} = \\sum_k \\delta_{B,ik} \\delta_{C,jk} \\beta_{B,k} \\beta_{C,k}\n\\]\n\n\\(\\delta_{X,ik}\\) is one if particle X from decay \\(k\\) is inside \\(x\\)-bin \\(i\\) and zero otherwise\n\nIf we only sum over decays \\(\\ell\\) with \\(\\beta_{\\ell,B} \\, \\beta_{\\ell,C} = 1\\), this further simplifies to \\[\nn_{ij} = \\sum_{\\ell} \\delta_{B,i\\ell} \\delta_{C,j\\ell}\n\\]\nCorrection factor for \\(A\\) then is with \\(n = \\sum_{i,j} n_{i,j}\\)\n\n\\[\n\\gamma_A = \\frac{\\epsilon'_A}{\\epsilon_A} = \\frac{\\sum_{i,j} n_{ij} \\, \\gamma_{i} \\, \\gamma_{`j}}{\\sum_{i,j} n_{ij}} = \\frac 1 n \\sum_{i,j} n_{ij} \\, \\gamma_{i} \\, \\gamma_{j}\n\\]" + }, + { + "objectID": "posts/tracking_efficiency.html#error-propagation", + "href": "posts/tracking_efficiency.html#error-propagation", + "title": "Tracking efficiency for two-body decay", + "section": "Error propagation", + "text": "Error propagation\n\nNeglect error on \\(n_{ij}\\) and \\(n\\) to simplify\nVariance of \\(\\gamma_i\\) is \\(V_{\\gamma,i}\\)\nVariance of \\(\\gamma_A\\) is \\[\nV_{\\gamma_A} = \\sum_k J_k^2 \\, V_{\\gamma,k} \\quad \\text{with} \\quad J_k = \\frac{\\partial \\epsilon'_A}{\\partial \\gamma_k} = \\frac{1}{n} \\sum_i \\gamma_i (n_{ki} + n_{ik})\n\\]\nCheck of analytical formula against numerical calculation\n\n\nfrom jacobi import propagate\nimport numpy as np\n\n\nrng = np.random.default_rng(1)\n\n# in general, n is not symmetric, n_ij != n_ji\nn = rng.random(size=(3, 3)) * 100\ng = 0.5 + rng.random(size=3)\nvg = 1 + rng.random(size=3)\n\nm = np.sum(n)\n\n\ndef compute(x):\n nx = np.reshape(x[:n.size], n.shape)\n gx = x[n.size:]\n return np.einsum(\"i,j,ij\", gx, gx, nx) / m\n\nx = np.append(n.reshape(-1), g)\nvx = np.append(np.zeros(n.size), vg)\n\ny, vy = propagate(compute, x, vx)\ny, vy\n\n(array(0.80687392), array(1.65110298))\n\n\n\ndef analytical(n, g, vg):\n j = g @ (n + n.T) / m\n vy = np.sum(j ** 2 * vg)\n return vy\n\nvy2 = analytical(n, g, vg)\nvy2\n\n1.6511029832764388\n\n\nNumerical and analytical calculations agree.\nNow I compare my formula with the formula on the TWiki page of the Tracking Group.\n\n# Formula on TWiki seems to assume that g ~ 1\ng = np.ones(3)\n\ndef analytical_twiki(n, vg):\n s = 0\n sg = vg ** 0.5\n for i in range(n.shape[0]):\n for j in range(n.shape[1]):\n s += n[i,j] ** 2 * (sg[i] + sg[j]) ** 2\n return s / m ** 2\n\nvy3 = analytical_twiki(n, vg)\nvy3\n\n0.802978765901557\n\n\n\nanalytical(n, g, vg)\n\n1.9900608390572314\n\n\nThe formula from the TWiki seems to give a variance that is much too small by more than a factor of two in this case." + }, + { + "objectID": "posts/duffing.html", + "href": "posts/duffing.html", + "title": "Solving the duffing oscillator", + "section": "", + "text": "Here I solve the differential equations for the duffing oscillator with scipy and animate the solution with matplotlib. The oscillator exhibits chaotic behavior.\n\n%matplotlib notebook\nimport numpy as np\nfrom scipy.integrate import solve_ivp\nimport matplotlib.pyplot as plt\nfrom matplotlib import animation\n\n\ndef V(x):\n return alpha * x ** 2 / 2 + beta * x ** 4 / 4\n\ndef ode(t, y): \n x, xdot = y\n xdotdot = -(gamma * xdot + alpha * x + beta * x ** 3)\n xdotdot += epsilon * np.cos(omega * t)\n return xdot, xdotdot\n\n\n# solve equation of motion\nalpha = -1\nbeta = 1\ngamma = 0.1\nomega = 1.4\n# interesting values for epsilon: 0.1, 0.4, 1.0\nepsilon = 0.4\n\nx0, v0 = 0, 0\nn_samples_per_period = 50\nn_period = 5000\nperiod = 2 * np.pi / omega\ntmax = n_period * period\n\nt = np.linspace(0, tmax, n_samples_per_period * n_period)\nsol = solve_ivp(ode, (0, tmax), (x0, v0), t_eval=t)\n\nx, xdot = sol.y\n\n# setup animation\nfig, ax = plt.subplots(2, 2)\n\nplt.sca(ax[0,0])\nplt.title(\"potential energy\")\nxext = max(np.max(np.abs(x)) * 1.05, 1.5)\nx1 = np.linspace(-xext, xext)\ny1 = V(x1)\nplt.plot(x1, y1)\nln1, = plt.plot([], [], 'mo')\nplt.xlabel(r'$x$')\nplt.ylabel(r'$V(x)$')\nplt.xlim(-xext - 0.05, xext + 0.05)\nplt.ylim(np.min(y1) - 0.05, np.max(y1))\n\nax2 = ax[1, 0]\nplt.sca(ax2)\nplt.title(\"position vs. time\")\nplt.xlabel(r'$t$')\nplt.ylabel(r'$x$')\nln2, = plt.plot([], [])\nplt.ylim(-xext, xext)\n\nplt.sca(ax[0, 1])\nplt.title(\"Poincaré section\")\nplt.xlabel(r'$x$')\nplt.ylabel(r'$\\dot{x}$')\nplt.scatter(x[::n_samples_per_period], xdot[::n_samples_per_period], s=1, lw=0)\nplt.xlim(-xext, xext)\nplt.ylim(np.min(xdot), np.max(xdot))\n\nplt.sca(ax[1,1])\nplt.title(\"phase space\")\nplt.xlabel(r'$x$')\nplt.ylabel(r'$\\dot{x}$')\nln3, = plt.plot([], [])\nln3a, = plt.plot([], [], 'mo')\nplt.xlim(-xext, xext)\nvext = np.max(np.abs(xdot))\nplt.ylim(-vext, vext)\n\nplt.tight_layout()\n\ndef animate(i): \n ln1.set_data(x[i], V(x[i]))\n j = max(0, i - 10 * n_samples_per_period)\n sl = slice(j, i+1)\n ln2.set_data(t[sl], x[sl])\n ax2.set_xlim(t[j], t[i+1])\n sl = slice(0, i+1)\n ln3.set_data(x[sl], xdot[sl])\n ln3a.set_data(x[i], xdot[i])\n return (ln1, ln2, ln3, ln3a)\n\nanim = animation.FuncAnimation(fig, animate, frames=len(x), interval=2)" + }, + { + "objectID": "posts/Leave-one-out cross-validation.html", + "href": "posts/Leave-one-out cross-validation.html", + "title": "Leave-one-out cross-validation", + "section": "", + "text": "General steps for LOO cross-validation: - Remove i-th datum from input data set - Fit model to remaining data set - Use fitted model to predict the i-th datum and store difference to original i-th datum - Do this for all i and compute variance of the differences - Select model with smallest variance\nI demonstrate below how to select the optimal order for a polynomial model with leave-one-out (LOO) cross validation.\n\n#!pip install numpy matplotlib iminuit tabulate\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom iminuit import Minuit \nfrom tabulate import tabulate\n\n\nrng = np.random.default_rng(seed=1)\n\n\n# create some toy data that follows polynomials of increasing degree\nx = np.linspace(0, 1, 100)\ny_set = np.empty((3, len(x)))\nfor poly_degree in (0, 1, 2):\n y_set[poly_degree] = np.polyval(np.ones(poly_degree + 1), x) + rng.normal(0, 0.1, len(x))\n \nfor y in y_set:\n plt.plot(x, y, \".\")\n\n\n\n\n\n# apply leave-one-out cross-validation\nfor true_poly_order, y in enumerate(y_set):\n\n variances = []\n tested_poly_orders = np.arange(5)\n for tested_poly_order in tested_poly_orders:\n deltas = []\n\n # leave out the i-th (x, y) pair in the fit\n for i in range(len(y)):\n mask = np.ones(len(y), dtype=bool)\n mask[i] = 0\n x_without_i = x[mask]\n y_without_i = y[mask]\n\n # least-squares cost function to fit a polynomial to the toy data\n def lsq(par):\n ym = np.polyval(par, x_without_i)\n return np.sum((y_without_i - ym) ** 2 / 0.1 ** 2)\n \n m = Minuit.from_array_func(lsq, np.zeros(tested_poly_order+1), pedantic=False)\n m.strategy = 0 # faster and does not compute errors automatically\n m.migrad()\n\n # predict the i-th y value with the fitted model and \n # store the delta to the real value\n assert m.migrad_ok()\n \n yi_predicted = np.polyval(m.np_values(), x[i])\n deltas.append(y[i] - yi_predicted)\n\n variances.append(np.var(deltas))\n\n print(f\"true polynomial order {true_poly_order}; selected order {np.argmin(variances)}\")\n print(tabulate(np.transpose((tested_poly_orders, variances)),\n headers=(\"model order\", \"LOO variance\")) + \"\\n\")\n\ntrue polynomial order 0; selected order 0\n model order LOO variance\n------------- --------------\n 0 0.0105525\n 1 0.0105614\n 2 0.0107262\n 3 0.0107479\n 4 0.0107608\n\ntrue polynomial order 1; selected order 1\n model order LOO variance\n------------- --------------\n 0 0.0903401\n 1 0.0121111\n 2 0.0122902\n 3 0.0124952\n 4 0.0124547\n\ntrue polynomial order 2; selected order 2\n model order LOO variance\n------------- --------------\n 0 0.347969\n 1 0.017877\n 2 0.0117249\n 3 0.0118761\n 4 0.011895" + }, + { + "objectID": "posts/plots_with_hyperlinks.html", + "href": "posts/plots_with_hyperlinks.html", + "title": "How to make a plot with legend entries that are hyperlinks", + "section": "", + "text": "This is a demo on what the title says, also see this matplotlib issue. The trick works with SVG and PDF. Try to click on “BBC” in the legend.\n\nfrom matplotlib import pyplot as plt\nimport numpy as np\n\n# generate SVG images instead of PNG\n%config InlineBackend.figure_formats = ['svg']\n\n# required for SVG to accept a click on the text area and not just on the text path\nplt.rcParams[\"svg.fonttype\"] = \"none\"\n\nplt.figure()\nplt.scatter([1, 2], [4, 6], label=\"BBC\")\nplt.scatter([1, 2, 3], [6, 5, 4], label=\"Google\")\n\nurls = {\"BBC\": 'https://www.bbc.com/news'}\n\nleg = plt.legend()\nfor ta in leg.texts:\n t = ta.get_text()\n try:\n url = urls[t]\n ta.set_url(url)\n except KeyError:\n pass\n\n# plt.savefig('scatter.svg')\n# plt.savefig('scatter.pdf')" + }, + { + "objectID": "posts/MCMC.html", + "href": "posts/MCMC.html", + "title": "MCMC demo", + "section": "", + "text": "This is a tiny demo of the MCMC algorithm from the emcee library, which computes the posterior for some parameters based on the likelihood function and priors on the parameters.\n\nimport numpy as np\nimport emcee\nimport matplotlib.pyplot as plt\nimport numba as nb\n\n\nrng = np.random.default_rng(1)\n\ny = 2.0\n\n@nb.njit\ndef model(mu):\n return mu + np.random.randn()\n\n@nb.njit\ndef log_prob(mu, y):\n diff = y - model(mu)\n return -0.5 * np.sum(diff ** 2)\n\nnwalkers = 1000\nndim = 1\nsampler = emcee.EnsembleSampler(nwalkers, ndim, log_prob, args=[y])\n\nstart = rng.normal(size=(nwalkers, ndim))\nr = sampler.run_mcmc(start, 100, progress=True)\n\nplt.hist(r.coords)\nprint(np.mean(r.coords), np.std(r.coords) / np.sqrt(len(r.coords)))\n\n100%|██████████| 100/100 [00:00<00:00, 168.91it/s]\n\n\n1.962706921253352 0.04570914440079373" + }, + { + "objectID": "posts/struct_of_arrays_vs_arrays_of_structs.html", + "href": "posts/struct_of_arrays_vs_arrays_of_structs.html", + "title": "Which data structure is faster: array of structs or struct of arrays?", + "section": "", + "text": "In the high-performance computing community, it is well known that certain algorithms run faster if the data are structured in such a way that the CPU can read them efficiently. Ideally, the CPU wants to process consecutive memory, not memory which is scattered. In other words, adding two numbers will be much faster on average if the two numbers are next to each other in memory than if they are distant in memory.\nThis means that, depending on the kind of computation, an array of structs (AoS) or a struct of arrays (SoA) performs better. Which one is better depends on the kind of calculation.\nI demonstrate this for making steps in a Monte-Carlo simulation of particles. Particles are stored on the stack of this simulation, which have a location and a momentum in 3D space, six numbers per particle. In the AoS case, the six numbers for each particle are consecutive in memory, while in the SoA case all x-components of the momentum are consecutive, then all y-components and so on. When the simulation makes a time step, the particles need to be moved along the direction of their momentum. To do the calculation we have to compute the unit vector from the momentum and then move the particle while taking its velocity into account. Whether AoS or SoA works better is not obvious, since arguments can be made for both access patterns.\nBelow, I define the data structures and implement the move algorithms. For move_aos I try out two alternatives, the second is an attempt to help the compiler with autovectorization.\n\nimport numpy as np\nimport numba as nb\nfrom numpy.testing import assert_allclose\nfrom matplotlib import pyplot as plt\nfrom timeit import Timer\n\n\nn = int(1e5)\naos = np.empty((n, 6))\nsoa = np.empty((6, n))\nfor i in range(6):\n aos[:, i] = np.arange(n) + i\n soa[i, :] = np.arange(n) + i\n\nassert_allclose(aos, soa.T)\n\naos_32 = aos.astype(np.float32)\nsoa_32 = soa.astype(np.float32)\n\n\n@nb.njit(error_model=\"numpy\")\ndef move_aos(aos, step):\n for i in range(len(aos)):\n r = aos[i, :3]\n p = aos[i, 3:]\n pn = np.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])\n ps = step / pn\n r += p * ps\n return aos\n\n\n@nb.njit(error_model=\"numpy\")\ndef move_aos_alt(aos, step):\n r = aos[:, :3]\n p = aos[:, 3:]\n pn = np.sqrt(p[:,0] * p[:,0] + p[:,1] * p[:,1] + p[:,2] * p[:,2])\n ps = step / pn\n for i in range(3):\n r[:, i] += p[:, i] * ps\n return aos\n\n\n@nb.njit(error_model=\"numpy\")\ndef move_soa(soa, step):\n r = soa[:3]\n p = soa[3:]\n pn = np.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])\n ps = step / pn\n for i in range(3):\n r[i] += p[i] * ps\n return soa\n\n\nassert_allclose(move_aos(aos, 3), move_soa(soa, 3).T)\nassert_allclose(move_aos_alt(aos, 3), move_soa(soa, 3).T)\nassert_allclose(move_aos(aos_32, 3), move_soa(soa_32, 3).T, rtol=1e-6)\nassert_allclose(move_aos_alt(aos_32, 3), move_soa(soa_32, 3).T, rtol=1e-6)\n\nfns = [x for x in locals() if x.startswith(\"move\")]\n\nI now run benchmarks for the calculations, using 32 and 64 bit floats and stacks of varying size. It turns out that move_soa in 32 bit performs best. Despite the fact that we need to compute the norm of the momentum for each particle, which in this case means adding momentum components squared from distant memory locations, computing the squares for each component in one go leads to better performance, probably because of pipelining in the CPU. The update of the particle locations can also be done with an efficient memory access pattern, since the respective inputs are consecutive in memory.\n\ndata = {}\nfor bits in (32, 64):\n data[bits] = {}\n for ifn, fn in enumerate(fns):\n times = []\n n = np.geomspace(1, 1e6, 10).astype(int)\n for ni in n:\n arg = \"aos\" if \"aos\" in fn else \"soa\"\n dtype = np.float32 if bits == 32 else np.float64\n aos = np.empty((ni, 6), dtype=dtype)\n soa = np.empty((6, ni), dtype=dtype)\n for i in range(6):\n aos[:, i] = np.arange(ni) + i\n soa[i, :] = np.arange(ni) + i\n t = Timer(f\"{fn}({arg}, 3)\", f\"from __main__ import {fn}, {arg}\")\n m, dt = t.autorange()\n tmin = min(dt, *t.repeat(3, m))\n times.append(tmin / m)\n data[bits][fn] = (n, times)\n\n\nfor bits in data:\n for ifn, fn in enumerate(fns):\n n, times = data[bits][fn]\n plt.plot(n, times, color=f\"C{ifn}\",\n ls=\"-\" if bits == 64 else \"--\",\n label=f\"{bits} bit: {fn}\")\nplt.legend()\nplt.loglog()\nplt.xlabel(\"number of particles\")\nplt.ylabel(\"run-time (smaller is better)\");\n\n\n\n\nAnother way to present this is to look at the speed-up factor compared to the worst performing AoS implementation. We gain a factor of 40 in the best case.\n\nfor bits in data:\n for ifn, fn in enumerate(fns):\n if fn == \"move_aos\":\n continue\n n, times = data[bits][fn]\n ratio = np.divide(data[bits][\"move_aos\"][1], times) \n plt.plot(n, ratio, color=f\"C{ifn}\",\n ls=\"-\" if bits == 64 else \"--\",\n label=f\"{bits} bit: {fn}\")\n i = np.argmax(ratio)\n plt.text(n[i], ratio[i] - 1, f\"{ratio[i]:.0f}\", ha=\"center\", va=\"top\")\nplt.plot(n, np.ones_like(n), color=\"C0\")\nplt.legend()\nplt.semilogx()\nplt.xlabel(\"number of particles\")\nplt.ylabel(\"speed-up\");\n\n\n\n\nTo understand where these improvements come from, we investigate the assembler code for each implementation and look for SIMD instructions (single input multiple data), which are often responsible for speed gains. We find that move_soa does not really stand out here. The speed does not come from the instructions used, it comes from optimized access patterns and pipelining.\n\nimport boost_histogram as bh\n\nh = bh.Histogram(bh.axis.StrCategory([], growth=True), bh.axis.Integer(0, 3))\n\nfor ifn, fn in enumerate(fns):\n cmds = []\n for line in list(globals()[fn].inspect_asm().values())[0].split(\"\\n\"):\n line = line.strip(\"\\t \")\n if line.startswith(\".\") or line.startswith(\"__\"):\n continue\n if \"\\t\" not in line:\n continue\n cmd = line.split(\"\\t\")[0]\n if not cmd.startswith(\"v\"):\n continue\n if cmd.endswith(\"sd\"):\n continue\n cmds.append(cmd)\n\n h.fill(cmds, ifn)\n\n\nimport matplotlib.pyplot as plt\n\nxlabels = []\nvalues = []\n\nfor x, v in sorted(zip(h.axes[0], h.values())):\n xlabels.append(x)\n values.append(v)\n\nvalues = np.array(values)\n\nxe = np.arange(len(xlabels) + 1)\nv = 0\nfor ifn, fn in enumerate((move_aos, move_aos_alt, move_soa)):\n before = v\n v = v + values[:, ifn]\n plt.stairs(v, 1-xe, baseline=before, fill=True, label=fn.__name__, orientation=\"horizontal\")\nplt.legend()\nplt.gca().yaxis.set_ticks(-np.arange(len(v)) + 0.5)\nplt.gca().yaxis.set_ticklabels(xlabels);" + }, + { + "objectID": "posts/p-value conversion.html", + "href": "posts/p-value conversion.html", + "title": "p-value computation and conversion: one-sided, two-sided?", + "section": "", + "text": "This is an attempt to clear up the confusion around p-value computation and based on the authorative source on that matter:\nG. Cowan, K. Cranmer, E. Gross, O. Vitells, Eur.Phys.J.C 71 (2011) 1554\nThe point of that paper is that one should use two different test statistics based on the likelihood-ratio for deviations that are always positive and for deviations where the sign is not known a priori. One gets two different p-values accordingly. The conversion to significance is then always the same.\nI generate a flat distribution from \\((-1,1)\\) which represents background-only samples, then I fit a Gaussian peak centered at 0 with width 0.1 to this data test the two hypotheses\n\n\\(H_0\\): there is only background\n\\(H_1\\): there is background and a signal\n\nThe \\(H_1\\) in these kinds of tests is never arbitrary in terms of the signal, we do not seek for any deviation, but a rather specific one. In our example, we seek for a signal in particular location (here centered at 0). This is important for the computation of the p-value later. The less constrained \\(H_1\\) is, the more of the Look-Elsewhere Effect I get, because more random fluctuations of the background can be confused with a signal.\nWhen fitting the toys, I allow the amplitude \\(\\mu\\) of the signal to be positive or negative (following Cowan et al.). As a test statistic for a deviation from the background-only case I use \\(t_0\\) (again following Cowan et al. in the notation) \\[\nt_0 = -2\\ln \\lambda(0) = -2 \\ln \\frac{L(0, \\hat {\\hat \\theta})}{L(\\hat \\mu, \\hat \\theta)}\n\\] where * \\(\\mu\\) is the amplitude of the hypothetical signal * \\(\\theta\\) are nuisance parameters * \\(\\hat \\mu\\) is the fitted value of \\(\\mu\\) * \\(\\hat\\theta\\) are the fitted values of \\(\\theta\\) when \\(\\mu\\) is also fitted * \\(\\hat{\\hat\\theta}\\) are the fitted values of \\(\\theta\\) under the condition \\(\\mu = 0\\).\nIf you search for \\(t_0\\) in the paper, it is not explicitly written there, \\(t_0\\) is \\(t_\\mu\\) from section 2.1 with \\(\\mu = 0\\).\n\nimport numpy as np\nfrom iminuit import Minuit\nfrom iminuit.cost import ExtendedUnbinnedNLL\nfrom numba_stats import uniform_pdf, norm_pdf\nimport matplotlib.pyplot as plt\nimport boost_histogram as bh\n\n\nrng = np.random.default_rng(1)\n\ndef model(x, mu, theta):\n return mu + theta, mu * norm_pdf(x, 0, 0.1) + theta * uniform_pdf(x, -1, 2)\n\npositive = [0]\nnegative = [0]\nt0 = []\nmu = []\nfor imc in range(1000):\n b = rng.uniform(-1, 1, size=1000)\n c = ExtendedUnbinnedNLL(b, model)\n m = Minuit(c, mu=0, theta=len(b))\n m.limits[\"theta\"] = (0, None)\n m.fixed[\"mu\"] = True\n m.migrad()\n assert m.valid\n lnL_b = -m.fval\n m.fixed[\"mu\"] = False\n m.migrad()\n mui = m.values[\"mu\"]\n lnL_sb = -m.fval\n t0i = 2 * (lnL_sb - lnL_b)\n t0.append(t0i)\n mu.append(mui)\n if mui > 0 and t0i > positive[0]:\n h = bh.Histogram(bh.axis.Regular(100, -1, 1))\n h.fill(b)\n positive = (t0i, h, m.values[:], m.covariance[:])\n elif mui < 0 and t0i > negative[0]:\n h = bh.Histogram(bh.axis.Regular(100, -1, 1))\n h.fill(b)\n negative = (t0i, h, m.values[:], m.covariance[:])\n \nt0 = np.array(t0)\nmu = np.array(mu)\n\nLet’s look at the two most extreme deviations from \\(H_0\\), the ones with the largest and smallest signal.\n\nfig, ax = plt.subplots(1, 2, figsize=(14, 5), sharex=True, sharey=True)\nfor (t0i, h, par, cov), axi in zip((negative, positive), ax):\n plt.sca(axi)\n scale = 1/h.axes[0].widths\n plt.errorbar(h.axes[0].centers, h.values() * scale, h.variances()**0.5 * scale, fmt=\"ok\", label=\"data\")\n scale = 1 / h.axes[0].widths\n xm = np.linspace(-1, 1)\n plt.plot(xm, model(xm, *par)[1], label=\"fit\")\n plt.legend()\n dev = par[0] / cov[0,0] ** 0.5\n plt.title(f\"$\\mu = {par[0]:.1f}$ $t_0 = {t0i:.2f}$\")\n plt.ylabel(\"density\")\n plt.xlabel(\"$x$\")\n\n\n\n\nBoth upward and downward fluctuations are unlikely events if \\(H_0\\) is true and therefore both get a large value of our test statistic \\(t_0\\), which does not know in which way the deviation went.\n\nSearches with \\(\\mu > 0\\) (for a new particle, new decay mode, etc.)\nIn most cases we look for a positive peak (new particle, new decay mode, etc). If we see a downward fluctuation, we know for sure that this cannot be what we are looking for. Therefore, when we compute the p-value for some observation from a simulation like this one, we must not regard large values of the test statistic when the fluctuation was negative, since this is not a deviation of \\(H_0\\) that mimics our signal. Instead we should set \\(t_0\\) to zero in such cases. Cowan et al. use \\(q_0\\) for this modified test statistic \\[\nq_0 = \\begin{cases}\nt_0 &\\text{if } \\hat\\mu \\ge 0 \\\\\n0 &\\text{otherwise}\n\\end{cases}\n\\]\n\nplt.hist(t0, alpha=0.5, bins=20, range=(0, 20), label=\"$t_0$\")\nq0 = t0.copy()\nq0 = np.where(mu < 0, 0, t0)\nplt.hist(q0, alpha=0.5, bins=20, range=(0, 20), label=\"$q_0$\")\nplt.legend()\nplt.axhline()\nplt.xlabel(\"$t_0, q_0$\")\nplt.semilogy();\n\n\n\n\nIf we observed \\(t_0 = q_0 = 15\\) in a real experiment, we need to compare it with the \\(q_0\\) distribution and not with the \\(t_0\\) distribution to compute the p-value.\n\nprint(f\"Wrong: p-value based on t0-distribution {np.mean(t0 > 15)}\")\nprint(f\"Right: p-value based on q0-distribution {np.mean(q0 > 15)}\")\n\nWrong: p-value based on t0-distribution 0.006\nRight: p-value based on q0-distribution 0.004\n\n\nAs we can see, the p-value is enhanced in this case, because we do not need to consider the negative fluctuations at all. The conversion to significance \\(Z\\) is done with a normal distribution.\n\nfrom scipy.stats import norm\n\np = np.mean(q0 > 15)\nprint(f\"Z = {norm.ppf(1 - p):.2f}\")\n\nZ = 2.65\n\n\n\n\nSearches for deviations where the sign of \\(\\mu\\) is not known a priori\nIf cannot exclude a priori that our signal has \\(\\mu < 0\\), we need to use the \\(t_0\\) distribution instead of \\(q_0\\).\nIf we observed \\(t_0 = 15\\) in a real experiment, we need to compare it with \\(t_0\\) distribution to compute the p-value.\n\nprint(f\"Right: p-value based on t0-distribution {np.mean(t0 > 15)}\")\n\nRight: p-value based on t0-distribution 0.006\n\n\nAs we can see, the p-value is diluted in this case, because we need to consider fluctuations in both directions. In other words, the Look-Elsewhere Effect is larger in this case, because more kinds of fluctuations in the background can be confused with a signal.\n\nfrom scipy.stats import norm\n\np = np.mean(t0 > 15)\nprint(f\"Z = {norm.ppf(1 - p):.2f}\")\n\nZ = 2.51\n\n\nAccordingly, the significance is a bit lower compared to the previous case." + }, + { + "objectID": "posts/all_KK_decays.html", + "href": "posts/all_KK_decays.html", + "title": "All decays into two Kaons", + "section": "", + "text": "In a recent paper review, the question came up how many resonances there are which decay into two kaons. It is not easy to look this up somewhere. My solution was to extract the information in the following way from the Pythia8 decay database. I have to first get the database, which is in XML format. I parse the XML database and then go through all entries and isolate the particles which exclusively decay into two kaons.\n\nimport xml.etree.ElementTree as ET\nfrom particle import literals as lp\nfrom matplotlib import pyplot as plt\nfrom pathlib import Path\nimport chromo\nfrom chromo.models import Pythia8\nfrom chromo.util import _cached_data_dir\nfrom chromo.kinematics import CenterOfMass, TeV\n\n# HACK: ensure that Pythia8 data is downloaded without running Pythia8\npythia8_data = Path(_cached_data_dir(Pythia8._data_url))\n\n\n# TODO: we should make this information accessible via the chromo API\nfn = pythia8_data / \"xmldoc\" / \"ParticleData.xml\"\nlines = [\"<root>\"]\nwith open(fn) as f:\n for iline, line in enumerate(f.readlines()):\n if iline < 166: continue\n if \"</chapter>\" in line: break\n lines.append(line)\nlines.append(\"</root>\")\n\ntree = ET.fromstring(\"\".join(lines))\n\nall_kk = []\n\npid_k = int(lp.K_plus.pdgid)\n\nfor p in tree.findall(\"particle\"):\n for ch in p.findall(\"channel\"):\n da = [int(x) for x in ch.attrib[\"products\"].split()]\n if len(da) != 2: continue\n if all(abs(pi) == pid_k for pi in da):\n pa = p.attrib\n all_kk.append((pa[\"name\"], float(pa[\"m0\"])))\n\nThe plot shows where the resonances are located in the invariant mass distribution. The y-axis has no meaning, it just separates the points vertically.\n\nfrom cycler import cycler\nfrom matplotlib.colors import TABLEAU_COLORS\n\nc = cycler(marker=\"osd\") * cycler(color=TABLEAU_COLORS)\n\nplt.gca().set_prop_cycle(c)\n\ni = 0\nfor kk in sorted(all_kk, key=lambda x: x[1]):\n plt.plot(kk[1] * 1e3, i, label=kk[0])\n i+=1\nplt.legend(ncol=3)\nplt.xlabel(\"mass / MeV$c^{-2}$\");\n\n\n\n\nAnd here is a sorted list of the masses.\n\nfor kk in sorted(all_kk, key=lambda x:x[1]):\n print(kk[0], kk[1] * 1e3)\n\na_0(980)0 983.5\nf_0(980) 1000.0\nphi 1019.46\nf_2 1275.1\na_20 1318.3\nf_0(1370) 1350.0\na_0(1450)0 1474.0\nf'_2(1525) 1525.0\nf_0(1710) 1720.0\nrho(1700)0 1720.0\nD0 1864.86\nchi_0c 3414.75\nchi_2c 3556.2\npsi(2S) 3686.11\nB0 5279.58\nB_s0 5366.7699999999995" + }, + { + "objectID": "posts/HESSE_vs_MINOS_TwoGauss.html", + "href": "posts/HESSE_vs_MINOS_TwoGauss.html", + "title": "Coverage of HESSE and MINOS intervals", + "section": "", + "text": "The MINUIT package provides two ways to compute uncertainty intervals for a fitted parameter, the HESSE method and the MINOS method. Ideally, these intervals should have 68 % coverage probability: if the experiment is repeated with new random data each time, the intervals should cover the true value in 68 % of the cases.\nThis only asymptotically correct, and if the samples are small, the actual coverage probability can deviate from the expected one. It is often thought that MINOS provides better coverage (closer to the expected value) because its intervals can be asymmetric and seem to retain more information.\nAs a toy example, I fit the sum of two normal distribution which have equal width but different location parameters. The fitted parameters are the two locations and the shared width. Here is one example fit.\n\nfrom iminuit import Minuit\nfrom iminuit.cost import UnbinnedNLL\nfrom numba_stats import norm\nimport numpy as np\nimport numba as nb\nfrom matplotlib import pyplot as plt\nfrom IPython.display import display\n\n\nsigma = 1.0\n\ndef gen_data(rng, n):\n n0 = rng.binomial(n, 0.5)\n n1 = n - n0\n return np.append(rng.normal(0, sigma, size=n0), rng.normal(1, sigma, size=n1))\n\n@nb.njit(nogil=True)\ndef model(x, mu_a, mu_b):\n return 0.5 * (norm.pdf(x, mu_a, sigma) + norm.pdf(x, mu_b, sigma))\n\nrng = np.random.default_rng(0)\n\ndata = gen_data(rng, 10000)\nm = Minuit(UnbinnedNLL(data, model), mu_a=0, mu_b=1)\nm.migrad()\n\n\n\n\nMigrad\n\n\nFCN = 3.065e+04\nNfcn = 31\n\n\nEDM = 9.53e-06 (Goal: 0.0002)\ntime = 4.5 sec\n\n\nValid Minimum\nBelow EDM threshold (goal x 10)\n\n\nNo parameters at limit\nBelow call limit\n\n\nHesse ok\nCovariance accurate\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nmu_a\n-0.014\n0.020\n\n\n\n\n\n\n\n1\nmu_b\n1.013\n0.020\n\n\n\n\n\n\n\n\n\n\n\n\nmu_a\nmu_b\n\n\nmu_a\n0.000412\n-0.2e-3 (-0.386)\n\n\nmu_b\n-0.2e-3 (-0.386)\n0.00041\n\n\n\n\n\n\n \n \n \n \n 2024-05-12T23:24:23.630891\n image/svg+xml\n \n \n Matplotlib v3.8.2, https://matplotlib.org/\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\nI now run this fit many times on newly drawn random samples. To speed things up, the calculation is done in parallel.\n\ndef compute(n):\n n_try = 1000\n rng = np.random.default_rng(n)\n n_hesse = 0\n n_minos = 0\n h_pcov = []\n m_pcov = []\n cor = []\n n_fail = 0\n for i_try in range(n_try):\n for sub in range(5):\n data = gen_data(rng, n)\n nll = UnbinnedNLL(data, model) \n m = Minuit(nll, mu_a=0, mu_b=1)\n m.migrad()\n m.hesse()\n m.minos()\n if m.valid and m.accurate:\n break\n n_fail += 1\n mu = m.values[\"mu_a\"]\n h_mu = m.errors[\"mu_a\"]\n m_mu = m.merrors[\"mu_a\"]\n cov = m.covariance\n c = cov[1,0] / np.sqrt(cov[0, 0] * cov[1, 1])\n if abs(mu) < h_mu:\n n_hesse += 1\n if (mu + m_mu.lower) < 0 < (mu + m_mu.upper):\n n_minos += 1\n cor.append(c)\n h_pcov.append(n_hesse / n_try)\n m_pcov.append(n_minos / n_try)\n print(f\"{n} hesse {h_pcov[-1]:.2f} minos {m_pcov[-1]:.2f} <corr> {np.mean(cor):.3f} {n_fail=}\", flush=True)\n return h_pcov, m_pcov, cor\n\nif __name__ == \"__main__\":\n from concurrent.futures import ThreadPoolExecutor as Pool\n\n n_pts = (10, 20, 50, 70, 100, 200, 300, 400, 500, 1000, 2000, 3000)\n\n h_pcov = []\n m_pcov = []\n corr = []\n with Pool() as pool:\n for (h, m, c) in pool.map(compute, n_pts):\n h_pcov.append(h)\n m_pcov.append(m)\n corr.append(c)\n\n10 hesse 0.76 minos 0.75 <corr> -0.388 n_fail=7\n20 hesse 0.77 minos 0.73 <corr> -0.450 n_fail=0\n50 hesse 0.75 minos 0.67 <corr> -0.458 n_fail=0\n100 hesse 0.77 minos 0.69 <corr> -0.461 n_fail=0\n70 hesse 0.75 minos 0.67 <corr> -0.472 n_fail=0\n200 hesse 0.69 minos 0.66 <corr> -0.435 n_fail=0\n1000 hesse 0.67 minos 0.67 <corr> -0.407 n_fail=0\n2000 hesse 0.69 minos 0.69 <corr> -0.401 n_fail=0\n500 hesse 0.70 minos 0.69 <corr> -0.413 n_fail=0\n3000 hesse 0.69 minos 0.69 <corr> -0.404 n_fail=0\n400 hesse 0.70 minos 0.69 <corr> -0.422 n_fail=0\n300 hesse 0.68 minos 0.67 <corr> -0.420 n_fail=0\n\n\nI plot the actual coverage probability of the HESSE and MINOS intervals. For small samples, both methods overcover, but the MINOS intervals converge faster to the correct value. So in this case, MINOS did perform better. This is not always the case, however.\n\nplt.plot(n_pts, h_pcov, label=\"HESSE\")\nplt.plot(n_pts, m_pcov, label=\"MINOS\")\nplt.axhline(0.68, ls=\":\", color=\"0.5\", zorder=0)\nplt.semilogx()\nplt.legend()\nplt.ylabel(\"coverage\")\nplt.xlabel(\"sample size\")\nplt.ylim(0, 1);" + }, + { + "objectID": "posts/UnbiasedEML.html", + "href": "posts/UnbiasedEML.html", + "title": "Unbiasedness of EML fit for a mixture model with fixed component pdfs", + "section": "", + "text": "The log-likelihood for one observed Poisson-distributed count is (without constants) \\[\n\\ell_i(\\lambda) := \\ln \\mathcal{L}_i(\\lambda) = -\\lambda + k_i \\ln \\lambda.\n\\]\nThe ML estimate is obtained by solving the score function \\(\\sum_i \\partial \\ell_i / \\partial \\lambda \\overset!= 0\\). The solution \\(\\hat\\lambda = \\frac1N \\sum_i k_i\\) is unbiased, \\[\nE[\\hat \\lambda] = \\frac1N \\sum_i E[k_i] = \\lambda.\n\\]\nThis can be generalised: if the estimator is a linear function of the observations, the estimator is unbiased.\nWe now consider a two component model, where the number density is \\(\\rho(x) = S f_s(x) + B f_b(x)\\).\nFor one bin \\(i\\), the expectation is \\(\\lambda_i = S s_i + B b_i\\) with constants \\(s_i = \\int_{a_i}^{b_i} f_s(x) dx\\) and \\(b_i = \\int_{a_i}^{b_i} f_b(x) dx\\).\nThe score functions are \\[\n\\ell_i(S, B) = -(S s_i + B b_i)/N + k_i \\ln(S s_i + B b_i)\n\\] \\[\n\\frac{\\partial \\sum_i \\ell_i}{\\partial S} = -1 + \\sum_i k_i \\frac{s_i}{S s_i + B b_i} \\overset!= 0 \\quad\\text{and}\\quad\n\\frac{\\partial \\sum_i \\ell_i}{\\partial B} = -1 + \\sum_i k_i \\frac{b_i}{S s_i + B b_i} \\overset!= 0.\n\\]\nWe calculate the solution explicitly for two bins. The calculation is fairly involved, although the solution is simple.\n\nfrom sympy import *\n\nS, B, s1, s2, b1, b2, k1, k2 = symbols(\"S B s_1 s_2 b_1 b_2 k_1 k_2\", real=True, nonnegative=True)\n\n\neq1 = Eq(k1 * s1 / (S * s1 + B * b1) + k2 * s2 / (S * s2 + B * b2), 1)\neq1\n\n\\(\\displaystyle \\frac{k_{1} s_{1}}{B b_{1} + S s_{1}} + \\frac{k_{2} s_{2}}{B b_{2} + S s_{2}} = 1\\)\n\n\n\neq2 = Eq(k1 * b1 / (S * s1 + B * b1) + k2 * b2 / (S * s2 + B * b2), 1)\neq2\n\n\\(\\displaystyle \\frac{b_{1} k_{1}}{B b_{1} + S s_{1}} + \\frac{b_{2} k_{2}}{B b_{2} + S s_{2}} = 1\\)\n\n\n\nsols = solve((eq1, eq2), (S, B))\nsols\n\n[((b_1*b_2*k_1 + b_1*b_2*k_2 - b_1*k_2*s_2 - b_2*k_1*s_1)/((b_1 - s_1)*(b_2 - s_2)),\n (-b_1*k_1*s_2 - b_2*k_2*s_1 + k_1*s_1*s_2 + k_2*s_1*s_2)/((b_1 - s_1)*(b_2 - s_2)))]\n\n\n\nfor sol in sols:\n for k, s in zip((S, B), sol):\n print(f\"Estimate for {k}:\")\n display(s)\n\nEstimate for S:\nEstimate for B:\n\n\n\\(\\displaystyle \\frac{b_{1} b_{2} k_{1} + b_{1} b_{2} k_{2} - b_{1} k_{2} s_{2} - b_{2} k_{1} s_{1}}{\\left(b_{1} - s_{1}\\right) \\left(b_{2} - s_{2}\\right)}\\)\n\n\n\\(\\displaystyle \\frac{- b_{1} k_{1} s_{2} - b_{2} k_{2} s_{1} + k_{1} s_{1} s_{2} + k_{2} s_{1} s_{2}}{\\left(b_{1} - s_{1}\\right) \\left(b_{2} - s_{2}\\right)}\\)\n\n\nThese estimates are linear functions of the counts \\(k_1\\) and \\(k_2\\) and therefore unbiased.\nWhat is left to show: that we also get a linear (and thus unbiased) solution for more than two bins.\nUnbiasedness for an unbinned EML fit follows, since the unbinned EML fit is a limiting case of the binned EML fit as the bin width goes to zero." + }, + { + "objectID": "posts/resample and numba.html", + "href": "posts/resample and numba.html", + "title": "Numba-accelerated Jackknife", + "section": "", + "text": "My resample library implements the jackknife and bootstrap resampling techniques. The implementations use pure-Python. Since resampling techniques are computationally expensive, it is important to ask whether we can speed up these methods with Numba. In this notebook I compare a JIT-compiled implementation of the Jackknife with the pure Python implementation from resample.\nSince Numba currently does not support JIT-compiled functions which accept other functions as arguments, the JIT-compiled version has to work a little differently from the pure-Python version. It creates a two-dimensional array with all leave-one-out samples at once inside the compiled code. This array is then passed to Python and the function is evaluated over this array. This has a downside: For large samples, creating an N x (N-1) array in memory becomes dangerous since the memory allocation may bring the computer down and very slow.\nFortunately, it turns out that there is nothing to gain from JIT-compiling and thus there is no need to accept this caveat.\n\nfrom resample.jackknife import jackknife\nimport numpy as np\nimport numba as nb\nfrom typing import Callable\nfrom timeit import timeit\nfrom matplotlib import pyplot as plt\n\n\n@nb.njit(nogil=True)\ndef _jackknife_jit(a: np.ndarray):\n n = len(a)\n x = np.empty((n, n - 1), dtype=a.dtype)\n for i in range(n):\n for j in range(n-1):\n shift = 0 if j < i else 1\n x[i, j] = a[j + shift]\n return x\n\n\n@nb.njit\ndef _jackknife_gen(a):\n n = len(a)\n x = np.empty(n - 1, a.dtype)\n for i in range(n - 1):\n x[i] = a[i + 1]\n yield x.view(x.dtype)\n\n # update needs to change values only up to i\n # for a = [0, 1, 2, 3]\n # x0 = [1, 2, 3]\n # x1 = [0, 2, 3]\n # x2 = [0, 1, 3]\n # x3 = [0, 1, 2]\n for i in range(1, n):\n for j in range(i):\n x[j] = a[j]\n yield x.view(x.dtype)\n\n\ndef jackknife_jit(fn: Callable, a: np.ndarray) -> np.ndarray:\n return np.asarray([fn(x) for x in _jackknife_gen(a)])\n\n# force compilation\njackknife_jit(np.mean, np.random.randn(10));\n\n\nx = np.random.rand(10000)\n\n\n%%timeit\njackknife(np.mean, x)\n\n105 ms ± 10.6 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n\n\n\n%%timeit\njackknife_jit( np.mean, x)\n\n119 ms ± 6.34 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n\n\nThe results are comparable. This also remains so when we benchmark samples of varying size.\n\nn = np.asarray(np.geomspace(3, 20000, 50), dtype=int)\ntimes = {\n (\"jackknife\", \"np.mean\"): np.empty(len(n)),\n (\"jackknife_jit\", \"np.mean\"): np.empty(len(n)),\n}\nfor i, ni in enumerate(n):\n x = np.random.randn(ni)\n for s_fcn, s_arg in times:\n t = timeit(f\"{s_fcn}({s_arg}, x)\", f\"from __main__ import {s_fcn}, x; import numpy as np\", number=1)\n times[(s_fcn, s_arg)][i] = t\n\n\nplt.figure()\nfor (s_fcn, s_arg), ts in times.items():\n plt.plot(n, ts, label=f\"{s_fcn} {s_arg}\")\nplt.loglog()\nplt.ylabel(\"time / sec\")\nplt.legend();" + }, + { + "objectID": "posts/chance_of_deviations_in_random_splits.html", + "href": "posts/chance_of_deviations_in_random_splits.html", + "title": "Chance of seeing a one-sigma deviation in random splits", + "section": "", + "text": "We randomly split a sample into two parts and check whether their respective arithmetic means deviate by more than one standard deviation from each other. The chance for this two happen appears to be independent of the distribution from which the original samples are drawn.\n\nimport numpy as np\n\n\nrng = np.random.default_rng(0)\n\nfor distribution in (rng.normal, rng.uniform, rng.exponential):\n x = rng.uniform(size=20)\n \n deltas = []\n ntry = 10000\n for itry in range(ntry):\n rng.shuffle(x)\n a = x[:int(len(x) / 2) ]\n b = x[int(len(x) / 2):]\n ma = np.mean(a)\n mb = np.mean(b)\n va = np.var(a) / len(a)\n vb = np.var(b) / len(b)\n v = va + vb\n deltas.append((ma - mb) / v ** 0.5)\n deltas = np.array(deltas)\n \n print(distribution.__name__, \"fraction with < 1 sigma deviation\", np.sum(deltas < 1) / len(deltas))\n\nnormal fraction with < 1 sigma deviation 0.8255\nuniform fraction with < 1 sigma deviation 0.8238\nexponential fraction with < 1 sigma deviation 0.8281" + }, + { + "objectID": "posts/error_propagation_ratio.html", + "href": "posts/error_propagation_ratio.html", + "title": "Error propagation for a ratio", + "section": "", + "text": "Error propagation also works for systematic uncertainties. I discuss along a simple example.\nWe want to error propagate to variable \\(c\\) with \\[\nc = \\frac a b,\n\\] where \\(a\\) is constant and \\(b\\) has uncertainty \\(\\sigma[b]\\). All variables shall be positive.\nError propagation yields\n\\[\n\\sigma^2[c] = \\big(\\frac{\\partial c}{\\partial b}\\big)^2 \\, \\sigma^2[b] = \\frac {a^2} {b^4} \\, \\sigma^2[b] = \\frac{c^2}{b^2} \\, \\sigma^2[b]\n\\]\nOr equivalent\n\\[\n\\frac{\\sigma[c]}{c} = \\frac{\\sigma[b]}{b}\n\\]\nThe \\(\\sigma\\) here means (as it always does) square-root of variance. Error propagation is correct for any distribution where the variance is defined.\nNumerical check:\n\nimport numpy as np\nfrom scipy import stats\n\na = 1.0 # result is independent of the numerical value of a\n\nrng = np.random.default_rng(1)\n\n# normally distributed numbers with 18 % std.dev.\nb_norm = 1 + 0.18 * rng.normal(size=10000)\nc_norm = a / b_norm\n\nprint(f\"normal distribution {np.std(c_norm) / np.mean(c_norm):.2f}\")\n\n# uniformly distributed numbers with 18 % std.dev.\nz = rng.uniform(size=10000) - 0.5\nb_flat = 1 + 0.18 / np.std(z) * z # scale z so that its std.dev. is 0.18\nc_flat = a / b_flat\n\nprint(f\"uniform distribution {np.std(c_flat) / np.mean(c_flat):.2f}\")\n\nnormal distribution 0.20\nuniform distribution 0.19\n\n\nThe numerical results of \\(\\pm 20\\,\\%\\) and \\(\\pm 19\\,\\%\\) differ a little from the analytical result \\(\\pm 18\\,\\%\\) because error propagation is approximate for non-linear functions. For small deviations, all functions are linear, but \\(\\pm 18\\,\\%\\) is not that small. In general, errors are usually only known up to 10 %, so this is not a shocking deviation.\nSince we don’t know the probability density of systematic offsets, we need to use a distribution-independent framework for the calculation to propagate errors, where systematic uncertainties are the square roots of variances of unknown distributions. The error propagation framework provides just that." + }, + { + "objectID": "unpublished/discrete profiling.html", + "href": "unpublished/discrete profiling.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "from iminuit import Minuit\nfrom iminuit.cost import ExtendedBinnedNLL\nfrom scipy.stats import norm, expon\nfrom scipy.interpolate import BPoly\nimport numpy as np\nfrom pyik.mplext import plot_hist\nimport matplotlib.pyplot as plt\n\n\ndef scaled_cdf_bernstein(x, pars):\n pars = np.atleast_1d(pars)\n c = BPoly(pars[..., np.newaxis], xrange).antiderivative()\n return c(x)\n\n\ndef scaled_cdf_expon(x, a, b):\n return a * (1 - np.exp(-b * x))\n\n\ndef model1(x, a, b, c):\n sig = a * norm(0.5, 0.05).cdf(x)\n bkg = scaled_cdf_expon(x, b, c)\n return bkg + sig\n\n\ndef model2(x, par):\n sig = par[0] * norm(0.5, 0.05).cdf(x)\n bkg = scaled_cdf_bernstein(x, par[1:])\n return bkg + sig\n\n\nrng = np.random.default_rng(1)\nxrange = np.array((0, 1))\nx_bkg = rng.exponential(0.4, size=10000)\nx_sig = norm(0.5, 0.05).rvs(5000)\nx = np.append(x_bkg, x_sig)\nx = x[(x > xrange[0]) & (x < xrange[1])]\nn, xe = np.histogram(x, bins=50, range=xrange)\nplot_hist(xe, n)\n\n\n\n\n\nm1 = Minuit(ExtendedBinnedNLL(n, xe, model1), 500, 1000, 1)\nm1.limits = (0, None)\nm1.migrad()\n\n\n\n\nFCN = 50.68\nNfcn = 98\n\n\nEDM = 4e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\na\n5.01e3\n0.08e3\n\n\n0\n\n\n\n\n1\nb\n10.06e3\n0.13e3\n\n\n0\n\n\n\n\n2\nc\n2.46\n0.04\n\n\n0\n\n\n\n\n\n\n\n\n\na\nb\nc\n\n\na\n7.23e+03\n-3.09e+03 (-0.280)\n0.703 (0.186)\n\n\nb\n-3.09e+03 (-0.280)\n1.68e+04\n-2.61 (-0.454)\n\n\nc\n0.703 (0.186)\n-2.61 (-0.454)\n0.00196\n\n\n\n\n\n\nplot_hist(xe, n)\nplot_hist(xe, np.diff(model1(xe, *m1.values)))\n\n\n\n\n\nm2 = Minuit(ExtendedBinnedNLL(n, xe, model2), (500, 2e4, 2e3, 3e3))\nm2.limits = (0, None)\nm2.migrad()\n\n\n\n\nFCN = 65.9\nNfcn = 126\n\n\nEDM = 1.1e-06 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n4.95e3\n0.09e3\n\n\n0\n\n\n\n\n1\nx1\n23.0e3\n0.4e3\n\n\n0\n\n\n\n\n2\nx2\n2.2e3\n0.5e3\n\n\n0\n\n\n\n\n3\nx3\n2.54e3\n0.19e3\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\nx2\nx3\n\n\nx0\n7.86e+03\n1.98e+03 (0.060)\n-1.41e+04 (-0.345)\n3.41e+03 (0.207)\n\n\nx1\n1.98e+03 (0.060)\n1.4e+05\n-1.02e+05 (-0.590)\n2.48e+04 (0.356)\n\n\nx2\n-1.41e+04 (-0.345)\n-1.02e+05 (-0.590)\n2.13e+05\n-6.22e+04 (-0.722)\n\n\nx3\n3.41e+03 (0.207)\n2.48e+04 (0.356)\n-6.22e+04 (-0.722)\n3.47e+04\n\n\n\n\n\n\nplot_hist(xe, n)\nplot_hist(xe, np.diff(model2(xe, m2.values)))\n\n\n\n\n\nm3 = Minuit(ExtendedBinnedNLL(n, xe, model2), (500, 1000, 1000, 1000, 1000))\nm3.limits = (0, None)\nm3.migrad()\n\n\n\n\nFCN = 52.27\nNfcn = 336\n\n\nEDM = 7.46e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n5.03e3\n0.09e3\n\n\n0\n\n\n\n\n1\nx1\n24.5e3\n0.6e3\n\n\n0\n\n\n\n\n2\nx2\n5.8e3\n0.9e3\n\n\n0\n\n\n\n\n3\nx3\n4.3e3\n0.6e3\n\n\n0\n\n\n\n\n4\nx4\n2.07e3\n0.21e3\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\nx2\nx3\nx4\n\n\nx0\n8.3e+03\n1.06e+04 (0.207)\n-2.7e+04 (-0.316)\n2.81e+03 (0.051)\n558 (0.029)\n\n\nx1\n1.06e+04 (0.207)\n3.16e+05\n-3.97e+05 (-0.752)\n1.63e+05 (0.480)\n-2.66e+04 (-0.223)\n\n\nx2\n-2.7e+04 (-0.316)\n-3.97e+05 (-0.752)\n8.82e+05\n-4.33e+05 (-0.761)\n7.95e+04 (0.400)\n\n\nx3\n2.81e+03 (0.051)\n1.63e+05 (0.480)\n-4.33e+05 (-0.761)\n3.68e+05\n-9.17e+04 (-0.714)\n\n\nx4\n558 (0.029)\n-2.66e+04 (-0.223)\n7.95e+04 (0.400)\n-9.17e+04 (-0.714)\n4.48e+04\n\n\n\n\n\n\nplot_hist(xe, n)\nplot_hist(xe, np.diff(model2(xe, m3.values)))\n\n\n\n\n\nm4 = Minuit(ExtendedBinnedNLL(n, xe, model2), (500, 1000, 1000, 1000, 1000, 1000))\nm4.limits = (0, None)\nm4.migrad()\n\n\n\n\nFCN = 49.14\nNfcn = 456\n\n\nEDM = 7.25e-05 (Goal: 0.0002)\n\n\n\nValid Minimum\nValid Parameters\nNo Parameters at limit\n\n\nBelow EDM threshold (goal x 10)\nBelow call limit\n\n\nCovariance\nHesse ok\nAccurate\nPos. def.\nNot forced\n\n\n\n\n\n\n\nName\nValue\nHesse Error\nMinos Error-\nMinos Error+\nLimit-\nLimit+\nFixed\n\n\n0\nx0\n4.97e3\n0.10e3\n\n\n0\n\n\n\n\n1\nx1\n25.2e3\n0.7e3\n\n\n0\n\n\n\n\n2\nx2\n8.0e3\n1.5e3\n\n\n0\n\n\n\n\n3\nx3\n8.6e3\n2.0e3\n\n\n0\n\n\n\n\n4\nx4\n2.1e3\n1.0e3\n\n\n0\n\n\n\n\n5\nx5\n2.38e3\n0.29e3\n\n\n0\n\n\n\n\n\n\n\n\n\nx0\nx1\nx2\nx3\nx4\nx5\n\n\nx0\n9.6e+03\n-4.21e+03 (-0.061)\n3.4e+04 (0.224)\n-8.4e+04 (-0.428)\n3.7e+04 (0.359)\n-5.97e+03 (-0.213)\n\n\nx1\n-4.21e+03 (-0.061)\n4.94e+05\n-8.14e+05 (-0.748)\n7e+05 (0.497)\n-2.8e+05 (-0.378)\n4.65e+04 (0.231)\n\n\nx2\n3.4e+04 (0.224)\n-8.14e+05 (-0.748)\n2.39e+06\n-2.67e+06 (-0.863)\n1.17e+06 (0.717)\n-2.02e+05 (-0.457)\n\n\nx3\n-8.4e+04 (-0.428)\n7e+05 (0.497)\n-2.67e+06 (-0.863)\n4.01e+06\n-1.93e+06 (-0.916)\n3.54e+05 (0.618)\n\n\nx4\n3.7e+04 (0.359)\n-2.8e+05 (-0.378)\n1.17e+06 (0.717)\n-1.93e+06 (-0.916)\n1.11e+06\n-2.39e+05 (-0.793)\n\n\nx5\n-5.97e+03 (-0.213)\n4.65e+04 (0.231)\n-2.02e+05 (-0.457)\n3.54e+05 (0.618)\n-2.39e+05 (-0.793)\n8.19e+04\n\n\n\n\n\n\nplot_hist(xe, n)\nplot_hist(xe, np.diff(model2(xe, m4.values)))\n\n\n\n\n\nsig1, fcn1, _ = m1.mnprofile(\"a\")\nsig2, fcn2, _ = m2.mnprofile(\"x0\")\nsig3, fcn3, _ = m3.mnprofile(\"x0\")\nsig4, fcn4, _ = m4.mnprofile(\"x0\")\nplt.plot(sig1, fcn1, label=\"model1\")\nplt.plot(sig2, fcn2, label=\"model2(2)\")\nplt.plot(sig3, fcn3, label=\"model2(3)\")\nplt.plot(sig4, fcn4, label=\"model2(4)\")\nplt.legend()\n\n<matplotlib.legend.Legend at 0x132c0bf70>\n\n\n\n\n\n\nsig, fcn1, ok = m1.mnprofile(\"a\", size=50, bound=(4500, 5500))\nassert np.all(ok)\n_, fcn2, ok = m2.mnprofile(\"x0\", size=50, bound=(4500, 5500))\nassert np.all(ok)\n_, fcn3, ok = m3.mnprofile(\"x0\", size=50, bound=(4500, 5500))\nassert np.all(ok)\n_, fcn4, ok = m4.mnprofile(\"x0\", size=50, bound=(4500, 5500))\nassert np.all(ok)\n\n\ndef akaike(npar, n):\n return 2 * npar / (1 - (npar + 1) / n)\n\ndef simple(npar, n):\n return npar\n\n\nfcn_min = np.inf\nfor fcn, npar in ((fcn1, m1.nfit), (fcn2, m2.nfit), (fcn3, m3.nfit), (fcn4, m4.nfit)):\n fcn_min = np.minimum(fcn_min, fcn + akaike(npar, len(n)))\n\n\nplt.plot(sig, fcn1-offset, label=\"model1\")\nplt.plot(sig, fcn2-offset, label=\"model2(2)\")\nplt.plot(sig, fcn3-offset, label=\"model2(3)\")\nplt.plot(sig, fcn4-offset, label=\"model2(4)\")\nplt.legend()\n\n<matplotlib.legend.Legend at 0x1331c2a00>\n\n\n\n\n\n\noffset = np.min(fcn_min)\n\nplt.plot(sig, fcn1-offset+simple(m1.nfit, len(n)), label=\"model1\")\nplt.plot(sig, fcn2-offset+simple(m2.nfit, len(n)), label=\"model2(2)\")\nplt.plot(sig, fcn3-offset+simple(m3.nfit, len(n)), label=\"model2(3)\")\nplt.plot(sig, fcn4-offset+simple(m4.nfit, len(n)), label=\"model2(4)\")\nplt.plot(sig, fcn_min-offset, color=\"k\", label=\"min\")\nplt.legend()\nplt.ylim(0, 20)\n\n(0.0, 20.0)" + }, + { + "objectID": "unpublished/Binning loss.html", + "href": "unpublished/Binning loss.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "from iminuit import Minuit\nimport numpy as np\nfrom scipy.stats import norm, uniform, poisson\nfrom matplotlib import pyplot as plt\nimport progressbar\n\n\nxrange = (0., 1.)\nz = 0.2\nmu = 0.5\nsigma = 0.05\n\ntruth = (z, mu, sigma)\n\n\ndef unbinned(z, mu, sigma):\n \"\"\"Unbinned likelihood\"\"\"\n probs = (1.0-z) / (xrange[1] - xrange[0]) + z * norm(mu, sigma).pdf(x)\n return -np.sum(np.log(probs))\n\n\ndef expected(z, mu, sigma):\n \"\"\"Return expected counts per bin in a histogram\"\"\"\n nx = len(x)\n b = uniform(*xrange).cdf(xe)\n s = norm(mu, sigma).cdf(xe)\n m = (1-z) * b + z * s\n return len(x) * (m[1:] - m[:-1])\n\n\ndef binned(z, mu, sigma):\n \"\"\"Binned likelihood for Poisson-distributed counts\"\"\"\n m = expected(z, mu, sigma)\n return np.sum(m - w * np.log(m))\n\n\n# check expected\nnp.random.seed(1)\ns = norm(mu, sigma).rvs(int(z * 1000))\nb = uniform(*xrange).rvs(1000 - len(s))\nx = np.append(s, b)\nassert len(x) == 1000\n\nfig, ax = plt.subplots(1, 3, figsize=(15, 4), sharex=True)\nfor axi, bins in zip(ax, (5, 50, 100)):\n plt.sca(axi)\n w, xe, *_ = plt.hist(x, bins=bins, range=xrange, label=\"data\")\n m = expected(*truth)\n plt.step(xe, np.append(0, m), label=\"expected\")\n plt.legend()\n plt.xlabel(\"x\")\n\n\n\n\n\nnp.random.seed(1)\n\nbins = (5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 30, 40, 50, 100)\nntot = 1000\nnmc = 100\n\npars_unbinned = []\npars_binned = []\ncovs_unbinned = []\ncovs_binned = []\n\nminuit_kwargs = {\n \"z\": 0.5, \"mu\": 0.5, \"sigma\": 0.05, \"pedantic\": False, \"errordef\": 0.5,\n \"limit_z\" : (0, 1), \"limit_mu\" : xrange, \"limit_sigma\" : (0, None)\n}\n\nfor imc in progressbar.progressbar(range(nmc)):\n xs = norm(mu, sigma).rvs(int(z * ntot))\n xb = uniform(*xrange).rvs(ntot - len(xs))\n x = np.append(xs, xb)\n \n m_un = Minuit(unbinned, **minuit_kwargs)\n m_un.migrad()\n m_un.migrad()\n\n valid = m_un.get_fmin().is_valid\n if not valid:\n continue\n\n pars_b = []\n covs_b = []\n for b in bins:\n w, xe = np.histogram(x, bins=b, range=xrange)\n\n m = Minuit(binned, **minuit_kwargs)\n m.migrad()\n m.migrad()\n\n valid = m.get_fmin().is_valid\n if not valid:\n break\n\n pars_b.append(m.np_values())\n covs_b.append(m.np_covariance())\n\n if not valid:\n continue\n\n pars_unbinned.append(m_un.np_values())\n covs_unbinned.append(m_un.np_covariance())\n \n pars_binned.append(pars_b)\n covs_binned.append(covs_b)\n\npars_unbinned = np.array(pars_unbinned)\npars_binned = np.array(pars_binned)\ncovs_unbinned = np.array(covs_unbinned)\ncovs_binned = np.array(covs_binned)\n\n100% (100 of 100) |######################| Elapsed Time: 0:03:22 Time: 0:03:22\n\n\n\n# bias\nfig, ax = plt.subplots(1, 3, figsize=(16, 4), sharex=True)\nplt.subplots_adjust(wspace=0.3)\nfor ipar, name in enumerate((\"z\", \"mu\", \"sigma\")):\n plt.sca(ax[ipar])\n y = pars_unbinned[:, ipar]\n m = np.mean(y)\n s = np.std(y) / n ** 0.5\n n = len(pars_unbinned)\n plt.axhline(m, label=\"unbinned\")\n plt.axhspan(m-s, m+s, alpha=0.2)\n y = pars_binned[:, :, ipar]\n m = np.mean(y, axis=0)\n s = np.std(y, axis=0) / n ** 0.5\n plt.errorbar(bins, m, s, fmt=\"o\", color=\"C1\", label=\"binned\")\n plt.legend()\n plt.xlabel(\"bins\")\n plt.semilogx()\n plt.ylabel(name)\n plt.axvline((xrange[1] - xrange[0])/sigma, ls=\"--\", color=\"0.5\")\n plt.axvline((xrange[1] - xrange[0])/(2 * sigma), ls=\":\", color=\"0.5\")\n plt.axhline(truth[ipar], color=\"k\", zorder=0)\n\n\n\n\n\n# coverage probability\nfig, ax = plt.subplots(1, 3, figsize=(16, 4), sharex=True)\nplt.subplots_adjust(wspace=0.3)\nfor ipar, name in enumerate((\"z\", \"mu\", \"sigma\")):\n plt.sca(ax[ipar])\n y = pars_unbinned[:, ipar]\n dy = covs_unbinned[:, ipar, ipar] ** 0.5\n prob = np.mean((truth[ipar] < y + dy) & (truth[ipar] > y - dy))\n plt.axhline(prob, label=\"unbinned\")\n y = pars_binned[:, :, ipar]\n dy = covs_binned[:, :, ipar, ipar] ** 0.5\n prob = np.mean((truth[ipar] < y + dy) & (truth[ipar] > y - dy), axis=0)\n plt.plot(bins, prob, \"o-\", color=\"C1\", label=\"binned\")\n plt.ylim(0, 1)\n plt.xlabel(\"bins\")\n plt.semilogx()\n plt.ylabel(name)\n plt.axhline(0.68, ls=\"-\", color=\"k\")\n plt.axvline((xrange[1] - xrange[0])/sigma, ls=\"--\", color=\"0.5\")\n plt.axvline((xrange[1] - xrange[0])/(2 * sigma), ls=\":\", color=\"0.5\")\n plt.legend(loc=\"lower right\")" + }, + { + "objectID": "unpublished/Heilter model, influence of alpha.html", + "href": "unpublished/Heilter model, influence of alpha.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "from sympy import *\n\n\nnmu, energy, xi, alpha, M = symbols(\"N_mu E xi alpha M\", real=True, positive=True)\n\n\nnmu1 = (energy/xi)**(1 + log(alpha) / log(M))\n\n\nd = diff(nmu1, alpha); d\n\n\\(\\displaystyle \\frac{\\left(\\frac{E}{\\xi}\\right)^{1 + \\frac{\\log{\\left(\\alpha \\right)}}{\\log{\\left(M \\right)}}} \\log{\\left(\\frac{E}{\\xi} \\right)}}{\\alpha \\log{\\left(M \\right)}}\\)\n\n\n\nd.subs(nmu1, nmu)\n\n\\(\\displaystyle \\frac{N_{\\mu} \\log{\\left(\\frac{E}{\\xi} \\right)}}{\\alpha \\log{\\left(M \\right)}}\\)" + }, + { + "objectID": "py311/lib/python3.11/site-packages/werkzeug/debug/shared/ICON_LICENSE.html", + "href": "py311/lib/python3.11/site-packages/werkzeug/debug/shared/ICON_LICENSE.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "Silk icon set 1.3 by Mark James mjames@gmail.com\nhttp://www.famfamfam.com/lab/icons/silk/\nLicense: CC-BY-2.5 or CC-BY-3.0" + }, + { + "objectID": "py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html", + "href": "py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "This project was started and continues to be led by Brian E. Granger (ellisonbg AT gmail DOT com). Min Ragan-Kelley (benjaminrk AT gmail DOT com) is the primary developer of pyzmq at this time.\nThe following people have contributed to the project:\n\nAlexander Else (alexander DOT else AT team DOT telstra DOT com)\nAlexander Pyhalov (apyhalov AT gmail DOT com)\nAlexandr Emelin (frvzmb AT gmail DOT com)\nAmr Ali (amr AT ledgerx DOT com)\nAndre Caron (andre DOT l DOT caron AT gmail DOT com)\nAndrea Crotti (andrea DOT crotti DOT 0 AT gmail DOT com)\nAndrew Gwozdziewycz (git AT apgwoz DOT com)\nBaptiste Lepilleur (baptiste DOT lepilleur AT gmail DOT com)\nBrandyn A. White (bwhite AT dappervision DOT com)\nBrian E. Granger (ellisonbg AT gmail DOT com)\nBrian Hoffman (hoffman_brian AT bah DOT com)\nCarlos A. Rocha (carlos DOT rocha AT gmail DOT com)\nChris Laws (clawsicus AT gmail DOT com)\nChristian Wyglendowski (christian AT bu DOT mp)\nChristoph Gohlke (cgohlke AT uci DOT edu)\nCurtis (curtis AT tinbrain DOT net)\nCyril Holweck (cyril DOT holweck AT free DOT fr)\nDan Colish (dcolish AT gmail DOT com)\nDaniel Lundin (dln AT eintr DOT org)\nDaniel Truemper (truemped AT googlemail DOT com)\nDouglas Creager (douglas DOT creager AT redjack DOT com)\nEduardo Stalinho (eduardooc DOT 86 AT gmail DOT com)\nEren Güven (erenguven0 AT gmail DOT com)\nErick Tryzelaar (erick DOT tryzelaar AT gmail DOT com)\nErik Tollerud (erik DOT tollerud AT gmail DOT com)\nFELD Boris (lothiraldan AT gmail DOT com)\nFantix King (fantix DOT king AT gmail DOT com)\nFelipe Cruz (felipecruz AT loogica DOT net)\nFernando Perez (Fernando DOT Perez AT berkeley DOT edu)\nFrank Wiles (frank AT revsys DOT com)\nFélix-Antoine Fortin (felix DOT antoine DOT fortin AT gmail DOT com)\nGavrie Philipson (gavriep AT il DOT ibm DOT com)\nGodefroid Chapelle (gotcha AT bubblenet DOT be)\nGreg Banks (gbanks AT mybasis DOT com)\nGreg Ward (greg AT gerg DOT ca)\nGuido Goldstein (github AT a-nugget DOT de)\nIan Lee (IanLee1521 AT gmail DOT com)\nIonuț Arțăriși (ionut AT artarisi DOT eu)\nIvo Danihelka (ivo AT danihelka DOT net)\nIyed (iyed DOT bennour AT gmail DOT com)\nJim Garrison (jim AT garrison DOT cc)\nJohn Gallagher (johnkgallagher AT gmail DOT com)\nJulian Taylor (jtaylor DOT debian AT googlemail DOT com)\nJustin Bronder (jsbronder AT gmail DOT com)\nJustin Riley (justin DOT t DOT riley AT gmail DOT com)\nMarc Abramowitz (marc AT marc-abramowitz DOT com)\nMatthew Aburn (mattja6 AT gmail DOT com)\nMichel Pelletier (pelletier DOT michel AT gmail DOT com)\nMichel Zou (xantares09 AT hotmail DOT com)\nMin Ragan-Kelley (benjaminrk AT gmail DOT com)\nNell Hardcastle (nell AT dev-nell DOT com)\nNicholas Pilkington (nicholas DOT pilkington AT gmail DOT com)\nNicholas Piël (nicholas AT nichol DOT as)\nNick Pellegrino (npellegrino AT mozilla DOT com)\nNicolas Delaby (nicolas DOT delaby AT ezeep DOT com)\nOndrej Certik (ondrej AT certik DOT cz)\nPaul Colomiets (paul AT colomiets DOT name)\nPawel Jasinski (pawel DOT jasinski AT gmail DOT com)\nPhus Lu (phus DOT lu AT gmail DOT com)\nRobert Buchholz (rbu AT goodpoint DOT de)\nRobert Jordens (jordens AT gmail DOT com)\nRyan Cox (ryan DOT a DOT cox AT gmail DOT com)\nRyan Kelly (ryan AT rfk DOT id DOT au)\nScott Maxwell (scott AT codecobblers DOT com)\nScott Sadler (github AT mashi DOT org)\nSimon Knight (simon DOT knight AT gmail DOT com)\nStefan Friesel (sf AT cloudcontrol DOT de)\nStefan van der Walt (stefan AT sun DOT ac DOT za)\nStephen Diehl (stephen DOT m DOT diehl AT gmail DOT com)\nSylvain Corlay (scorlay AT bloomberg DOT net)\nThomas Kluyver (takowl AT gmail DOT com)\nThomas Spura (tomspur AT fedoraproject DOT org)\nTigger Bear (Tigger AT Tiggers-Mac-mini DOT local)\nTorsten Landschoff (torsten DOT landschoff AT dynamore DOT de)\nVadim Markovtsev (v DOT markovtsev AT samsung DOT com)\nYannick Hold (yannickhold AT gmail DOT com)\nZbigniew Jędrzejewski-Szmek (zbyszek AT in DOT waw DOT pl)\nhugo shi (hugoshi AT bleb2 DOT (none))\njdgleeson (jdgleeson AT mac DOT com)\nkyledj (kyle AT bucebuce DOT com)\nspez (steve AT hipmunk DOT com)\nstu (stuart DOT axon AT jpcreative DOT co DOT uk)\nxantares (xantares AT fujitsu-l64 DOT (none))\n\nas reported by:\ngit log --all --format='- %aN (%aE)' | sort -u | sed 's/@/ AT /1' | sed -e 's/\\.\\([^ ]\\)/ DOT \\1/g'\nwith some adjustments.\n\n\n\nBrandon Craig-Rhodes (brandon AT rhodesmill DOT org)\nEugene Chernyshov (chernyshov DOT eugene AT gmail DOT com)\nCraig Austin (craig DOT austin AT gmail DOT com)\n\n\n\n\n\nTravis Cline (travis DOT cline AT gmail DOT com)\nRyan Kelly (ryan AT rfk DOT id DOT au)\nZachary Voase (z AT zacharyvoase DOT com)" + }, + { + "objectID": "py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html#authors", + "href": "py311/lib/python3.11/site-packages/pyzmq-25.1.1.dist-info/AUTHORS.html#authors", + "title": "Hans Dembinski's blog", + "section": "", + "text": "This project was started and continues to be led by Brian E. Granger (ellisonbg AT gmail DOT com). Min Ragan-Kelley (benjaminrk AT gmail DOT com) is the primary developer of pyzmq at this time.\nThe following people have contributed to the project:\n\nAlexander Else (alexander DOT else AT team DOT telstra DOT com)\nAlexander Pyhalov (apyhalov AT gmail DOT com)\nAlexandr Emelin (frvzmb AT gmail DOT com)\nAmr Ali (amr AT ledgerx DOT com)\nAndre Caron (andre DOT l DOT caron AT gmail DOT com)\nAndrea Crotti (andrea DOT crotti DOT 0 AT gmail DOT com)\nAndrew Gwozdziewycz (git AT apgwoz DOT com)\nBaptiste Lepilleur (baptiste DOT lepilleur AT gmail DOT com)\nBrandyn A. White (bwhite AT dappervision DOT com)\nBrian E. Granger (ellisonbg AT gmail DOT com)\nBrian Hoffman (hoffman_brian AT bah DOT com)\nCarlos A. Rocha (carlos DOT rocha AT gmail DOT com)\nChris Laws (clawsicus AT gmail DOT com)\nChristian Wyglendowski (christian AT bu DOT mp)\nChristoph Gohlke (cgohlke AT uci DOT edu)\nCurtis (curtis AT tinbrain DOT net)\nCyril Holweck (cyril DOT holweck AT free DOT fr)\nDan Colish (dcolish AT gmail DOT com)\nDaniel Lundin (dln AT eintr DOT org)\nDaniel Truemper (truemped AT googlemail DOT com)\nDouglas Creager (douglas DOT creager AT redjack DOT com)\nEduardo Stalinho (eduardooc DOT 86 AT gmail DOT com)\nEren Güven (erenguven0 AT gmail DOT com)\nErick Tryzelaar (erick DOT tryzelaar AT gmail DOT com)\nErik Tollerud (erik DOT tollerud AT gmail DOT com)\nFELD Boris (lothiraldan AT gmail DOT com)\nFantix King (fantix DOT king AT gmail DOT com)\nFelipe Cruz (felipecruz AT loogica DOT net)\nFernando Perez (Fernando DOT Perez AT berkeley DOT edu)\nFrank Wiles (frank AT revsys DOT com)\nFélix-Antoine Fortin (felix DOT antoine DOT fortin AT gmail DOT com)\nGavrie Philipson (gavriep AT il DOT ibm DOT com)\nGodefroid Chapelle (gotcha AT bubblenet DOT be)\nGreg Banks (gbanks AT mybasis DOT com)\nGreg Ward (greg AT gerg DOT ca)\nGuido Goldstein (github AT a-nugget DOT de)\nIan Lee (IanLee1521 AT gmail DOT com)\nIonuț Arțăriși (ionut AT artarisi DOT eu)\nIvo Danihelka (ivo AT danihelka DOT net)\nIyed (iyed DOT bennour AT gmail DOT com)\nJim Garrison (jim AT garrison DOT cc)\nJohn Gallagher (johnkgallagher AT gmail DOT com)\nJulian Taylor (jtaylor DOT debian AT googlemail DOT com)\nJustin Bronder (jsbronder AT gmail DOT com)\nJustin Riley (justin DOT t DOT riley AT gmail DOT com)\nMarc Abramowitz (marc AT marc-abramowitz DOT com)\nMatthew Aburn (mattja6 AT gmail DOT com)\nMichel Pelletier (pelletier DOT michel AT gmail DOT com)\nMichel Zou (xantares09 AT hotmail DOT com)\nMin Ragan-Kelley (benjaminrk AT gmail DOT com)\nNell Hardcastle (nell AT dev-nell DOT com)\nNicholas Pilkington (nicholas DOT pilkington AT gmail DOT com)\nNicholas Piël (nicholas AT nichol DOT as)\nNick Pellegrino (npellegrino AT mozilla DOT com)\nNicolas Delaby (nicolas DOT delaby AT ezeep DOT com)\nOndrej Certik (ondrej AT certik DOT cz)\nPaul Colomiets (paul AT colomiets DOT name)\nPawel Jasinski (pawel DOT jasinski AT gmail DOT com)\nPhus Lu (phus DOT lu AT gmail DOT com)\nRobert Buchholz (rbu AT goodpoint DOT de)\nRobert Jordens (jordens AT gmail DOT com)\nRyan Cox (ryan DOT a DOT cox AT gmail DOT com)\nRyan Kelly (ryan AT rfk DOT id DOT au)\nScott Maxwell (scott AT codecobblers DOT com)\nScott Sadler (github AT mashi DOT org)\nSimon Knight (simon DOT knight AT gmail DOT com)\nStefan Friesel (sf AT cloudcontrol DOT de)\nStefan van der Walt (stefan AT sun DOT ac DOT za)\nStephen Diehl (stephen DOT m DOT diehl AT gmail DOT com)\nSylvain Corlay (scorlay AT bloomberg DOT net)\nThomas Kluyver (takowl AT gmail DOT com)\nThomas Spura (tomspur AT fedoraproject DOT org)\nTigger Bear (Tigger AT Tiggers-Mac-mini DOT local)\nTorsten Landschoff (torsten DOT landschoff AT dynamore DOT de)\nVadim Markovtsev (v DOT markovtsev AT samsung DOT com)\nYannick Hold (yannickhold AT gmail DOT com)\nZbigniew Jędrzejewski-Szmek (zbyszek AT in DOT waw DOT pl)\nhugo shi (hugoshi AT bleb2 DOT (none))\njdgleeson (jdgleeson AT mac DOT com)\nkyledj (kyle AT bucebuce DOT com)\nspez (steve AT hipmunk DOT com)\nstu (stuart DOT axon AT jpcreative DOT co DOT uk)\nxantares (xantares AT fujitsu-l64 DOT (none))\n\nas reported by:\ngit log --all --format='- %aN (%aE)' | sort -u | sed 's/@/ AT /1' | sed -e 's/\\.\\([^ ]\\)/ DOT \\1/g'\nwith some adjustments.\n\n\n\nBrandon Craig-Rhodes (brandon AT rhodesmill DOT org)\nEugene Chernyshov (chernyshov DOT eugene AT gmail DOT com)\nCraig Austin (craig DOT austin AT gmail DOT com)\n\n\n\n\n\nTravis Cline (travis DOT cline AT gmail DOT com)\nRyan Kelly (ryan AT rfk DOT id DOT au)\nZachary Voase (z AT zacharyvoase DOT com)" + }, + { + "objectID": "py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html", + "href": "py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html", + "title": "UAT for NbAgg backend.", + "section": "", + "text": "from imp import reload\nThe first line simply reloads matplotlib, uses the nbagg backend and then reloads the backend, just to ensure we have the latest modification to the backend code. Note: The underlying JavaScript will not be updated by this process, so a refresh of the browser after clearing the output and saving is necessary to clear everything fully.\nimport matplotlib\nreload(matplotlib)\n\nmatplotlib.use('nbagg')\n\nimport matplotlib.backends.backend_nbagg\nreload(matplotlib.backends.backend_nbagg)" + }, + { + "objectID": "py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html#uat-13---animation", + "href": "py311/lib/python3.11/site-packages/matplotlib/backends/web_backend/nbagg_uat.html#uat-13---animation", + "title": "UAT for NbAgg backend.", + "section": "UAT 13 - Animation", + "text": "UAT 13 - Animation\nThe following should generate an animated line:\n\nimport matplotlib.animation as animation\nimport numpy as np\n\nfig, ax = plt.subplots()\n\nx = np.arange(0, 2*np.pi, 0.01) # x-array\nline, = ax.plot(x, np.sin(x))\n\ndef animate(i):\n line.set_ydata(np.sin(x+i/10.0)) # update the data\n return line,\n\n#Init only required for blitting to give a clean slate.\ndef init():\n line.set_ydata(np.ma.array(x, mask=True))\n return line,\n\nani = animation.FuncAnimation(fig, animate, np.arange(1, 200), init_func=init,\n interval=100., blit=True)\nplt.show()\n\n\nUAT 14 - Keyboard shortcuts in IPython after close of figure\nAfter closing the previous figure (with the close button above the figure) the IPython keyboard shortcuts should still function.\n\n\nUAT 15 - Figure face colours\nThe nbagg honours all colours apart from that of the figure.patch. The two plots below should produce a figure with a red background. There should be no yellow figure.\n\nimport matplotlib\nmatplotlib.rcParams.update({'figure.facecolor': 'red',\n 'savefig.facecolor': 'yellow'})\nplt.figure()\nplt.plot([3, 2, 1])\n\nplt.show()\n\n\n\nUAT 16 - Events\nPressing any keyboard key or mouse button (or scrolling) should cycle the line while the figure has focus. The figure should have focus by default when it is created and re-gain it by clicking on the canvas. Clicking anywhere outside of the figure should release focus, but moving the mouse out of the figure should not release focus.\n\nimport itertools\nfig, ax = plt.subplots()\nx = np.linspace(0,10,10000)\ny = np.sin(x)\nln, = ax.plot(x,y)\nevt = []\ncolors = iter(itertools.cycle(['r', 'g', 'b', 'k', 'c']))\ndef on_event(event):\n if event.name.startswith('key'):\n fig.suptitle('%s: %s' % (event.name, event.key))\n elif event.name == 'scroll_event':\n fig.suptitle('%s: %s' % (event.name, event.step))\n else:\n fig.suptitle('%s: %s' % (event.name, event.button))\n evt.append(event)\n ln.set_color(next(colors))\n fig.canvas.draw()\n fig.canvas.draw_idle()\n\nfig.canvas.mpl_connect('button_press_event', on_event)\nfig.canvas.mpl_connect('button_release_event', on_event)\nfig.canvas.mpl_connect('scroll_event', on_event)\nfig.canvas.mpl_connect('key_press_event', on_event)\nfig.canvas.mpl_connect('key_release_event', on_event)\n\nplt.show()\n\n\n\nUAT 17 - Timers\nSingle-shot timers follow a completely different code path in the nbagg backend than regular timers (such as those used in the animation example above.) The next set of tests ensures that both “regular” and “single-shot” timers work properly.\nThe following should show a simple clock that updates twice a second:\n\nimport time\n\nfig, ax = plt.subplots()\ntext = ax.text(0.5, 0.5, '', ha='center')\n\ndef update(text):\n text.set(text=time.ctime())\n text.axes.figure.canvas.draw()\n \ntimer = fig.canvas.new_timer(500, [(update, [text], {})])\ntimer.start()\nplt.show()\n\nHowever, the following should only update once and then stop:\n\nfig, ax = plt.subplots()\ntext = ax.text(0.5, 0.5, '', ha='center') \ntimer = fig.canvas.new_timer(500, [(update, [text], {})])\n\ntimer.single_shot = True\ntimer.start()\n\nplt.show()\n\nAnd the next two examples should never show any visible text at all:\n\nfig, ax = plt.subplots()\ntext = ax.text(0.5, 0.5, '', ha='center')\ntimer = fig.canvas.new_timer(500, [(update, [text], {})])\n\ntimer.start()\ntimer.stop()\n\nplt.show()\n\n\nfig, ax = plt.subplots()\ntext = ax.text(0.5, 0.5, '', ha='center')\ntimer = fig.canvas.new_timer(500, [(update, [text], {})])\n\ntimer.single_shot = True\ntimer.start()\ntimer.stop()\n\nplt.show()\n\n\n\nUAT 18 - stopping figure when removed from DOM\nWhen the div that contains from the figure is removed from the DOM the figure should shut down it’s comm, and if the python-side figure has no more active comms, it should destroy the figure. Repeatedly running the cell below should always have the same figure number\n\nfig, ax = plt.subplots()\nax.plot(range(5))\nplt.show()\n\nRunning the cell below will re-show the figure. After this, re-running the cell above should result in a new figure number.\n\nfig.canvas.manager.reshow()\n\n\n\nUAT 19 - Blitting\nClicking on the figure should plot a green horizontal line moving up the axes.\n\nimport itertools\n\ncnt = itertools.count()\nbg = None\n\ndef onclick_handle(event):\n \"\"\"Should draw elevating green line on each mouse click\"\"\"\n global bg\n if bg is None:\n bg = ax.figure.canvas.copy_from_bbox(ax.bbox) \n ax.figure.canvas.restore_region(bg)\n\n cur_y = (next(cnt) % 10) * 0.1\n ln.set_ydata([cur_y, cur_y])\n ax.draw_artist(ln)\n ax.figure.canvas.blit(ax.bbox)\n\nfig, ax = plt.subplots()\nax.plot([0, 1], [0, 1], 'r')\nln, = ax.plot([0, 1], [0, 0], 'g', animated=True)\nplt.show()\nax.figure.canvas.draw()\n\nax.figure.canvas.mpl_connect('button_press_event', onclick_handle)" + }, + { + "objectID": "py311/lib/python3.11/site-packages/idna-3.6.dist-info/LICENSE.html", + "href": "py311/lib/python3.11/site-packages/idna-3.6.dist-info/LICENSE.html", + "title": "Hans Dembinski's blog", + "section": "", + "text": "BSD 3-Clause License\nCopyright (c) 2013-2023, Kim Davies and contributors. All rights reserved.\nRedistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:\n\nRedistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.\nRedistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.\nNeither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.\n\nTHIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE." + }, + { + "objectID": "about.html", + "href": "about.html", + "title": "About", + "section": "", + "text": "I have curious mind and a broad range of interests, professionally they circle around data analysis, statistics, software development, machine learning, and artificial intelligence. In this blog, I write about these topics.\nI have a PhD in astroparticle physics. If you want to know more about my professional career, read more about me on LinkedIn. I wrote and maintain a couple of data science libraries. Check out my Github page if you want to know more. I am a member of the Boost C++ Project, to which I contributed the Histogram library. My C++ skill is in the grandmaster area, but recently I only program in Python, since it is so much more enjoyable and with Numba one can write numerical code in Python with performance on par with C++ in a fraction of the time." + } +] \ No newline at end of file diff --git a/site_libs/bootstrap/bootstrap-icons.css b/site_libs/bootstrap/bootstrap-icons.css new file mode 100644 index 0000000..94f1940 --- /dev/null +++ b/site_libs/bootstrap/bootstrap-icons.css @@ -0,0 +1,2018 @@ +@font-face { + font-display: block; + font-family: "bootstrap-icons"; + src: +url("./bootstrap-icons.woff?2ab2cbbe07fcebb53bdaa7313bb290f2") format("woff"); +} + +.bi::before, +[class^="bi-"]::before, +[class*=" bi-"]::before { + display: inline-block; + font-family: bootstrap-icons !important; + font-style: normal; + font-weight: normal !important; + font-variant: normal; + text-transform: none; + line-height: 1; + vertical-align: -.125em; + -webkit-font-smoothing: antialiased; + -moz-osx-font-smoothing: 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"\f859"; } +.bi-sign-intersection::before { content: "\f85a"; } +.bi-sign-merge-left-fill::before { content: "\f85b"; } +.bi-sign-merge-left::before { content: "\f85c"; } +.bi-sign-merge-right-fill::before { content: "\f85d"; } +.bi-sign-merge-right::before { content: "\f85e"; } +.bi-sign-no-left-turn-fill::before { content: "\f85f"; } +.bi-sign-no-left-turn::before { content: "\f860"; } +.bi-sign-no-parking-fill::before { content: "\f861"; } +.bi-sign-no-parking::before { content: "\f862"; } +.bi-sign-no-right-turn-fill::before { content: "\f863"; } +.bi-sign-no-right-turn::before { content: "\f864"; } +.bi-sign-railroad-fill::before { content: "\f865"; } +.bi-sign-railroad::before { content: "\f866"; } +.bi-building-add::before { content: "\f867"; } +.bi-building-check::before { content: "\f868"; } +.bi-building-dash::before { content: "\f869"; } +.bi-building-down::before { content: "\f86a"; } +.bi-building-exclamation::before { content: "\f86b"; } +.bi-building-fill-add::before { content: "\f86c"; } +.bi-building-fill-check::before { content: "\f86d"; } +.bi-building-fill-dash::before { content: "\f86e"; } +.bi-building-fill-down::before { content: "\f86f"; } +.bi-building-fill-exclamation::before { content: "\f870"; } +.bi-building-fill-gear::before { content: "\f871"; } +.bi-building-fill-lock::before { content: "\f872"; } +.bi-building-fill-slash::before { content: "\f873"; } +.bi-building-fill-up::before { content: "\f874"; } +.bi-building-fill-x::before { content: "\f875"; } +.bi-building-fill::before { content: "\f876"; } +.bi-building-gear::before { content: "\f877"; } +.bi-building-lock::before { content: "\f878"; } +.bi-building-slash::before { content: "\f879"; } +.bi-building-up::before { content: "\f87a"; } +.bi-building-x::before { content: "\f87b"; } +.bi-buildings-fill::before { content: "\f87c"; } +.bi-buildings::before { content: "\f87d"; } +.bi-bus-front-fill::before { content: "\f87e"; } +.bi-bus-front::before { content: "\f87f"; } +.bi-ev-front-fill::before { content: "\f880"; } +.bi-ev-front::before { content: "\f881"; } +.bi-globe-americas::before { content: "\f882"; } +.bi-globe-asia-australia::before { content: "\f883"; } +.bi-globe-central-south-asia::before { content: "\f884"; } +.bi-globe-europe-africa::before { content: "\f885"; } +.bi-house-add-fill::before { content: "\f886"; } +.bi-house-add::before { content: "\f887"; } +.bi-house-check-fill::before { content: "\f888"; } +.bi-house-check::before { content: "\f889"; } +.bi-house-dash-fill::before { content: "\f88a"; } +.bi-house-dash::before { content: "\f88b"; } +.bi-house-down-fill::before { content: "\f88c"; } +.bi-house-down::before { content: "\f88d"; } +.bi-house-exclamation-fill::before { content: "\f88e"; } +.bi-house-exclamation::before { content: "\f88f"; } +.bi-house-gear-fill::before { content: "\f890"; } +.bi-house-gear::before { content: "\f891"; } +.bi-house-lock-fill::before { content: "\f892"; } +.bi-house-lock::before { content: "\f893"; } +.bi-house-slash-fill::before { content: "\f894"; } +.bi-house-slash::before { content: "\f895"; } +.bi-house-up-fill::before { content: "\f896"; } +.bi-house-up::before { content: "\f897"; } +.bi-house-x-fill::before { content: "\f898"; } +.bi-house-x::before { content: "\f899"; } +.bi-person-add::before { content: "\f89a"; } +.bi-person-down::before { content: "\f89b"; } +.bi-person-exclamation::before { content: "\f89c"; } +.bi-person-fill-add::before { content: "\f89d"; } +.bi-person-fill-check::before { content: "\f89e"; } +.bi-person-fill-dash::before { content: "\f89f"; } +.bi-person-fill-down::before { content: "\f8a0"; } +.bi-person-fill-exclamation::before { content: "\f8a1"; } +.bi-person-fill-gear::before { content: "\f8a2"; } +.bi-person-fill-lock::before { content: "\f8a3"; } +.bi-person-fill-slash::before { content: "\f8a4"; } +.bi-person-fill-up::before { content: "\f8a5"; } +.bi-person-fill-x::before { content: "\f8a6"; } +.bi-person-gear::before { content: "\f8a7"; } +.bi-person-lock::before { content: "\f8a8"; } +.bi-person-slash::before { content: "\f8a9"; } +.bi-person-up::before { content: "\f8aa"; } +.bi-scooter::before { content: "\f8ab"; } +.bi-taxi-front-fill::before { content: "\f8ac"; } +.bi-taxi-front::before { content: "\f8ad"; } +.bi-amd::before { content: "\f8ae"; } +.bi-database-add::before { content: "\f8af"; } +.bi-database-check::before { content: "\f8b0"; } +.bi-database-dash::before { content: "\f8b1"; } +.bi-database-down::before { content: "\f8b2"; } +.bi-database-exclamation::before { content: "\f8b3"; } +.bi-database-fill-add::before { content: "\f8b4"; } +.bi-database-fill-check::before { content: "\f8b5"; } +.bi-database-fill-dash::before { content: "\f8b6"; } +.bi-database-fill-down::before { content: "\f8b7"; } +.bi-database-fill-exclamation::before { content: "\f8b8"; } +.bi-database-fill-gear::before { content: "\f8b9"; } +.bi-database-fill-lock::before { content: "\f8ba"; } +.bi-database-fill-slash::before { content: "\f8bb"; } +.bi-database-fill-up::before { content: "\f8bc"; } +.bi-database-fill-x::before { content: "\f8bd"; } +.bi-database-fill::before { content: "\f8be"; } +.bi-database-gear::before { content: "\f8bf"; } +.bi-database-lock::before { content: "\f8c0"; } +.bi-database-slash::before { content: "\f8c1"; } +.bi-database-up::before { content: "\f8c2"; } +.bi-database-x::before { content: "\f8c3"; } +.bi-database::before { content: "\f8c4"; } +.bi-houses-fill::before { content: "\f8c5"; } +.bi-houses::before { content: "\f8c6"; } +.bi-nvidia::before { content: "\f8c7"; } +.bi-person-vcard-fill::before { content: "\f8c8"; } +.bi-person-vcard::before { content: "\f8c9"; } +.bi-sina-weibo::before { content: "\f8ca"; } +.bi-tencent-qq::before { content: "\f8cb"; } +.bi-wikipedia::before { content: "\f8cc"; } diff --git a/site_libs/bootstrap/bootstrap-icons.woff b/site_libs/bootstrap/bootstrap-icons.woff new file mode 100644 index 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t?t:{}},a(K,t,Y),t}_handleSwipe(){const t=Math.abs(this.touchDeltaX);if(t<=40)return;const e=t/this.touchDeltaX;this.touchDeltaX=0,e&&this._slide(e>0?J:Z)}_addEventListeners(){this._config.keyboard&&j.on(this._element,"keydown.bs.carousel",(t=>this._keydown(t))),"hover"===this._config.pause&&(j.on(this._element,"mouseenter.bs.carousel",(t=>this.pause(t))),j.on(this._element,"mouseleave.bs.carousel",(t=>this.cycle(t)))),this._config.touch&&this._touchSupported&&this._addTouchEventListeners()}_addTouchEventListeners(){const t=t=>this._pointerEvent&&("pen"===t.pointerType||"touch"===t.pointerType),e=e=>{t(e)?this.touchStartX=e.clientX:this._pointerEvent||(this.touchStartX=e.touches[0].clientX)},i=t=>{this.touchDeltaX=t.touches&&t.touches.length>1?0:t.touches[0].clientX-this.touchStartX},n=e=>{t(e)&&(this.touchDeltaX=e.clientX-this.touchStartX),this._handleSwipe(),"hover"===this._config.pause&&(this.pause(),this.touchTimeout&&clearTimeout(this.touchTimeout),this.touchTimeout=setTimeout((t=>this.cycle(t)),500+this._config.interval))};V.find(".carousel-item img",this._element).forEach((t=>{j.on(t,"dragstart.bs.carousel",(t=>t.preventDefault()))})),this._pointerEvent?(j.on(this._element,"pointerdown.bs.carousel",(t=>e(t))),j.on(this._element,"pointerup.bs.carousel",(t=>n(t))),this._element.classList.add("pointer-event")):(j.on(this._element,"touchstart.bs.carousel",(t=>e(t))),j.on(this._element,"touchmove.bs.carousel",(t=>i(t))),j.on(this._element,"touchend.bs.carousel",(t=>n(t))))}_keydown(t){if(/input|textarea/i.test(t.target.tagName))return;const e=tt[t.key];e&&(t.preventDefault(),this._slide(e))}_getItemIndex(t){return this._items=t&&t.parentNode?V.find(".carousel-item",t.parentNode):[],this._items.indexOf(t)}_getItemByOrder(t,e){const i=t===Q;return v(this._items,e,i,this._config.wrap)}_triggerSlideEvent(t,e){const i=this._getItemIndex(t),n=this._getItemIndex(V.findOne(nt,this._element));return j.trigger(this._element,"slide.bs.carousel",{relatedTarget:t,direction:e,from:n,to:i})}_setActiveIndicatorElement(t){if(this._indicatorsElement){const e=V.findOne(".active",this._indicatorsElement);e.classList.remove(it),e.removeAttribute("aria-current");const i=V.find("[data-bs-target]",this._indicatorsElement);for(let e=0;e{j.trigger(this._element,et,{relatedTarget:o,direction:d,from:s,to:r})};if(this._element.classList.contains("slide")){o.classList.add(h),u(o),n.classList.add(c),o.classList.add(c);const t=()=>{o.classList.remove(c,h),o.classList.add(it),n.classList.remove(it,h,c),this._isSliding=!1,setTimeout(f,0)};this._queueCallback(t,n,!0)}else n.classList.remove(it),o.classList.add(it),this._isSliding=!1,f();a&&this.cycle()}_directionToOrder(t){return[J,Z].includes(t)?m()?t===Z?G:Q:t===Z?Q:G:t}_orderToDirection(t){return[Q,G].includes(t)?m()?t===G?Z:J:t===G?J:Z:t}static carouselInterface(t,e){const i=st.getOrCreateInstance(t,e);let{_config:n}=i;"object"==typeof e&&(n={...n,...e});const s="string"==typeof e?e:n.slide;if("number"==typeof e)i.to(e);else if("string"==typeof s){if(void 0===i[s])throw new TypeError(`No method named "${s}"`);i[s]()}else n.interval&&n.ride&&(i.pause(),i.cycle())}static jQueryInterface(t){return this.each((function(){st.carouselInterface(this,t)}))}static dataApiClickHandler(t){const e=n(this);if(!e||!e.classList.contains("carousel"))return;const i={...U.getDataAttributes(e),...U.getDataAttributes(this)},s=this.getAttribute("data-bs-slide-to");s&&(i.interval=!1),st.carouselInterface(e,i),s&&st.getInstance(e).to(s),t.preventDefault()}}j.on(document,"click.bs.carousel.data-api","[data-bs-slide], [data-bs-slide-to]",st.dataApiClickHandler),j.on(window,"load.bs.carousel.data-api",(()=>{const t=V.find('[data-bs-ride="carousel"]');for(let e=0,i=t.length;et===this._element));null!==s&&o.length&&(this._selector=s,this._triggerArray.push(e))}this._initializeChildren(),this._config.parent||this._addAriaAndCollapsedClass(this._triggerArray,this._isShown()),this._config.toggle&&this.toggle()}static get Default(){return rt}static get NAME(){return ot}toggle(){this._isShown()?this.hide():this.show()}show(){if(this._isTransitioning||this._isShown())return;let t,e=[];if(this._config.parent){const t=V.find(ut,this._config.parent);e=V.find(".collapse.show, .collapse.collapsing",this._config.parent).filter((e=>!t.includes(e)))}const i=V.findOne(this._selector);if(e.length){const n=e.find((t=>i!==t));if(t=n?pt.getInstance(n):null,t&&t._isTransitioning)return}if(j.trigger(this._element,"show.bs.collapse").defaultPrevented)return;e.forEach((e=>{i!==e&&pt.getOrCreateInstance(e,{toggle:!1}).hide(),t||H.set(e,"bs.collapse",null)}));const n=this._getDimension();this._element.classList.remove(ct),this._element.classList.add(ht),this._element.style[n]=0,this._addAriaAndCollapsedClass(this._triggerArray,!0),this._isTransitioning=!0;const s=`scroll${n[0].toUpperCase()+n.slice(1)}`;this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(ht),this._element.classList.add(ct,lt),this._element.style[n]="",j.trigger(this._element,"shown.bs.collapse")}),this._element,!0),this._element.style[n]=`${this._element[s]}px`}hide(){if(this._isTransitioning||!this._isShown())return;if(j.trigger(this._element,"hide.bs.collapse").defaultPrevented)return;const t=this._getDimension();this._element.style[t]=`${this._element.getBoundingClientRect()[t]}px`,u(this._element),this._element.classList.add(ht),this._element.classList.remove(ct,lt);const e=this._triggerArray.length;for(let t=0;t{this._isTransitioning=!1,this._element.classList.remove(ht),this._element.classList.add(ct),j.trigger(this._element,"hidden.bs.collapse")}),this._element,!0)}_isShown(t=this._element){return t.classList.contains(lt)}_getConfig(t){return(t={...rt,...U.getDataAttributes(this._element),...t}).toggle=Boolean(t.toggle),t.parent=r(t.parent),a(ot,t,at),t}_getDimension(){return this._element.classList.contains("collapse-horizontal")?"width":"height"}_initializeChildren(){if(!this._config.parent)return;const t=V.find(ut,this._config.parent);V.find(ft,this._config.parent).filter((e=>!t.includes(e))).forEach((t=>{const e=n(t);e&&this._addAriaAndCollapsedClass([t],this._isShown(e))}))}_addAriaAndCollapsedClass(t,e){t.length&&t.forEach((t=>{e?t.classList.remove(dt):t.classList.add(dt),t.setAttribute("aria-expanded",e)}))}static jQueryInterface(t){return this.each((function(){const e={};"string"==typeof t&&/show|hide/.test(t)&&(e.toggle=!1);const 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Be={placement:"bottom",modifiers:[],strategy:"absolute"};function Re(){for(var t=arguments.length,e=new Array(t),i=0;ij.on(t,"mouseover",d))),this._element.focus(),this._element.setAttribute("aria-expanded",!0),this._menu.classList.add(Je),this._element.classList.add(Je),j.trigger(this._element,"shown.bs.dropdown",t)}hide(){if(c(this._element)||!this._isShown(this._menu))return;const t={relatedTarget:this._element};this._completeHide(t)}dispose(){this._popper&&this._popper.destroy(),super.dispose()}update(){this._inNavbar=this._detectNavbar(),this._popper&&this._popper.update()}_completeHide(t){j.trigger(this._element,"hide.bs.dropdown",t).defaultPrevented||("ontouchstart"in 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t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return"static"===this._config.display&&(t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,..."function"==typeof this._config.popperConfig?this._config.popperConfig(t):this._config.popperConfig}}_selectMenuItem({key:t,target:e}){const i=V.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter(l);i.length&&v(i,e,t===Ye,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(t&&(2===t.button||"keyup"===t.type&&"Tab"!==t.key))return;const e=V.find(ti);for(let i=0,n=e.length;ie+t)),this._setElementAttributes(di,"paddingRight",(e=>e+t)),this._setElementAttributes(ui,"marginRight",(e=>e-t))}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t)[e];t.style[e]=`${i(Number.parseFloat(s))}px`}))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,"paddingRight"),this._resetElementAttributes(di,"paddingRight"),this._resetElementAttributes(ui,"marginRight")}_saveInitialAttribute(t,e){const i=t.style[e];i&&U.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=U.getDataAttribute(t,e);void 0===i?t.style.removeProperty(e):(U.removeDataAttribute(t,e),t.style[e]=i)}))}_applyManipulationCallback(t,e){o(t)?e(t):V.find(t,this._element).forEach(e)}isOverflowing(){return this.getWidth()>0}}const pi={className:"modal-backdrop",isVisible:!0,isAnimated:!1,rootElement:"body",clickCallback:null},mi={className:"string",isVisible:"boolean",isAnimated:"boolean",rootElement:"(element|string)",clickCallback:"(function|null)"},gi="show",_i="mousedown.bs.backdrop";class bi{constructor(t){this._config=this._getConfig(t),this._isAppended=!1,this._element=null}show(t){this._config.isVisible?(this._append(),this._config.isAnimated&&u(this._getElement()),this._getElement().classList.add(gi),this._emulateAnimation((()=>{_(t)}))):_(t)}hide(t){this._config.isVisible?(this._getElement().classList.remove(gi),this._emulateAnimation((()=>{this.dispose(),_(t)}))):_(t)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_getConfig(t){return(t={...pi,..."object"==typeof t?t:{}}).rootElement=r(t.rootElement),a("backdrop",t,mi),t}_append(){this._isAppended||(this._config.rootElement.append(this._getElement()),j.on(this._getElement(),_i,(()=>{_(this._config.clickCallback)})),this._isAppended=!0)}dispose(){this._isAppended&&(j.off(this._element,_i),this._element.remove(),this._isAppended=!1)}_emulateAnimation(t){b(t,this._getElement(),this._config.isAnimated)}}const vi={trapElement:null,autofocus:!0},yi={trapElement:"element",autofocus:"boolean"},wi=".bs.focustrap",Ei="backward";class Ai{constructor(t){this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}activate(){const{trapElement:t,autofocus:e}=this._config;this._isActive||(e&&t.focus(),j.off(document,wi),j.on(document,"focusin.bs.focustrap",(t=>this._handleFocusin(t))),j.on(document,"keydown.tab.bs.focustrap",(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,j.off(document,wi))}_handleFocusin(t){const{target:e}=t,{trapElement:i}=this._config;if(e===document||e===i||i.contains(e))return;const n=V.focusableChildren(i);0===n.length?i.focus():this._lastTabNavDirection===Ei?n[n.length-1].focus():n[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?Ei:"forward")}_getConfig(t){return t={...vi,..."object"==typeof t?t:{}},a("focustrap",t,yi),t}}const Ti="modal",Oi="Escape",Ci={backdrop:!0,keyboard:!0,focus:!0},ki={backdrop:"(boolean|string)",keyboard:"boolean",focus:"boolean"},Li="hidden.bs.modal",xi="show.bs.modal",Di="resize.bs.modal",Si="click.dismiss.bs.modal",Ni="keydown.dismiss.bs.modal",Ii="mousedown.dismiss.bs.modal",Pi="modal-open",ji="show",Mi="modal-static";class Hi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._dialog=V.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._ignoreBackdropClick=!1,this._isTransitioning=!1,this._scrollBar=new fi}static get Default(){return Ci}static get NAME(){return Ti}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||j.trigger(this._element,xi,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isAnimated()&&(this._isTransitioning=!0),this._scrollBar.hide(),document.body.classList.add(Pi),this._adjustDialog(),this._setEscapeEvent(),this._setResizeEvent(),j.on(this._dialog,Ii,(()=>{j.one(this._element,"mouseup.dismiss.bs.modal",(t=>{t.target===this._element&&(this._ignoreBackdropClick=!0)}))})),this._showBackdrop((()=>this._showElement(t))))}hide(){if(!this._isShown||this._isTransitioning)return;if(j.trigger(this._element,"hide.bs.modal").defaultPrevented)return;this._isShown=!1;const t=this._isAnimated();t&&(this._isTransitioning=!0),this._setEscapeEvent(),this._setResizeEvent(),this._focustrap.deactivate(),this._element.classList.remove(ji),j.off(this._element,Si),j.off(this._dialog,Ii),this._queueCallback((()=>this._hideModal()),this._element,t)}dispose(){[window,this._dialog].forEach((t=>j.off(t,".bs.modal"))),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new bi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_getConfig(t){return t={...Ci,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Ti,t,ki),t}_showElement(t){const e=this._isAnimated(),i=V.findOne(".modal-body",this._dialog);this._element.parentNode&&this._element.parentNode.nodeType===Node.ELEMENT_NODE||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0,i&&(i.scrollTop=0),e&&u(this._element),this._element.classList.add(ji),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,j.trigger(this._element,"shown.bs.modal",{relatedTarget:t})}),this._dialog,e)}_setEscapeEvent(){this._isShown?j.on(this._element,Ni,(t=>{this._config.keyboard&&t.key===Oi?(t.preventDefault(),this.hide()):this._config.keyboard||t.key!==Oi||this._triggerBackdropTransition()})):j.off(this._element,Ni)}_setResizeEvent(){this._isShown?j.on(window,Di,(()=>this._adjustDialog())):j.off(window,Di)}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(Pi),this._resetAdjustments(),this._scrollBar.reset(),j.trigger(this._element,Li)}))}_showBackdrop(t){j.on(this._element,Si,(t=>{this._ignoreBackdropClick?this._ignoreBackdropClick=!1:t.target===t.currentTarget&&(!0===this._config.backdrop?this.hide():"static"===this._config.backdrop&&this._triggerBackdropTransition())})),this._backdrop.show(t)}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(j.trigger(this._element,"hidePrevented.bs.modal").defaultPrevented)return;const{classList:t,scrollHeight:e,style:i}=this._element,n=e>document.documentElement.clientHeight;!n&&"hidden"===i.overflowY||t.contains(Mi)||(n||(i.overflowY="hidden"),t.add(Mi),this._queueCallback((()=>{t.remove(Mi),n||this._queueCallback((()=>{i.overflowY=""}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;(!i&&t&&!m()||i&&!t&&m())&&(this._element.style.paddingLeft=`${e}px`),(i&&!t&&!m()||!i&&t&&m())&&(this._element.style.paddingRight=`${e}px`)}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=Hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}j.on(document,"click.bs.modal.data-api",'[data-bs-toggle="modal"]',(function(t){const e=n(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),j.one(e,xi,(t=>{t.defaultPrevented||j.one(e,Li,(()=>{l(this)&&this.focus()}))}));const i=V.findOne(".modal.show");i&&Hi.getInstance(i).hide(),Hi.getOrCreateInstance(e).toggle(this)})),R(Hi),g(Hi);const Bi="offcanvas",Ri={backdrop:!0,keyboard:!0,scroll:!1},Wi={backdrop:"boolean",keyboard:"boolean",scroll:"boolean"},$i="show",zi=".offcanvas.show",qi="hidden.bs.offcanvas";class Fi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get NAME(){return Bi}static get Default(){return Ri}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||j.trigger(this._element,"show.bs.offcanvas",{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._element.style.visibility="visible",this._backdrop.show(),this._config.scroll||(new fi).hide(),this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add($i),this._queueCallback((()=>{this._config.scroll||this._focustrap.activate(),j.trigger(this._element,"shown.bs.offcanvas",{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(j.trigger(this._element,"hide.bs.offcanvas").defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.remove($i),this._backdrop.hide(),this._queueCallback((()=>{this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._element.style.visibility="hidden",this._config.scroll||(new fi).reset(),j.trigger(this._element,qi)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_getConfig(t){return t={...Ri,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Bi,t,Wi),t}_initializeBackDrop(){return new bi({className:"offcanvas-backdrop",isVisible:this._config.backdrop,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:()=>this.hide()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_addEventListeners(){j.on(this._element,"keydown.dismiss.bs.offcanvas",(t=>{this._config.keyboard&&"Escape"===t.key&&this.hide()}))}static jQueryInterface(t){return this.each((function(){const e=Fi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}j.on(document,"click.bs.offcanvas.data-api",'[data-bs-toggle="offcanvas"]',(function(t){const e=n(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),c(this))return;j.one(e,qi,(()=>{l(this)&&this.focus()}));const i=V.findOne(zi);i&&i!==e&&Fi.getInstance(i).hide(),Fi.getOrCreateInstance(e).toggle(this)})),j.on(window,"load.bs.offcanvas.data-api",(()=>V.find(zi).forEach((t=>Fi.getOrCreateInstance(t).show())))),R(Fi),g(Fi);const Ui=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Vi=/^(?:(?:https?|mailto|ftp|tel|file|sms):|[^#&/:?]*(?:[#/?]|$))/i,Ki=/^data:(?:image\/(?:bmp|gif|jpeg|jpg|png|tiff|webp)|video\/(?:mpeg|mp4|ogg|webm)|audio\/(?:mp3|oga|ogg|opus));base64,[\d+/a-z]+=*$/i,Xi=(t,e)=>{const i=t.nodeName.toLowerCase();if(e.includes(i))return!Ui.has(i)||Boolean(Vi.test(t.nodeValue)||Ki.test(t.nodeValue));const n=e.filter((t=>t instanceof RegExp));for(let t=0,e=n.length;t{Xi(t,r)||i.removeAttribute(t.nodeName)}))}return n.body.innerHTML}const Qi="tooltip",Gi=new Set(["sanitize","allowList","sanitizeFn"]),Zi={animation:"boolean",template:"string",title:"(string|element|function)",trigger:"string",delay:"(number|object)",html:"boolean",selector:"(string|boolean)",placement:"(string|function)",offset:"(array|string|function)",container:"(string|element|boolean)",fallbackPlacements:"array",boundary:"(string|element)",customClass:"(string|function)",sanitize:"boolean",sanitizeFn:"(null|function)",allowList:"object",popperConfig:"(null|object|function)"},Ji={AUTO:"auto",TOP:"top",RIGHT:m()?"left":"right",BOTTOM:"bottom",LEFT:m()?"right":"left"},tn={animation:!0,template:'

    ',trigger:"hover focus",title:"",delay:0,html:!1,selector:!1,placement:"top",offset:[0,0],container:!1,fallbackPlacements:["top","right","bottom","left"],boundary:"clippingParents",customClass:"",sanitize:!0,sanitizeFn:null,allowList:{"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},popperConfig:null},en={HIDE:"hide.bs.tooltip",HIDDEN:"hidden.bs.tooltip",SHOW:"show.bs.tooltip",SHOWN:"shown.bs.tooltip",INSERTED:"inserted.bs.tooltip",CLICK:"click.bs.tooltip",FOCUSIN:"focusin.bs.tooltip",FOCUSOUT:"focusout.bs.tooltip",MOUSEENTER:"mouseenter.bs.tooltip",MOUSELEAVE:"mouseleave.bs.tooltip"},nn="fade",sn="show",on="show",rn="out",an=".tooltip-inner",ln=".modal",cn="hide.bs.modal",hn="hover",dn="focus";class un extends B{constructor(t,e){if(void 0===Fe)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t),this._isEnabled=!0,this._timeout=0,this._hoverState="",this._activeTrigger={},this._popper=null,this._config=this._getConfig(e),this.tip=null,this._setListeners()}static get Default(){return tn}static get NAME(){return Qi}static get Event(){return en}static get DefaultType(){return Zi}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(t){if(this._isEnabled)if(t){const e=this._initializeOnDelegatedTarget(t);e._activeTrigger.click=!e._activeTrigger.click,e._isWithActiveTrigger()?e._enter(null,e):e._leave(null,e)}else{if(this.getTipElement().classList.contains(sn))return void this._leave(null,this);this._enter(null,this)}}dispose(){clearTimeout(this._timeout),j.off(this._element.closest(ln),cn,this._hideModalHandler),this.tip&&this.tip.remove(),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this.isWithContent()||!this._isEnabled)return;const t=j.trigger(this._element,this.constructor.Event.SHOW),e=h(this._element),i=null===e?this._element.ownerDocument.documentElement.contains(this._element):e.contains(this._element);if(t.defaultPrevented||!i)return;"tooltip"===this.constructor.NAME&&this.tip&&this.getTitle()!==this.tip.querySelector(an).innerHTML&&(this._disposePopper(),this.tip.remove(),this.tip=null);const n=this.getTipElement(),s=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME);n.setAttribute("id",s),this._element.setAttribute("aria-describedby",s),this._config.animation&&n.classList.add(nn);const o="function"==typeof this._config.placement?this._config.placement.call(this,n,this._element):this._config.placement,r=this._getAttachment(o);this._addAttachmentClass(r);const{container:a}=this._config;H.set(n,this.constructor.DATA_KEY,this),this._element.ownerDocument.documentElement.contains(this.tip)||(a.append(n),j.trigger(this._element,this.constructor.Event.INSERTED)),this._popper?this._popper.update():this._popper=qe(this._element,n,this._getPopperConfig(r)),n.classList.add(sn);const l=this._resolvePossibleFunction(this._config.customClass);l&&n.classList.add(...l.split(" ")),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>{j.on(t,"mouseover",d)}));const c=this.tip.classList.contains(nn);this._queueCallback((()=>{const t=this._hoverState;this._hoverState=null,j.trigger(this._element,this.constructor.Event.SHOWN),t===rn&&this._leave(null,this)}),this.tip,c)}hide(){if(!this._popper)return;const t=this.getTipElement();if(j.trigger(this._element,this.constructor.Event.HIDE).defaultPrevented)return;t.classList.remove(sn),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>j.off(t,"mouseover",d))),this._activeTrigger.click=!1,this._activeTrigger.focus=!1,this._activeTrigger.hover=!1;const e=this.tip.classList.contains(nn);this._queueCallback((()=>{this._isWithActiveTrigger()||(this._hoverState!==on&&t.remove(),this._cleanTipClass(),this._element.removeAttribute("aria-describedby"),j.trigger(this._element,this.constructor.Event.HIDDEN),this._disposePopper())}),this.tip,e),this._hoverState=""}update(){null!==this._popper&&this._popper.update()}isWithContent(){return Boolean(this.getTitle())}getTipElement(){if(this.tip)return this.tip;const t=document.createElement("div");t.innerHTML=this._config.template;const e=t.children[0];return this.setContent(e),e.classList.remove(nn,sn),this.tip=e,this.tip}setContent(t){this._sanitizeAndSetContent(t,this.getTitle(),an)}_sanitizeAndSetContent(t,e,i){const n=V.findOne(i,t);e||!n?this.setElementContent(n,e):n.remove()}setElementContent(t,e){if(null!==t)return o(e)?(e=r(e),void(this._config.html?e.parentNode!==t&&(t.innerHTML="",t.append(e)):t.textContent=e.textContent)):void(this._config.html?(this._config.sanitize&&(e=Yi(e,this._config.allowList,this._config.sanitizeFn)),t.innerHTML=e):t.textContent=e)}getTitle(){const t=this._element.getAttribute("data-bs-original-title")||this._config.title;return this._resolvePossibleFunction(t)}updateAttachment(t){return"right"===t?"end":"left"===t?"start":t}_initializeOnDelegatedTarget(t,e){return e||this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return"function"==typeof t?t.call(this._element):t}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"onChange",enabled:!0,phase:"afterWrite",fn:t=>this._handlePopperPlacementChange(t)}],onFirstUpdate:t=>{t.options.placement!==t.placement&&this._handlePopperPlacementChange(t)}};return{...e,..."function"==typeof this._config.popperConfig?this._config.popperConfig(e):this._config.popperConfig}}_addAttachmentClass(t){this.getTipElement().classList.add(`${this._getBasicClassPrefix()}-${this.updateAttachment(t)}`)}_getAttachment(t){return Ji[t.toUpperCase()]}_setListeners(){this._config.trigger.split(" ").forEach((t=>{if("click"===t)j.on(this._element,this.constructor.Event.CLICK,this._config.selector,(t=>this.toggle(t)));else if("manual"!==t){const e=t===hn?this.constructor.Event.MOUSEENTER:this.constructor.Event.FOCUSIN,i=t===hn?this.constructor.Event.MOUSELEAVE:this.constructor.Event.FOCUSOUT;j.on(this._element,e,this._config.selector,(t=>this._enter(t))),j.on(this._element,i,this._config.selector,(t=>this._leave(t)))}})),this._hideModalHandler=()=>{this._element&&this.hide()},j.on(this._element.closest(ln),cn,this._hideModalHandler),this._config.selector?this._config={...this._config,trigger:"manual",selector:""}:this._fixTitle()}_fixTitle(){const t=this._element.getAttribute("title"),e=typeof this._element.getAttribute("data-bs-original-title");(t||"string"!==e)&&(this._element.setAttribute("data-bs-original-title",t||""),!t||this._element.getAttribute("aria-label")||this._element.textContent||this._element.setAttribute("aria-label",t),this._element.setAttribute("title",""))}_enter(t,e){e=this._initializeOnDelegatedTarget(t,e),t&&(e._activeTrigger["focusin"===t.type?dn:hn]=!0),e.getTipElement().classList.contains(sn)||e._hoverState===on?e._hoverState=on:(clearTimeout(e._timeout),e._hoverState=on,e._config.delay&&e._config.delay.show?e._timeout=setTimeout((()=>{e._hoverState===on&&e.show()}),e._config.delay.show):e.show())}_leave(t,e){e=this._initializeOnDelegatedTarget(t,e),t&&(e._activeTrigger["focusout"===t.type?dn:hn]=e._element.contains(t.relatedTarget)),e._isWithActiveTrigger()||(clearTimeout(e._timeout),e._hoverState=rn,e._config.delay&&e._config.delay.hide?e._timeout=setTimeout((()=>{e._hoverState===rn&&e.hide()}),e._config.delay.hide):e.hide())}_isWithActiveTrigger(){for(const t in this._activeTrigger)if(this._activeTrigger[t])return!0;return!1}_getConfig(t){const e=U.getDataAttributes(this._element);return Object.keys(e).forEach((t=>{Gi.has(t)&&delete e[t]})),(t={...this.constructor.Default,...e,..."object"==typeof t&&t?t:{}}).container=!1===t.container?document.body:r(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),a(Qi,t,this.constructor.DefaultType),t.sanitize&&(t.template=Yi(t.template,t.allowList,t.sanitizeFn)),t}_getDelegateConfig(){const t={};for(const e in this._config)this.constructor.Default[e]!==this._config[e]&&(t[e]=this._config[e]);return t}_cleanTipClass(){const t=this.getTipElement(),e=new RegExp(`(^|\\s)${this._getBasicClassPrefix()}\\S+`,"g"),i=t.getAttribute("class").match(e);null!==i&&i.length>0&&i.map((t=>t.trim())).forEach((e=>t.classList.remove(e)))}_getBasicClassPrefix(){return"bs-tooltip"}_handlePopperPlacementChange(t){const{state:e}=t;e&&(this.tip=e.elements.popper,this._cleanTipClass(),this._addAttachmentClass(this._getAttachment(e.placement)))}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null)}static jQueryInterface(t){return this.each((function(){const e=un.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}g(un);const fn={...un.Default,placement:"right",offset:[0,8],trigger:"click",content:"",template:''},pn={...un.DefaultType,content:"(string|element|function)"},mn={HIDE:"hide.bs.popover",HIDDEN:"hidden.bs.popover",SHOW:"show.bs.popover",SHOWN:"shown.bs.popover",INSERTED:"inserted.bs.popover",CLICK:"click.bs.popover",FOCUSIN:"focusin.bs.popover",FOCUSOUT:"focusout.bs.popover",MOUSEENTER:"mouseenter.bs.popover",MOUSELEAVE:"mouseleave.bs.popover"};class gn extends un{static get Default(){return fn}static get NAME(){return"popover"}static get Event(){return mn}static get DefaultType(){return pn}isWithContent(){return this.getTitle()||this._getContent()}setContent(t){this._sanitizeAndSetContent(t,this.getTitle(),".popover-header"),this._sanitizeAndSetContent(t,this._getContent(),".popover-body")}_getContent(){return this._resolvePossibleFunction(this._config.content)}_getBasicClassPrefix(){return"bs-popover"}static jQueryInterface(t){return this.each((function(){const e=gn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}g(gn);const _n="scrollspy",bn={offset:10,method:"auto",target:""},vn={offset:"number",method:"string",target:"(string|element)"},yn="active",wn=".nav-link, .list-group-item, .dropdown-item",En="position";class An extends B{constructor(t,e){super(t),this._scrollElement="BODY"===this._element.tagName?window:this._element,this._config=this._getConfig(e),this._offsets=[],this._targets=[],this._activeTarget=null,this._scrollHeight=0,j.on(this._scrollElement,"scroll.bs.scrollspy",(()=>this._process())),this.refresh(),this._process()}static get Default(){return bn}static get NAME(){return _n}refresh(){const t=this._scrollElement===this._scrollElement.window?"offset":En,e="auto"===this._config.method?t:this._config.method,n=e===En?this._getScrollTop():0;this._offsets=[],this._targets=[],this._scrollHeight=this._getScrollHeight(),V.find(wn,this._config.target).map((t=>{const s=i(t),o=s?V.findOne(s):null;if(o){const t=o.getBoundingClientRect();if(t.width||t.height)return[U[e](o).top+n,s]}return null})).filter((t=>t)).sort(((t,e)=>t[0]-e[0])).forEach((t=>{this._offsets.push(t[0]),this._targets.push(t[1])}))}dispose(){j.off(this._scrollElement,".bs.scrollspy"),super.dispose()}_getConfig(t){return(t={...bn,...U.getDataAttributes(this._element),..."object"==typeof t&&t?t:{}}).target=r(t.target)||document.documentElement,a(_n,t,vn),t}_getScrollTop(){return this._scrollElement===window?this._scrollElement.pageYOffset:this._scrollElement.scrollTop}_getScrollHeight(){return this._scrollElement.scrollHeight||Math.max(document.body.scrollHeight,document.documentElement.scrollHeight)}_getOffsetHeight(){return this._scrollElement===window?window.innerHeight:this._scrollElement.getBoundingClientRect().height}_process(){const t=this._getScrollTop()+this._config.offset,e=this._getScrollHeight(),i=this._config.offset+e-this._getOffsetHeight();if(this._scrollHeight!==e&&this.refresh(),t>=i){const t=this._targets[this._targets.length-1];this._activeTarget!==t&&this._activate(t)}else{if(this._activeTarget&&t0)return this._activeTarget=null,void this._clear();for(let e=this._offsets.length;e--;)this._activeTarget!==this._targets[e]&&t>=this._offsets[e]&&(void 0===this._offsets[e+1]||t`${e}[data-bs-target="${t}"],${e}[href="${t}"]`)),i=V.findOne(e.join(","),this._config.target);i.classList.add(yn),i.classList.contains("dropdown-item")?V.findOne(".dropdown-toggle",i.closest(".dropdown")).classList.add(yn):V.parents(i,".nav, .list-group").forEach((t=>{V.prev(t,".nav-link, .list-group-item").forEach((t=>t.classList.add(yn))),V.prev(t,".nav-item").forEach((t=>{V.children(t,".nav-link").forEach((t=>t.classList.add(yn)))}))})),j.trigger(this._scrollElement,"activate.bs.scrollspy",{relatedTarget:t})}_clear(){V.find(wn,this._config.target).filter((t=>t.classList.contains(yn))).forEach((t=>t.classList.remove(yn)))}static jQueryInterface(t){return this.each((function(){const e=An.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}j.on(window,"load.bs.scrollspy.data-api",(()=>{V.find('[data-bs-spy="scroll"]').forEach((t=>new An(t)))})),g(An);const Tn="active",On="fade",Cn="show",kn=".active",Ln=":scope > li > .active";class xn extends B{static get NAME(){return"tab"}show(){if(this._element.parentNode&&this._element.parentNode.nodeType===Node.ELEMENT_NODE&&this._element.classList.contains(Tn))return;let t;const 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t=e.state,n=e.name,r=t.rects.reference,o=t.rects.popper,i=t.modifiersData.preventOverflow,a=Y(t,{elementContext:"reference"}),s=Y(t,{altBoundary:!0}),f=he(a,r),c=he(s,o,i),p=me(f),u=me(c);t.modifiersData[n]={referenceClippingOffsets:f,popperEscapeOffsets:c,isReferenceHidden:p,hasPopperEscaped:u},t.attributes.popper=Object.assign({},t.attributes.popper,{"data-popper-reference-hidden":p,"data-popper-escaped":u})}},ge=K({defaultModifiers:[Z,$,ne,re]}),ye=[Z,$,ne,re,oe,pe,le,de,ve],be=K({defaultModifiers:ye});e.applyStyles=re,e.arrow=de,e.computeStyles=ne,e.createPopper=be,e.createPopperLite=ge,e.defaultModifiers=ye,e.detectOverflow=Y,e.eventListeners=Z,e.flip=pe,e.hide=ve,e.offset=oe,e.popperGenerator=K,e.popperOffsets=$,e.preventOverflow=le,Object.defineProperty(e,"__esModule",{value:!0})})); + diff --git a/site_libs/quarto-html/quarto-syntax-highlighting.css b/site_libs/quarto-html/quarto-syntax-highlighting.css new file mode 100644 index 0000000..d9fd98f --- /dev/null +++ b/site_libs/quarto-html/quarto-syntax-highlighting.css @@ -0,0 +1,203 @@ +/* quarto syntax highlight colors */ +:root { + --quarto-hl-ot-color: #003B4F; + --quarto-hl-at-color: #657422; + --quarto-hl-ss-color: #20794D; + --quarto-hl-an-color: #5E5E5E; + --quarto-hl-fu-color: #4758AB; + --quarto-hl-st-color: #20794D; + --quarto-hl-cf-color: #003B4F; + --quarto-hl-op-color: #5E5E5E; + --quarto-hl-er-color: #AD0000; + --quarto-hl-bn-color: #AD0000; + --quarto-hl-al-color: #AD0000; + --quarto-hl-va-color: #111111; + --quarto-hl-bu-color: inherit; + --quarto-hl-ex-color: inherit; + --quarto-hl-pp-color: #AD0000; + --quarto-hl-in-color: #5E5E5E; + --quarto-hl-vs-color: #20794D; + --quarto-hl-wa-color: #5E5E5E; + --quarto-hl-do-color: #5E5E5E; + --quarto-hl-im-color: #00769E; + --quarto-hl-ch-color: #20794D; + --quarto-hl-dt-color: #AD0000; + --quarto-hl-fl-color: #AD0000; + --quarto-hl-co-color: #5E5E5E; + --quarto-hl-cv-color: #5E5E5E; + --quarto-hl-cn-color: #8f5902; + --quarto-hl-sc-color: #5E5E5E; + --quarto-hl-dv-color: #AD0000; + --quarto-hl-kw-color: #003B4F; +} + +/* other quarto variables */ +:root { + --quarto-font-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace; +} + +pre > code.sourceCode > span { + color: #003B4F; +} + +code span { + color: #003B4F; +} + +code.sourceCode > span { + color: #003B4F; +} + +div.sourceCode, +div.sourceCode pre.sourceCode { + color: #003B4F; +} + +code span.ot { + color: #003B4F; + font-style: inherit; +} + +code span.at { + color: #657422; + font-style: inherit; +} + +code span.ss { + color: #20794D; + font-style: inherit; +} + +code span.an { + color: #5E5E5E; + font-style: inherit; +} + +code span.fu { + color: #4758AB; + font-style: inherit; +} + +code span.st { + color: #20794D; + font-style: inherit; +} + +code span.cf { + color: #003B4F; + font-style: inherit; +} + +code span.op { + color: #5E5E5E; + font-style: inherit; +} + +code span.er { + color: #AD0000; + font-style: inherit; +} + +code span.bn { + color: #AD0000; + font-style: inherit; +} + +code span.al { + color: #AD0000; + font-style: inherit; +} + +code span.va { + color: #111111; + font-style: inherit; +} + +code span.bu { + font-style: inherit; +} + +code span.ex { + font-style: inherit; +} + +code span.pp { + color: #AD0000; + font-style: inherit; +} + +code span.in { + color: #5E5E5E; + font-style: inherit; +} + +code span.vs { + color: #20794D; + font-style: inherit; +} + +code span.wa { + color: #5E5E5E; + font-style: italic; +} + +code span.do { + color: #5E5E5E; + font-style: italic; +} + +code span.im { + color: #00769E; + font-style: inherit; +} + +code span.ch { + color: #20794D; + font-style: inherit; +} + +code span.dt { + color: #AD0000; + font-style: inherit; +} + +code span.fl { + color: #AD0000; + font-style: inherit; +} + +code span.co { + color: #5E5E5E; + font-style: inherit; +} + +code span.cv { + color: #5E5E5E; + font-style: italic; +} + +code span.cn { + color: #8f5902; + font-style: inherit; +} + +code span.sc { + color: #5E5E5E; + font-style: inherit; +} + +code span.dv { + color: #AD0000; + font-style: inherit; +} + +code span.kw { + color: #003B4F; + font-style: inherit; +} + +.prevent-inlining { + content: " { + // Find any conflicting margin elements and add margins to the + // top to prevent overlap + const marginChildren = window.document.querySelectorAll( + ".column-margin.column-container > * " + ); + + let lastBottom = 0; + for (const marginChild of marginChildren) { + if (marginChild.offsetParent !== null) { + // clear the top margin so we recompute it + marginChild.style.marginTop = null; + const top = marginChild.getBoundingClientRect().top + window.scrollY; + console.log({ + childtop: marginChild.getBoundingClientRect().top, + scroll: window.scrollY, + top, + lastBottom, + }); + if (top < lastBottom) { + const margin = lastBottom - top; + marginChild.style.marginTop = `${margin}px`; + } + const styles = window.getComputedStyle(marginChild); + const marginTop = parseFloat(styles["marginTop"]); + + console.log({ + top, + height: marginChild.getBoundingClientRect().height, + marginTop, + total: top + marginChild.getBoundingClientRect().height + marginTop, + }); + lastBottom = top + marginChild.getBoundingClientRect().height + marginTop; + } + } +}; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Recompute the position of margin elements anytime the body size changes + if (window.ResizeObserver) { + const resizeObserver = new window.ResizeObserver( + throttle(layoutMarginEls, 50) + ); + resizeObserver.observe(window.document.body); + } + + const tocEl = window.document.querySelector('nav.toc-active[role="doc-toc"]'); + const sidebarEl = window.document.getElementById("quarto-sidebar"); + const leftTocEl = window.document.getElementById("quarto-sidebar-toc-left"); + const marginSidebarEl = window.document.getElementById( + "quarto-margin-sidebar" + ); + // function to determine whether the element has a previous sibling that is active + const prevSiblingIsActiveLink = (el) => { + const sibling = el.previousElementSibling; + if (sibling && sibling.tagName === "A") { + return sibling.classList.contains("active"); + } else { + return false; + } + }; + + // fire slideEnter for bootstrap tab activations (for htmlwidget resize behavior) + function fireSlideEnter(e) { + const event = window.document.createEvent("Event"); + event.initEvent("slideenter", true, true); + window.document.dispatchEvent(event); + } + const tabs = window.document.querySelectorAll('a[data-bs-toggle="tab"]'); + tabs.forEach((tab) => { + tab.addEventListener("shown.bs.tab", fireSlideEnter); + }); + + // fire slideEnter for tabby tab activations (for htmlwidget resize behavior) + document.addEventListener("tabby", fireSlideEnter, false); + + // Track scrolling and mark TOC links as active + // get table of contents and sidebar (bail if we don't have at least one) + const tocLinks = tocEl + ? [...tocEl.querySelectorAll("a[data-scroll-target]")] + : []; + const makeActive = (link) => tocLinks[link].classList.add("active"); + const removeActive = (link) => tocLinks[link].classList.remove("active"); + const removeAllActive = () => + [...Array(tocLinks.length).keys()].forEach((link) => removeActive(link)); + + // activate the anchor for a section associated with this TOC entry + tocLinks.forEach((link) => { + link.addEventListener("click", () => { + if (link.href.indexOf("#") !== -1) { + const anchor = link.href.split("#")[1]; + const heading = window.document.querySelector( + `[data-anchor-id=${anchor}]` + ); + if (heading) { + // Add the class + heading.classList.add("reveal-anchorjs-link"); + + // function to show the anchor + const handleMouseout = () => { + heading.classList.remove("reveal-anchorjs-link"); + heading.removeEventListener("mouseout", handleMouseout); + }; + + // add a function to clear the anchor when the user mouses out of it + heading.addEventListener("mouseout", handleMouseout); + } + } + }); + }); + + const sections = tocLinks.map((link) => { + const target = link.getAttribute("data-scroll-target"); + if (target.startsWith("#")) { + return window.document.getElementById(decodeURI(`${target.slice(1)}`)); + } else { + return window.document.querySelector(decodeURI(`${target}`)); + } + }); + + const sectionMargin = 200; + let currentActive = 0; + // track whether we've initialized state the first time + let init = false; + + const updateActiveLink = () => { + // The index from bottom to top (e.g. reversed list) + let sectionIndex = -1; + if ( + window.innerHeight + window.pageYOffset >= + window.document.body.offsetHeight + ) { + sectionIndex = 0; + } else { + sectionIndex = [...sections].reverse().findIndex((section) => { + if (section) { + return window.pageYOffset >= section.offsetTop - sectionMargin; + } else { + return false; + } + }); + } + if (sectionIndex > -1) { + const current = sections.length - sectionIndex - 1; + if (current !== currentActive) { + removeAllActive(); + currentActive = current; + makeActive(current); + if (init) { + window.dispatchEvent(sectionChanged); + } + init = true; + } + } + }; + + const inHiddenRegion = (top, bottom, hiddenRegions) => { + for (const region of hiddenRegions) { + if (top <= region.bottom && bottom >= region.top) { + return true; + } + } + return false; + }; + + const categorySelector = "header.quarto-title-block .quarto-category"; + const activateCategories = (href) => { + // Find any categories + // Surround them with a link pointing back to: + // #category=Authoring + try { + const categoryEls = window.document.querySelectorAll(categorySelector); + for (const categoryEl of categoryEls) { + const categoryText = categoryEl.textContent; + if (categoryText) { + const link = `${href}#category=${encodeURIComponent(categoryText)}`; + const linkEl = window.document.createElement("a"); + linkEl.setAttribute("href", link); + for (const child of categoryEl.childNodes) { + linkEl.append(child); + } + categoryEl.appendChild(linkEl); + } + } + } catch { + // Ignore errors + } + }; + function hasTitleCategories() { + return window.document.querySelector(categorySelector) !== null; + } + + function offsetRelativeUrl(url) { + const offset = getMeta("quarto:offset"); + return offset ? offset + url : url; + } + + function offsetAbsoluteUrl(url) { + const offset = getMeta("quarto:offset"); + const baseUrl = new URL(offset, window.location); + + const projRelativeUrl = url.replace(baseUrl, ""); + if (projRelativeUrl.startsWith("/")) { + return projRelativeUrl; + } else { + return "/" + projRelativeUrl; + } + } + + // read a meta tag value + function getMeta(metaName) { + const metas = window.document.getElementsByTagName("meta"); + for (let i = 0; i < metas.length; i++) { + if (metas[i].getAttribute("name") === metaName) { + return metas[i].getAttribute("content"); + } + } + return ""; + } + + async function findAndActivateCategories() { + const currentPagePath = offsetAbsoluteUrl(window.location.href); + const response = await fetch(offsetRelativeUrl("listings.json")); + if (response.status == 200) { + return response.json().then(function (listingPaths) { + const listingHrefs = []; + for (const listingPath of listingPaths) { + const pathWithoutLeadingSlash = listingPath.listing.substring(1); + for (const item of listingPath.items) { + if ( + item === currentPagePath || + item === currentPagePath + "index.html" + ) { + // Resolve this path against the offset to be sure + // we already are using the correct path to the listing + // (this adjusts the listing urls to be rooted against + // whatever root the page is actually running against) + const relative = offsetRelativeUrl(pathWithoutLeadingSlash); + const baseUrl = window.location; + const resolvedPath = new URL(relative, baseUrl); + listingHrefs.push(resolvedPath.pathname); + break; + } + } + } + + // Look up the tree for a nearby linting and use that if we find one + const nearestListing = findNearestParentListing( + offsetAbsoluteUrl(window.location.pathname), + listingHrefs + ); + if (nearestListing) { + activateCategories(nearestListing); + } else { + // See if the referrer is a listing page for this item + const referredRelativePath = offsetAbsoluteUrl(document.referrer); + const referrerListing = listingHrefs.find((listingHref) => { + const isListingReferrer = + listingHref === referredRelativePath || + listingHref === referredRelativePath + "index.html"; + return isListingReferrer; + }); + + if (referrerListing) { + // Try to use the referrer if possible + activateCategories(referrerListing); + } else if (listingHrefs.length > 0) { + // Otherwise, just fall back to the first listing + activateCategories(listingHrefs[0]); + } + } + }); + } + } + if (hasTitleCategories()) { + findAndActivateCategories(); + } + + const findNearestParentListing = (href, listingHrefs) => { + if (!href || !listingHrefs) { + return undefined; + } + // Look up the tree for a nearby linting and use that if we find one + const relativeParts = href.substring(1).split("/"); + while (relativeParts.length > 0) { + const path = relativeParts.join("/"); + for (const listingHref of listingHrefs) { + if (listingHref.startsWith(path)) { + return listingHref; + } + } + relativeParts.pop(); + } + + return undefined; + }; + + const manageSidebarVisiblity = (el, placeholderDescriptor) => { + let isVisible = true; + let elRect; + + return (hiddenRegions) => { + if (el === null) { + return; + } + + // Find the last element of the TOC + const lastChildEl = el.lastElementChild; + + if (lastChildEl) { + // Converts the sidebar to a menu + const convertToMenu = () => { + for (const child of el.children) { + child.style.opacity = 0; + child.style.overflow = "hidden"; + } + + nexttick(() => { + const toggleContainer = window.document.createElement("div"); + toggleContainer.style.width = "100%"; + toggleContainer.classList.add("zindex-over-content"); + toggleContainer.classList.add("quarto-sidebar-toggle"); + toggleContainer.classList.add("headroom-target"); // Marks this to be managed by headeroom + toggleContainer.id = placeholderDescriptor.id; + toggleContainer.style.position = "fixed"; + + const toggleIcon = window.document.createElement("i"); + toggleIcon.classList.add("quarto-sidebar-toggle-icon"); + toggleIcon.classList.add("bi"); + toggleIcon.classList.add("bi-caret-down-fill"); + + const toggleTitle = window.document.createElement("div"); + const titleEl = window.document.body.querySelector( + placeholderDescriptor.titleSelector + ); + if (titleEl) { + toggleTitle.append( + titleEl.textContent || titleEl.innerText, + toggleIcon + ); + } + toggleTitle.classList.add("zindex-over-content"); + toggleTitle.classList.add("quarto-sidebar-toggle-title"); + toggleContainer.append(toggleTitle); + + const toggleContents = window.document.createElement("div"); + toggleContents.classList = el.classList; + toggleContents.classList.add("zindex-over-content"); + toggleContents.classList.add("quarto-sidebar-toggle-contents"); + for (const child of el.children) { + if (child.id === "toc-title") { + continue; + } + + const clone = child.cloneNode(true); + clone.style.opacity = 1; + clone.style.display = null; + toggleContents.append(clone); + } + toggleContents.style.height = "0px"; + const positionToggle = () => { + // position the element (top left of parent, same width as parent) + if (!elRect) { + elRect = el.getBoundingClientRect(); + } + toggleContainer.style.left = `${elRect.left}px`; + toggleContainer.style.top = `${elRect.top}px`; + toggleContainer.style.width = `${elRect.width}px`; + }; + positionToggle(); + + toggleContainer.append(toggleContents); + el.parentElement.prepend(toggleContainer); + + // Process clicks + let tocShowing = false; + // Allow the caller to control whether this is dismissed + // when it is clicked (e.g. sidebar navigation supports + // opening and closing the nav tree, so don't dismiss on click) + const clickEl = placeholderDescriptor.dismissOnClick + ? toggleContainer + : toggleTitle; + + const closeToggle = () => { + if (tocShowing) { + toggleContainer.classList.remove("expanded"); + toggleContents.style.height = "0px"; + tocShowing = false; + } + }; + + // Get rid of any expanded toggle if the user scrolls + window.document.addEventListener( + "scroll", + throttle(() => { + closeToggle(); + }, 50) + ); + + // Handle positioning of the toggle + window.addEventListener( + "resize", + throttle(() => { + elRect = undefined; + positionToggle(); + }, 50) + ); + + window.addEventListener("quarto-hrChanged", () => { + elRect = undefined; + }); + + // Process the click + clickEl.onclick = () => { + if (!tocShowing) { + toggleContainer.classList.add("expanded"); + toggleContents.style.height = null; + tocShowing = true; + } else { + closeToggle(); + } + }; + }); + }; + + // Converts a sidebar from a menu back to a sidebar + const convertToSidebar = () => { + for (const child of el.children) { + child.style.opacity = 1; + child.style.overflow = null; + } + + const placeholderEl = window.document.getElementById( + placeholderDescriptor.id + ); + if (placeholderEl) { + placeholderEl.remove(); + } + + el.classList.remove("rollup"); + }; + + if (isReaderMode()) { + convertToMenu(); + isVisible = false; + } else { + // Find the top and bottom o the element that is being managed + const elTop = el.offsetTop; + const elBottom = + elTop + lastChildEl.offsetTop + lastChildEl.offsetHeight; + + if (!isVisible) { + // If the element is current not visible reveal if there are + // no conflicts with overlay regions + if (!inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToSidebar(); + isVisible = true; + } + } else { + // If the element is visible, hide it if it conflicts with overlay regions + // and insert a placeholder toggle (or if we're in reader mode) + if (inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToMenu(); + isVisible = false; + } + } + } + } + }; + }; + + const tabEls = document.querySelectorAll('a[data-bs-toggle="tab"]'); + for (const tabEl of tabEls) { + const id = tabEl.getAttribute("data-bs-target"); + if (id) { + const columnEl = document.querySelector( + `${id} .column-margin, .tabset-margin-content` + ); + if (columnEl) + tabEl.addEventListener("shown.bs.tab", function (event) { + const el = event.srcElement; + if (el) { + const visibleCls = `${el.id}-margin-content`; + // walk up until we find a parent tabset + let panelTabsetEl = el.parentElement; + while (panelTabsetEl) { + if (panelTabsetEl.classList.contains("panel-tabset")) { + break; + } + panelTabsetEl = panelTabsetEl.parentElement; + } + + if (panelTabsetEl) { + const prevSib = panelTabsetEl.previousElementSibling; + if ( + prevSib && + prevSib.classList.contains("tabset-margin-container") + ) { + const childNodes = prevSib.querySelectorAll( + ".tabset-margin-content" + ); + for (const childEl of childNodes) { + if (childEl.classList.contains(visibleCls)) { + childEl.classList.remove("collapse"); + } else { + childEl.classList.add("collapse"); + } + } + } + } + } + + layoutMarginEls(); + }); + } + } + + // Manage the visibility of the toc and the sidebar + const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, { + id: "quarto-toc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + const sidebarScrollVisiblity = manageSidebarVisiblity(sidebarEl, { + id: "quarto-sidebarnav-toggle", + titleSelector: ".title", + dismissOnClick: false, + }); + let tocLeftScrollVisibility; + if (leftTocEl) { + tocLeftScrollVisibility = manageSidebarVisiblity(leftTocEl, { + id: "quarto-lefttoc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + } + + // Find the first element that uses formatting in special columns + const conflictingEls = window.document.body.querySelectorAll( + '[class^="column-"], [class*=" column-"], aside, [class*="margin-caption"], [class*=" margin-caption"], [class*="margin-ref"], [class*=" margin-ref"]' + ); + + // Filter all the possibly conflicting elements into ones + // the do conflict on the left or ride side + const arrConflictingEls = Array.from(conflictingEls); + const leftSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return false; + } + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + className.startsWith("column-") && + !className.endsWith("right") && + !className.endsWith("container") && + className !== "column-margin" + ); + }); + }); + const rightSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return true; + } + + const hasMarginCaption = Array.from(el.classList).find((className) => { + return className == "margin-caption"; + }); + if (hasMarginCaption) { + return true; + } + + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + !className.endsWith("container") && + className.startsWith("column-") && + !className.endsWith("left") + ); + }); + }); + + const kOverlapPaddingSize = 10; + function toRegions(els) { + return els.map((el) => { + const boundRect = el.getBoundingClientRect(); + const top = + boundRect.top + + document.documentElement.scrollTop - + kOverlapPaddingSize; + return { + top, + bottom: top + el.scrollHeight + 2 * kOverlapPaddingSize, + }; + }); + } + + let hasObserved = false; + const visibleItemObserver = (els) => { + let visibleElements = [...els]; + const intersectionObserver = new IntersectionObserver( + (entries, _observer) => { + entries.forEach((entry) => { + if (entry.isIntersecting) { + if (visibleElements.indexOf(entry.target) === -1) { + visibleElements.push(entry.target); + } + } else { + visibleElements = visibleElements.filter((visibleEntry) => { + return visibleEntry !== entry; + }); + } + }); + + if (!hasObserved) { + hideOverlappedSidebars(); + } + hasObserved = true; + }, + {} + ); + els.forEach((el) => { + intersectionObserver.observe(el); + }); + + return { + getVisibleEntries: () => { + return visibleElements; + }, + }; + }; + + const rightElementObserver = visibleItemObserver(rightSideConflictEls); + const leftElementObserver = visibleItemObserver(leftSideConflictEls); + + const hideOverlappedSidebars = () => { + marginScrollVisibility(toRegions(rightElementObserver.getVisibleEntries())); + sidebarScrollVisiblity(toRegions(leftElementObserver.getVisibleEntries())); + if (tocLeftScrollVisibility) { + tocLeftScrollVisibility( + toRegions(leftElementObserver.getVisibleEntries()) + ); + } + }; + + window.quartoToggleReader = () => { + // Applies a slow class (or removes it) + // to update the transition speed + const slowTransition = (slow) => { + const manageTransition = (id, slow) => { + const el = document.getElementById(id); + if (el) { + if (slow) { + el.classList.add("slow"); + } else { + el.classList.remove("slow"); + } + } + }; + + manageTransition("TOC", slow); + manageTransition("quarto-sidebar", slow); + }; + const readerMode = !isReaderMode(); + setReaderModeValue(readerMode); + + // If we're entering reader mode, slow the transition + if (readerMode) { + slowTransition(readerMode); + } + highlightReaderToggle(readerMode); + hideOverlappedSidebars(); + + // If we're exiting reader mode, restore the non-slow transition + if (!readerMode) { + slowTransition(!readerMode); + } + }; + + const highlightReaderToggle = (readerMode) => { + const els = document.querySelectorAll(".quarto-reader-toggle"); + if (els) { + els.forEach((el) => { + if (readerMode) { + el.classList.add("reader"); + } else { + el.classList.remove("reader"); + } + }); + } + }; + + const setReaderModeValue = (val) => { + if (window.location.protocol !== "file:") { + window.localStorage.setItem("quarto-reader-mode", val); + } else { + localReaderMode = val; + } + }; + + const isReaderMode = () => { + if (window.location.protocol !== "file:") { + return window.localStorage.getItem("quarto-reader-mode") === "true"; + } else { + return localReaderMode; + } + }; + let localReaderMode = null; + + const tocOpenDepthStr = tocEl?.getAttribute("data-toc-expanded"); + const tocOpenDepth = tocOpenDepthStr ? Number(tocOpenDepthStr) : 1; + + // Walk the TOC and collapse/expand nodes + // Nodes are expanded if: + // - they are top level + // - they have children that are 'active' links + // - they are directly below an link that is 'active' + const walk = (el, depth) => { + // Tick depth when we enter a UL + if (el.tagName === "UL") { + depth = depth + 1; + } + + // It this is active link + let isActiveNode = false; + if (el.tagName === "A" && el.classList.contains("active")) { + isActiveNode = true; + } + + // See if there is an active child to this element + let hasActiveChild = false; + for (child of el.children) { + hasActiveChild = walk(child, depth) || hasActiveChild; + } + + // Process the collapse state if this is an UL + if (el.tagName === "UL") { + if (tocOpenDepth === -1 && depth > 1) { + el.classList.add("collapse"); + } else if ( + depth <= tocOpenDepth || + hasActiveChild || + prevSiblingIsActiveLink(el) + ) { + el.classList.remove("collapse"); + } else { + el.classList.add("collapse"); + } + + // untick depth when we leave a UL + depth = depth - 1; + } + return hasActiveChild || isActiveNode; + }; + + // walk the TOC and expand / collapse any items that should be shown + + if (tocEl) { + walk(tocEl, 0); + updateActiveLink(); + } + + // Throttle the scroll event and walk peridiocally + window.document.addEventListener( + "scroll", + throttle(() => { + if (tocEl) { + updateActiveLink(); + walk(tocEl, 0); + } + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 5) + ); + window.addEventListener( + "resize", + throttle(() => { + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 10) + ); + hideOverlappedSidebars(); + highlightReaderToggle(isReaderMode()); +}); + +// grouped tabsets +window.addEventListener("pageshow", (_event) => { + function getTabSettings() { + const data = localStorage.getItem("quarto-persistent-tabsets-data"); + if (!data) { + localStorage.setItem("quarto-persistent-tabsets-data", "{}"); + return {}; + } + if (data) { + return JSON.parse(data); + } + } + + function setTabSettings(data) { + localStorage.setItem( + "quarto-persistent-tabsets-data", + JSON.stringify(data) + ); + } + + function setTabState(groupName, groupValue) { + const data = getTabSettings(); + data[groupName] = groupValue; + setTabSettings(data); + } + + function toggleTab(tab, active) { + const tabPanelId = tab.getAttribute("aria-controls"); + const tabPanel = document.getElementById(tabPanelId); + if (active) { + tab.classList.add("active"); + tabPanel.classList.add("active"); + } else { + tab.classList.remove("active"); + tabPanel.classList.remove("active"); + } + } + + function toggleAll(selectedGroup, selectorsToSync) { + for (const [thisGroup, tabs] of Object.entries(selectorsToSync)) { + const active = selectedGroup === thisGroup; + for (const tab of tabs) { + toggleTab(tab, active); + } + } + } + + function findSelectorsToSyncByLanguage() { + const result = {}; + const tabs = Array.from( + document.querySelectorAll(`div[data-group] a[id^='tabset-']`) + ); + for (const item of tabs) { + const div = item.parentElement.parentElement.parentElement; + const group = div.getAttribute("data-group"); + if (!result[group]) { + result[group] = {}; + } + const selectorsToSync = result[group]; + const value = item.innerHTML; + if (!selectorsToSync[value]) { + selectorsToSync[value] = []; + } + selectorsToSync[value].push(item); + } + return result; + } + + function setupSelectorSync() { + const selectorsToSync = findSelectorsToSyncByLanguage(); + Object.entries(selectorsToSync).forEach(([group, tabSetsByValue]) => { + Object.entries(tabSetsByValue).forEach(([value, items]) => { + items.forEach((item) => { + item.addEventListener("click", (_event) => { + setTabState(group, value); + toggleAll(value, selectorsToSync[group]); + }); + }); + }); + }); + return selectorsToSync; + } + + const selectorsToSync = setupSelectorSync(); + for (const [group, selectedName] of 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    ",valueNames:["page","dotted"],searchClass:"pagination-search-that-is-not-supposed-to-exist",sortClass:"pagination-sort-that-is-not-supposed-to-exist"});s.bind(n.listContainer,"click",(function(e){var r=e.target||e.srcElement,n=t.utils.getAttribute(r,"data-page"),s=t.utils.getAttribute(r,"data-i");s&&t.show((s-1)*n+1,n)})),t.on("updated",(function(){r(n,e)})),r(n,e)}}},"./src/parse.js":function(t,e,r){t.exports=function(t){var e=r("./src/item.js")(t),n=function(r,n){for(var s=0,i=r.length;s0?setTimeout((function(){e(r,s)}),1):(t.update(),t.trigger("parseComplete"))};return t.handlers.parseComplete=t.handlers.parseComplete||[],function(){var e=function(t){for(var e=t.childNodes,r=[],n=0,s=e.length;n]/g.exec(t)){var e=document.createElement("tbody");return e.innerHTML=t,e.firstElementChild}if(-1!==t.indexOf("<")){var r=document.createElement("div");return r.innerHTML=t,r.firstElementChild}}},a=function(e,r,n){var s=void 0,i=function(e){for(var r=0,n=t.valueNames.length;r=1;)t.list.removeChild(t.list.firstChild)},function(){var r;if("function"!=typeof t.item){if(!(r="string"==typeof t.item?-1===t.item.indexOf("<")?document.getElementById(t.item):i(t.item):s()))throw new Error("The list needs to have at least one item on init otherwise you'll have to add a template.");r=n(r,t.valueNames),e=function(){return r.cloneNode(!0)}}else e=function(e){var r=t.item(e);return i(r)}}()};t.exports=function(t){return new e(t)}},"./src/utils/classes.js":function(t,e,r){var n=r("./src/utils/index-of.js"),s=/\s+/;Object.prototype.toString;function i(t){if(!t||!t.nodeType)throw new Error("A DOM element reference is required");this.el=t,this.list=t.classList}t.exports=function(t){return new i(t)},i.prototype.add=function(t){if(this.list)return this.list.add(t),this;var e=this.array();return~n(e,t)||e.push(t),this.el.className=e.join(" "),this},i.prototype.remove=function(t){if(this.list)return this.list.remove(t),this;var e=this.array(),r=n(e,t);return~r&&e.splice(r,1),this.el.className=e.join(" "),this},i.prototype.toggle=function(t,e){return this.list?(void 0!==e?e!==this.list.toggle(t,e)&&this.list.toggle(t):this.list.toggle(t),this):(void 0!==e?e?this.add(t):this.remove(t):this.has(t)?this.remove(t):this.add(t),this)},i.prototype.array=function(){var t=(this.el.getAttribute("class")||"").replace(/^\s+|\s+$/g,"").split(s);return""===t[0]&&t.shift(),t},i.prototype.has=i.prototype.contains=function(t){return this.list?this.list.contains(t):!!~n(this.array(),t)}},"./src/utils/events.js":function(t,e,r){var n=window.addEventListener?"addEventListener":"attachEvent",s=window.removeEventListener?"removeEventListener":"detachEvent",i="addEventListener"!==n?"on":"",a=r("./src/utils/to-array.js");e.bind=function(t,e,r,s){for(var o=0,l=(t=a(t)).length;o32)return!1;var a=n,o=function(){var t,r={};for(t=0;t=p;b--){var j=o[t.charAt(b-1)];if(C[b]=0===m?(C[b+1]<<1|1)&j:(C[b+1]<<1|1)&j|(v[b+1]|v[b])<<1|1|v[b+1],C[b]&d){var x=l(m,b-1);if(x<=u){if(u=x,!((c=b-1)>a))break;p=Math.max(1,2*a-c)}}}if(l(m+1,a)>u)break;v=C}return!(c<0)}},"./src/utils/get-attribute.js":function(t){t.exports=function(t,e){var r=t.getAttribute&&t.getAttribute(e)||null;if(!r)for(var n=t.attributes,s=n.length,i=0;i=48&&t<=57}function i(t,e){for(var i=(t+="").length,a=(e+="").length,o=0,l=0;o=i&&l=a?-1:l>=a&&o=i?1:i-a}i.caseInsensitive=i.i=function(t,e){return i((""+t).toLowerCase(),(""+e).toLowerCase())},Object.defineProperties(i,{alphabet:{get:function(){return e},set:function(t){r=[];var s=0;if(e=t)for(;s { + if (categoriesLoaded) { + activateCategory(category); + setCategoryHash(category); + } +}; + +window["quarto-listing-loaded"] = () => { + // Process any existing hash + const hash = getHash(); + + if (hash) { + // If there is a category, switch to that + if (hash.category) { + activateCategory(hash.category); + } + // Paginate a specific listing + const listingIds = Object.keys(window["quarto-listings"]); + for (const listingId of listingIds) { + const page = hash[getListingPageKey(listingId)]; + if (page) { + showPage(listingId, page); + } + } + } + + const listingIds = Object.keys(window["quarto-listings"]); + for (const listingId of listingIds) { + // The actual list + const list = window["quarto-listings"][listingId]; + + // Update the handlers for pagination events + refreshPaginationHandlers(listingId); + + // Render any visible items that need it + renderVisibleProgressiveImages(list); + + // Whenever the list is updated, we also need to + // attach handlers to the new pagination elements + // and refresh any newly visible items. + list.on("updated", function () { + renderVisibleProgressiveImages(list); + setTimeout(() => refreshPaginationHandlers(listingId)); + + // Show or hide the no matching message + toggleNoMatchingMessage(list); + }); + } +}; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Attach click handlers to categories + const categoryEls = window.document.querySelectorAll( + ".quarto-listing-category .category" + ); + + for (const categoryEl of categoryEls) { + const category = categoryEl.getAttribute("data-category"); + categoryEl.onclick = () => { + activateCategory(category); + setCategoryHash(category); + }; + } + + // Attach a click handler to the category title + // (there should be only one, but since it is a class name, handle N) + const categoryTitleEls = window.document.querySelectorAll( + ".quarto-listing-category-title" + ); + for (const categoryTitleEl of categoryTitleEls) { + categoryTitleEl.onclick = () => { + activateCategory(""); + setCategoryHash(""); + }; + } + + categoriesLoaded = true; +}); + +function toggleNoMatchingMessage(list) { + const selector = `#${list.listContainer.id} .listing-no-matching`; + const noMatchingEl = window.document.querySelector(selector); + if (noMatchingEl) { + if (list.visibleItems.length === 0) { + noMatchingEl.classList.remove("d-none"); + } else { + if (!noMatchingEl.classList.contains("d-none")) { + noMatchingEl.classList.add("d-none"); + } + } + } +} + +function setCategoryHash(category) { + setHash({ category }); +} + +function setPageHash(listingId, page) { + const currentHash = getHash() || {}; + currentHash[getListingPageKey(listingId)] = page; + setHash(currentHash); +} + +function getListingPageKey(listingId) { + return `${listingId}-page`; +} + +function refreshPaginationHandlers(listingId) { + const listingEl = window.document.getElementById(listingId); + const paginationEls = listingEl.querySelectorAll( + ".pagination li.page-item:not(.disabled) .page.page-link" + ); + for (const paginationEl of paginationEls) { + paginationEl.onclick = (sender) => { + setPageHash(listingId, sender.target.getAttribute("data-i")); + showPage(listingId, sender.target.getAttribute("data-i")); + return false; + }; + } +} + +function renderVisibleProgressiveImages(list) { + // Run through the visible items and render any progressive images + for (const item of list.visibleItems) { + const itemEl = item.elm; + if (itemEl) { + const progressiveImgs = itemEl.querySelectorAll( + `img[${kProgressiveAttr}]` + ); + for (const progressiveImg of progressiveImgs) { + const srcValue = progressiveImg.getAttribute(kProgressiveAttr); + if (srcValue) { + progressiveImg.setAttribute("src", srcValue); + } + progressiveImg.removeAttribute(kProgressiveAttr); + } + } + } +} + +function getHash() { + // Hashes are of the form + // #name:value|name1:value1|name2:value2 + const currentUrl = new URL(window.location); + const hashRaw = currentUrl.hash ? currentUrl.hash.slice(1) : undefined; + return parseHash(hashRaw); +} + +const kAnd = "&"; +const kEquals = "="; + +function parseHash(hash) { + if (!hash) { + return undefined; + } + const hasValuesStrs = hash.split(kAnd); + const hashValues = hasValuesStrs + .map((hashValueStr) => { + const vals = hashValueStr.split(kEquals); + if (vals.length === 2) { + return { name: vals[0], value: vals[1] }; + } else { + return undefined; + } + }) + .filter((value) => { + return value !== undefined; + }); + + const hashObj = {}; + hashValues.forEach((hashValue) => { + hashObj[hashValue.name] = decodeURIComponent(hashValue.value); + }); + return hashObj; +} + +function makeHash(obj) { + return Object.keys(obj) + .map((key) => { + return `${key}${kEquals}${obj[key]}`; + }) + .join(kAnd); +} + +function setHash(obj) { + const hash = makeHash(obj); + window.history.pushState(null, null, `#${hash}`); +} + +function showPage(listingId, page) { + const list = window["quarto-listings"][listingId]; + if (list) { + list.show((page - 1) * list.page + 1, list.page); + } +} + +function activateCategory(category) { + // Deactivate existing categories + const activeEls = window.document.querySelectorAll( + ".quarto-listing-category .category.active" + ); + for (const activeEl of activeEls) { + activeEl.classList.remove("active"); + } + + // Activate this category + const categoryEl = window.document.querySelector( + `.quarto-listing-category .category[data-category='${category}'` + ); + if (categoryEl) { + categoryEl.classList.add("active"); + } + + // Filter the listings to this category + filterListingCategory(category); +} + +function filterListingCategory(category) { + const listingIds = Object.keys(window["quarto-listings"]); + for (const listingId of listingIds) { + const list = window["quarto-listings"][listingId]; + if (list) { + if (category === "") { + // resets the filter + list.filter(); + } else { + // filter to this category + list.filter(function (item) { + const itemValues = item.values(); + if (itemValues.categories !== null) { + const categories = itemValues.categories.split(","); + return categories.includes(category); + } else { + return false; + } + }); + } + } + } +} diff --git a/site_libs/quarto-nav/headroom.min.js b/site_libs/quarto-nav/headroom.min.js new file mode 100644 index 0000000..b08f1df --- /dev/null +++ b/site_libs/quarto-nav/headroom.min.js @@ -0,0 +1,7 @@ +/*! + * headroom.js v0.12.0 - Give your page some headroom. Hide your header until you need it + * Copyright (c) 2020 Nick Williams - http://wicky.nillia.ms/headroom.js + * License: MIT + */ + +!function(t,n){"object"==typeof exports&&"undefined"!=typeof module?module.exports=n():"function"==typeof define&&define.amd?define(n):(t=t||self).Headroom=n()}(this,function(){"use strict";function t(){return"undefined"!=typeof window}function d(t){return function(t){return t&&t.document&&function(t){return 9===t.nodeType}(t.document)}(t)?function(t){var n=t.document,o=n.body,s=n.documentElement;return{scrollHeight:function(){return Math.max(o.scrollHeight,s.scrollHeight,o.offsetHeight,s.offsetHeight,o.clientHeight,s.clientHeight)},height:function(){return t.innerHeight||s.clientHeight||o.clientHeight},scrollY:function(){return void 0!==t.pageYOffset?t.pageYOffset:(s||o.parentNode||o).scrollTop}}}(t):function(t){return{scrollHeight:function(){return Math.max(t.scrollHeight,t.offsetHeight,t.clientHeight)},height:function(){return 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s.cutsTheMustard&&!this.initialised&&(this.addClass("initial"),this.initialised=!0,setTimeout(function(t){t.scrollTracker=n(t.scroller,{offset:t.offset,tolerance:t.tolerance},t.update.bind(t))},100,this)),this},destroy:function(){this.initialised=!1,Object.keys(this.classes).forEach(this.removeClass,this),this.scrollTracker.destroy()},unpin:function(){!this.hasClass("pinned")&&this.hasClass("unpinned")||(this.addClass("unpinned"),this.removeClass("pinned"),this.onUnpin&&this.onUnpin.call(this))},pin:function(){this.hasClass("unpinned")&&(this.addClass("pinned"),this.removeClass("unpinned"),this.onPin&&this.onPin.call(this))},freeze:function(){this.frozen=!0,this.addClass("frozen")},unfreeze:function(){this.frozen=!1,this.removeClass("frozen")},top:function(){this.hasClass("top")||(this.addClass("top"),this.removeClass("notTop"),this.onTop&&this.onTop.call(this))},notTop:function(){this.hasClass("notTop")||(this.addClass("notTop"),this.removeClass("top"),this.onNotTop&&this.onNotTop.call(this))},bottom:function(){this.hasClass("bottom")||(this.addClass("bottom"),this.removeClass("notBottom"),this.onBottom&&this.onBottom.call(this))},notBottom:function(){this.hasClass("notBottom")||(this.addClass("notBottom"),this.removeClass("bottom"),this.onNotBottom&&this.onNotBottom.call(this))},shouldUnpin:function(t){return"down"===t.direction&&!t.top&&t.toleranceExceeded},shouldPin:function(t){return"up"===t.direction&&t.toleranceExceeded||t.top},addClass:function(t){this.elem.classList.add.apply(this.elem.classList,this.classes[t].split(" 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b/site_libs/quarto-nav/quarto-nav.js @@ -0,0 +1,277 @@ +const headroomChanged = new CustomEvent("quarto-hrChanged", { + detail: {}, + bubbles: true, + cancelable: false, + composed: false, +}); + +window.document.addEventListener("DOMContentLoaded", function () { + let init = false; + + // Manage the back to top button, if one is present. + let lastScrollTop = window.pageYOffset || document.documentElement.scrollTop; + const scrollDownBuffer = 5; + const scrollUpBuffer = 35; + const btn = document.getElementById("quarto-back-to-top"); + const hideBackToTop = () => { + btn.style.display = "none"; + }; + const showBackToTop = () => { + btn.style.display = "inline-block"; + }; + if (btn) { + window.document.addEventListener( + "scroll", + function () { + const currentScrollTop = + window.pageYOffset || document.documentElement.scrollTop; + + // Shows and hides the button 'intelligently' as the user scrolls + if (currentScrollTop - scrollDownBuffer > lastScrollTop) { + hideBackToTop(); + lastScrollTop = currentScrollTop <= 0 ? 0 : currentScrollTop; + } else if (currentScrollTop < lastScrollTop - scrollUpBuffer) { + showBackToTop(); + lastScrollTop = currentScrollTop <= 0 ? 0 : currentScrollTop; + } + + // Show the button at the bottom, hides it at the top + if (currentScrollTop <= 0) { + hideBackToTop(); + } else if ( + window.innerHeight + currentScrollTop >= + document.body.offsetHeight + ) { + showBackToTop(); + } + }, + false + ); + } + + function throttle(func, wait) { + var timeout; + return function () { + const context = this; + const args = arguments; + const later = function () { + clearTimeout(timeout); + timeout = null; + func.apply(context, args); + }; + + if (!timeout) { + timeout = setTimeout(later, wait); + } + }; + } + + function headerOffset() { + // Set an offset if there is are fixed top navbar + const headerEl = window.document.querySelector("header.fixed-top"); + if (headerEl) { + return headerEl.clientHeight; + } else { + return 0; + } + } + + function footerOffset() { + const footerEl = window.document.querySelector("footer.footer"); + if (footerEl) { + return footerEl.clientHeight; + } else { + return 0; + } + } + + function updateDocumentOffsetWithoutAnimation() { + updateDocumentOffset(false); + } + + function updateDocumentOffset(animated) { + // set body offset + const topOffset = headerOffset(); + const bodyOffset = topOffset + footerOffset(); + const bodyEl = window.document.body; + bodyEl.setAttribute("data-bs-offset", topOffset); + bodyEl.style.paddingTop = topOffset + "px"; + + // deal with sidebar offsets + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + if (!animated) { + sidebar.classList.add("notransition"); + // Remove the no transition class after the animation has time to complete + setTimeout(function () { + sidebar.classList.remove("notransition"); + }, 201); + } + + if (window.Headroom && sidebar.classList.contains("sidebar-unpinned")) { + sidebar.style.top = "0"; + sidebar.style.maxHeight = "100vh"; + } else { + sidebar.style.top = topOffset + "px"; + sidebar.style.maxHeight = "calc(100vh - " + topOffset + "px)"; + } + }); + + // allow space for footer + const mainContainer = window.document.querySelector(".quarto-container"); + if (mainContainer) { + mainContainer.style.minHeight = "calc(100vh - " + bodyOffset + "px)"; + } + + // link offset + let linkStyle = window.document.querySelector("#quarto-target-style"); + if (!linkStyle) { + linkStyle = window.document.createElement("style"); + linkStyle.setAttribute("id", "quarto-target-style"); + window.document.head.appendChild(linkStyle); + } + while (linkStyle.firstChild) { + linkStyle.removeChild(linkStyle.firstChild); + } + if (topOffset > 0) { + linkStyle.appendChild( + window.document.createTextNode(` + section:target::before { + content: ""; + display: block; + height: ${topOffset}px; + margin: -${topOffset}px 0 0; + }`) + ); + } + if (init) { + window.dispatchEvent(headroomChanged); + } + init = true; + } + + // initialize headroom + var header = window.document.querySelector("#quarto-header"); + if (header && window.Headroom) { + const headroom = new window.Headroom(header, { + tolerance: 5, + onPin: function () { + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + sidebar.classList.remove("sidebar-unpinned"); + }); + updateDocumentOffset(); + }, + onUnpin: function () { + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + sidebar.classList.add("sidebar-unpinned"); + }); + updateDocumentOffset(); + }, + }); + headroom.init(); + + let frozen = false; + window.quartoToggleHeadroom = function () { + if (frozen) { + headroom.unfreeze(); + frozen = false; + } else { + headroom.freeze(); + frozen = true; + } + }; + } + + window.addEventListener( + "hashchange", + function (e) { + if ( + getComputedStyle(document.documentElement).scrollBehavior !== "smooth" + ) { + window.scrollTo(0, window.pageYOffset - headerOffset()); + } + }, + false + ); + + // Observe size changed for the header + const headerEl = window.document.querySelector("header.fixed-top"); + if (headerEl && window.ResizeObserver) { + const observer = new window.ResizeObserver( + updateDocumentOffsetWithoutAnimation + ); + observer.observe(headerEl, { + attributes: true, + childList: true, + characterData: true, + }); + } else { + window.addEventListener( + "resize", + throttle(updateDocumentOffsetWithoutAnimation, 50) + ); + } + setTimeout(updateDocumentOffsetWithoutAnimation, 250); + + // fixup index.html links if we aren't on the filesystem + if (window.location.protocol !== "file:") { + const links = window.document.querySelectorAll("a"); + for (let i = 0; i < links.length; i++) { + if (links[i].href) { + links[i].href = links[i].href.replace(/\/index\.html/, "/"); + } + } + + // Fixup any sharing links that require urls + // Append url to any sharing urls + const sharingLinks = window.document.querySelectorAll( + "a.sidebar-tools-main-item" + ); + for (let i = 0; i < sharingLinks.length; i++) { + const sharingLink = sharingLinks[i]; + const href = sharingLink.getAttribute("href"); + if (href) { + sharingLink.setAttribute( + "href", + href.replace("|url|", window.location.href) + ); + } + } + + // Scroll the active navigation item into view, if necessary + const navSidebar = window.document.querySelector("nav#quarto-sidebar"); + if (navSidebar) { + // Find the active item + const activeItem = navSidebar.querySelector("li.sidebar-item a.active"); + if (activeItem) { + // Wait for the scroll height and height to resolve by observing size changes on the + // nav element that is scrollable + const resizeObserver = new ResizeObserver((_entries) => { + // The bottom of the element + const elBottom = activeItem.offsetTop; + const viewBottom = navSidebar.scrollTop + navSidebar.clientHeight; + + // The element height and scroll height are the same, then we are still loading + if (viewBottom !== navSidebar.scrollHeight) { + // Determine if the item isn't visible and scroll to it + if (elBottom >= viewBottom) { + navSidebar.scrollTop = elBottom; + } + + // stop observing now since we've completed the scroll + resizeObserver.unobserve(navSidebar); + } + }); + resizeObserver.observe(navSidebar); + } + } + } +}); diff --git a/site_libs/quarto-search/autocomplete.umd.js b/site_libs/quarto-search/autocomplete.umd.js new file mode 100644 index 0000000..619c57c --- /dev/null +++ b/site_libs/quarto-search/autocomplete.umd.js @@ -0,0 +1,3 @@ +/*! @algolia/autocomplete-js 1.7.3 | MIT License | © Algolia, Inc. and contributors | https://github.com/algolia/autocomplete */ +!function(e,t){"object"==typeof exports&&"undefined"!=typeof module?t(exports):"function"==typeof define&&define.amd?define(["exports"],t):t((e="undefined"!=typeof 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0===r?I.includeMatches:r,o=n.includeScore,c=void 0===o?I.includeScore:o,a=[];return i&&a.push(de),c&&a.push(ve),e.map((function(e){var n=e.idx,r={item:t[n],refIndex:n};return a.length&&a.forEach((function(t){t(e,r)})),r}))}var ye=function(){function e(n){var i=arguments.length>1&&void 0!==arguments[1]?arguments[1]:{},o=arguments.length>2?arguments[2]:void 0;r(this,e),this.options=t(t({},I),i),this.options.useExtendedSearch,this._keyStore=new S(this.options.keys),this.setCollection(n,o)}return o(e,[{key:"setCollection",value:function(e,t){if(this._docs=e,t&&!(t instanceof $))throw new Error("Incorrect 'index' type");this._myIndex=t||F(this.options.keys,this._docs,{getFn:this.options.getFn,fieldNormWeight:this.options.fieldNormWeight})}},{key:"add",value:function(e){k(e)&&(this._docs.push(e),this._myIndex.add(e))}},{key:"remove",value:function(){for(var e=arguments.length>0&&void 0!==arguments[0]?arguments[0]:function(){return!1},t=[],n=0,r=this._docs.length;n1&&void 0!==arguments[1]?arguments[1]:{},n=t.limit,r=void 0===n?-1:n,i=this.options,o=i.includeMatches,c=i.includeScore,a=i.shouldSort,s=i.sortFn,u=i.ignoreFieldNorm,h=g(e)?g(this._docs[0])?this._searchStringList(e):this._searchObjectList(e):this._searchLogical(e);return fe(h,{ignoreFieldNorm:u}),a&&h.sort(s),y(r)&&r>-1&&(h=h.slice(0,r)),ge(h,this._docs,{includeMatches:o,includeScore:c})}},{key:"_searchStringList",value:function(e){var t=re(e,this.options),n=this._myIndex.records,r=[];return n.forEach((function(e){var n=e.v,i=e.i,o=e.n;if(k(n)){var c=t.searchIn(n),a=c.isMatch,s=c.score,u=c.indices;a&&r.push({item:n,idx:i,matches:[{score:s,value:n,norm:o,indices:u}]})}})),r}},{key:"_searchLogical",value:function(e){var t=this,n=function(e,t){var n=(arguments.length>2&&void 0!==arguments[2]?arguments[2]:{}).auto,r=void 0===n||n,i=function e(n){var i=Object.keys(n),o=ue(n);if(!o&&i.length>1&&!se(n))return e(le(n));if(he(n)){var c=o?n[ce]:i[0],a=o?n[ae]:n[c];if(!g(a))throw new Error(x(c));var s={keyId:j(c),pattern:a};return r&&(s.searcher=re(a,t)),s}var u={children:[],operator:i[0]};return i.forEach((function(t){var r=n[t];v(r)&&r.forEach((function(t){u.children.push(e(t))}))})),u};return se(e)||(e=le(e)),i(e)}(e,this.options),r=function e(n,r,i){if(!n.children){var o=n.keyId,c=n.searcher,a=t._findMatches({key:t._keyStore.get(o),value:t._myIndex.getValueForItemAtKeyId(r,o),searcher:c});return a&&a.length?[{idx:i,item:r,matches:a}]:[]}for(var s=[],u=0,h=n.children.length;u1&&void 0!==arguments[1]?arguments[1]:{},n=t.getFn,r=void 0===n?I.getFn:n,i=t.fieldNormWeight,o=void 0===i?I.fieldNormWeight:i,c=e.keys,a=e.records,s=new $({getFn:r,fieldNormWeight:o});return s.setKeys(c),s.setIndexRecords(a),s},ye.config=I,function(){ne.push.apply(ne,arguments)}(te),ye},"object"==typeof exports&&"undefined"!=typeof module?module.exports=t():"function"==typeof define&&define.amd?define(t):(e="undefined"!=typeof globalThis?globalThis:e||self).Fuse=t(); \ No newline at end of file diff --git a/site_libs/quarto-search/quarto-search.js b/site_libs/quarto-search/quarto-search.js new file mode 100644 index 0000000..f5d852d --- /dev/null +++ b/site_libs/quarto-search/quarto-search.js @@ -0,0 +1,1140 @@ +const kQueryArg = "q"; +const kResultsArg = "show-results"; + +// If items don't provide a URL, then both the navigator and the onSelect +// function aren't called (and therefore, the default implementation is used) +// +// We're using this sentinel URL to signal to those handlers that this +// item is a more item (along with the type) and can be handled appropriately +const kItemTypeMoreHref = "0767FDFD-0422-4E5A-BC8A-3BE11E5BBA05"; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Ensure that search is available on this page. If it isn't, + // should return early and not do anything + var searchEl = window.document.getElementById("quarto-search"); + if (!searchEl) return; + + const { autocomplete } = window["@algolia/autocomplete-js"]; + + let quartoSearchOptions = {}; + let language = {}; + const searchOptionEl = window.document.getElementById( + "quarto-search-options" + ); + if (searchOptionEl) { + const jsonStr = searchOptionEl.textContent; + quartoSearchOptions = JSON.parse(jsonStr); + language = quartoSearchOptions.language; + } + + // note the search mode + if (quartoSearchOptions.type === "overlay") { + searchEl.classList.add("type-overlay"); + } else { + searchEl.classList.add("type-textbox"); + } + + // Used to determine highlighting behavior for this page + // A `q` query param is expected when the user follows a search + // to this page + const currentUrl = new URL(window.location); + const query = currentUrl.searchParams.get(kQueryArg); + const showSearchResults = currentUrl.searchParams.get(kResultsArg); + const mainEl = window.document.querySelector("main"); + + // highlight matches on the page + if (query !== null && mainEl) { + // perform any highlighting + highlight(escapeRegExp(query), mainEl); + + // fix up the URL to remove the q query param + const replacementUrl = new URL(window.location); + replacementUrl.searchParams.delete(kQueryArg); + window.history.replaceState({}, "", replacementUrl); + } + + // function to clear highlighting on the page when the search query changes + // (e.g. if the user edits the query or clears it) + let highlighting = true; + const resetHighlighting = (searchTerm) => { + if (mainEl && highlighting && query !== null && searchTerm !== query) { + clearHighlight(query, mainEl); + highlighting = false; + } + }; + + // Clear search highlighting when the user scrolls sufficiently + const resetFn = () => { + resetHighlighting(""); + window.removeEventListener("quarto-hrChanged", resetFn); + window.removeEventListener("quarto-sectionChanged", resetFn); + }; + + // Register this event after the initial scrolling and settling of events + // on the page + window.addEventListener("quarto-hrChanged", resetFn); + window.addEventListener("quarto-sectionChanged", resetFn); + + // Responsively switch to overlay mode if the search is present on the navbar + // Note that switching the sidebar to overlay mode requires more coordinate (not just + // the media query since we generate different HTML for sidebar overlays than we do + // for sidebar input UI) + const detachedMediaQuery = + quartoSearchOptions.type === "overlay" ? "all" : "(max-width: 991px)"; + + // If configured, include the analytics client to send insights + const plugins = configurePlugins(quartoSearchOptions); + + let lastState = null; + const { setIsOpen, setQuery, setCollections } = autocomplete({ + container: searchEl, + detachedMediaQuery: detachedMediaQuery, + defaultActiveItemId: 0, + panelContainer: "#quarto-search-results", + panelPlacement: quartoSearchOptions["panel-placement"], + debug: false, + openOnFocus: true, + plugins, + classNames: { + form: "d-flex", + }, + translations: { + clearButtonTitle: language["search-clear-button-title"], + detachedCancelButtonText: language["search-detached-cancel-button-title"], + submitButtonTitle: language["search-submit-button-title"], + }, + initialState: { + query, + }, + getItemUrl({ item }) { + return item.href; + }, + onStateChange({ state }) { + // Perhaps reset highlighting + resetHighlighting(state.query); + + // If the panel just opened, ensure the panel is positioned properly + if (state.isOpen) { + if (lastState && !lastState.isOpen) { + setTimeout(() => { + positionPanel(quartoSearchOptions["panel-placement"]); + }, 150); + } + } + + // Perhaps show the copy link + showCopyLink(state.query, quartoSearchOptions); + + lastState = state; + }, + reshape({ sources, state }) { + return sources.map((source) => { + try { + const items = source.getItems(); + + // Validate the items + validateItems(items); + + // group the items by document + const groupedItems = new Map(); + items.forEach((item) => { + const hrefParts = item.href.split("#"); + const baseHref = hrefParts[0]; + const isDocumentItem = hrefParts.length === 1; + + const items = groupedItems.get(baseHref); + if (!items) { + groupedItems.set(baseHref, [item]); + } else { + // If the href for this item matches the document + // exactly, place this item first as it is the item that represents + // the document itself + if (isDocumentItem) { + items.unshift(item); + } else { + items.push(item); + } + groupedItems.set(baseHref, items); + } + }); + + const reshapedItems = []; + let count = 1; + for (const [_key, value] of groupedItems) { + const firstItem = value[0]; + reshapedItems.push({ + ...firstItem, + type: kItemTypeDoc, + }); + + const collapseMatches = quartoSearchOptions["collapse-after"]; + const collapseCount = + typeof collapseMatches === "number" ? collapseMatches : 1; + + if (value.length > 1) { + const target = `search-more-${count}`; + const isExpanded = + state.context.expanded && + state.context.expanded.includes(target); + + const remainingCount = value.length - collapseCount; + + for (let i = 1; i < value.length; i++) { + if (collapseMatches && i === collapseCount) { + reshapedItems.push({ + target, + title: isExpanded + ? language["search-hide-matches-text"] + : remainingCount === 1 + ? `${remainingCount} ${language["search-more-match-text"]}` + : `${remainingCount} ${language["search-more-matches-text"]}`, + type: kItemTypeMore, + href: kItemTypeMoreHref, + }); + } + + if (isExpanded || !collapseMatches || i < collapseCount) { + reshapedItems.push({ + ...value[i], + type: kItemTypeItem, + target, + }); + } + } + } + count += 1; + } + + return { + ...source, + getItems() { + return reshapedItems; + }, + }; + } catch (error) { + // Some form of error occurred + return { + ...source, + getItems() { + return [ + { + title: error.name || "An Error Occurred While Searching", + text: + error.message || + "An unknown error occurred while attempting to perform the requested search.", + type: kItemTypeError, + }, + ]; + }, + }; + } + }); + }, + navigator: { + navigate({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + window.location.assign(itemUrl); + } + }, + navigateNewTab({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + const windowReference = window.open(itemUrl, "_blank", "noopener"); + if (windowReference) { + windowReference.focus(); + } + } + }, + navigateNewWindow({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + window.open(itemUrl, "_blank", "noopener"); + } + }, + }, + getSources({ state, setContext, setActiveItemId, refresh }) { + return [ + { + sourceId: "documents", + getItemUrl({ item }) { + if (item.href) { + return offsetURL(item.href); + } else { + return undefined; + } + }, + onSelect({ + item, + state, + setContext, + setIsOpen, + setActiveItemId, + refresh, + }) { + if (item.type === kItemTypeMore) { + toggleExpanded(item, state, setContext, setActiveItemId, refresh); + + // Toggle more + setIsOpen(true); + } + }, + getItems({ query }) { + if (query === null || query === "") { + return []; + } + + const limit = quartoSearchOptions.limit; + if (quartoSearchOptions.algolia) { + return algoliaSearch(query, limit, quartoSearchOptions.algolia); + } else { + // Fuse search options + const fuseSearchOptions = { + isCaseSensitive: false, + shouldSort: true, + minMatchCharLength: 2, + limit: limit, + }; + + return readSearchData().then(function (fuse) { + return fuseSearch(query, fuse, fuseSearchOptions); + }); + } + }, + templates: { + noResults({ createElement }) { + const hasQuery = lastState.query; + + return createElement( + "div", + { + class: `quarto-search-no-results${ + hasQuery ? "" : " no-query" + }`, + }, + language["search-no-results-text"] + ); + }, + header({ items, createElement }) { + // count the documents + const count = items.filter((item) => { + return item.type === kItemTypeDoc; + }).length; + + if (count > 0) { + return createElement( + "div", + { class: "search-result-header" }, + `${count} ${language["search-matching-documents-text"]}` + ); + } else { + return createElement( + "div", + { class: "search-result-header-no-results" }, + `` + ); + } + }, + footer({ _items, createElement }) { + if ( + quartoSearchOptions.algolia && + quartoSearchOptions.algolia["show-logo"] + ) { + const libDir = quartoSearchOptions.algolia["libDir"]; + const logo = createElement("img", { + src: offsetURL( + `${libDir}/quarto-search/search-by-algolia.svg` + ), + class: "algolia-search-logo", + }); + return createElement( + "a", + { href: "http://www.algolia.com/" }, + logo + ); + } + }, + + item({ item, createElement }) { + return renderItem( + item, + createElement, + state, + setActiveItemId, + setContext, + refresh + ); + }, + }, + }, + ]; + }, + }); + + window.quartoOpenSearch = () => { + setIsOpen(false); + setIsOpen(true); + focusSearchInput(); + }; + + // Remove the labeleledby attribute since it is pointing + // to a non-existent label + if (quartoSearchOptions.type === "overlay") { + const inputEl = window.document.querySelector( + "#quarto-search .aa-Autocomplete" + ); + if (inputEl) { + inputEl.removeAttribute("aria-labelledby"); + } + } + + // If the main document scrolls dismiss the search results + // (otherwise, since they're floating in the document they can scroll with the document) + window.document.body.onscroll = () => { + setIsOpen(false); + }; + + if (showSearchResults) { + setIsOpen(true); + focusSearchInput(); + } +}); + +function configurePlugins(quartoSearchOptions) { + const autocompletePlugins = []; + const algoliaOptions = quartoSearchOptions.algolia; + if ( + algoliaOptions && + algoliaOptions["analytics-events"] && + algoliaOptions["search-only-api-key"] && + algoliaOptions["application-id"] + ) { + const apiKey = algoliaOptions["search-only-api-key"]; + const appId = algoliaOptions["application-id"]; + + // Aloglia insights may not be loaded because they require cookie consent + // Use deferred loading so events will start being recorded when/if consent + // is granted. + const algoliaInsightsDeferredPlugin = deferredLoadPlugin(() => { + if ( + window.aa && + window["@algolia/autocomplete-plugin-algolia-insights"] + ) { + window.aa("init", { + appId, + apiKey, + useCookie: true, + }); + + const { createAlgoliaInsightsPlugin } = + window["@algolia/autocomplete-plugin-algolia-insights"]; + // Register the insights client + const algoliaInsightsPlugin = createAlgoliaInsightsPlugin({ + insightsClient: window.aa, + onItemsChange({ insights, insightsEvents }) { + const events = insightsEvents.map((event) => { + const maxEvents = event.objectIDs.slice(0, 20); + return { + ...event, + objectIDs: maxEvents, + }; + }); + + insights.viewedObjectIDs(...events); + }, + }); + return algoliaInsightsPlugin; + } + }); + + // Add the plugin + autocompletePlugins.push(algoliaInsightsDeferredPlugin); + return autocompletePlugins; + } +} + +// For plugins that may not load immediately, create a wrapper +// plugin and forward events and plugin data once the plugin +// is initialized. This is useful for cases like cookie consent +// which may prevent the analytics insights event plugin from initializing +// immediately. +function deferredLoadPlugin(createPlugin) { + let plugin = undefined; + let subscribeObj = undefined; + const wrappedPlugin = () => { + if (!plugin && subscribeObj) { + plugin = createPlugin(); + if (plugin && plugin.subscribe) { + plugin.subscribe(subscribeObj); + } + } + return plugin; + }; + + return { + subscribe: (obj) => { + subscribeObj = obj; + }, + onStateChange: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onStateChange) { + plugin.onStateChange(obj); + } + }, + onSubmit: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onSubmit) { + plugin.onSubmit(obj); + } + }, + onReset: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onReset) { + plugin.onReset(obj); + } + }, + getSources: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.getSources) { + return plugin.getSources(obj); + } else { + return Promise.resolve([]); + } + }, + data: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.data) { + plugin.data(obj); + } + }, + }; +} + +function validateItems(items) { + // Validate the first item + if (items.length > 0) { + const item = items[0]; + const missingFields = []; + if (item.href == undefined) { + missingFields.push("href"); + } + if (!item.title == undefined) { + missingFields.push("title"); + } + if (!item.text == undefined) { + missingFields.push("text"); + } + + if (missingFields.length === 1) { + throw { + name: `Error: Search index is missing the ${missingFields[0]} field.`, + message: `The items being returned for this search do not include all the required fields. Please ensure that your index items include the ${missingFields[0]} field or use index-fields in your _quarto.yml file to specify the field names.`, + }; + } else if (missingFields.length > 1) { + const missingFieldList = missingFields + .map((field) => { + return `${field}`; + }) + .join(", "); + + throw { + name: `Error: Search index is missing the following fields: ${missingFieldList}.`, + message: `The items being returned for this search do not include all the required fields. Please ensure that your index items includes the following fields: ${missingFieldList}, or use index-fields in your _quarto.yml file to specify the field names.`, + }; + } + } +} + +let lastQuery = null; +function showCopyLink(query, options) { + const language = options.language; + lastQuery = query; + // Insert share icon + const inputSuffixEl = window.document.body.querySelector( + ".aa-Form .aa-InputWrapperSuffix" + ); + + if (inputSuffixEl) { + let copyButtonEl = window.document.body.querySelector( + ".aa-Form .aa-InputWrapperSuffix .aa-CopyButton" + ); + + if (copyButtonEl === null) { + copyButtonEl = window.document.createElement("button"); + copyButtonEl.setAttribute("class", "aa-CopyButton"); + copyButtonEl.setAttribute("type", "button"); + copyButtonEl.setAttribute("title", language["search-copy-link-title"]); + copyButtonEl.onmousedown = (e) => { + e.preventDefault(); + e.stopPropagation(); + }; + + const linkIcon = "bi-clipboard"; + const checkIcon = "bi-check2"; + + const shareIconEl = window.document.createElement("i"); + shareIconEl.setAttribute("class", `bi ${linkIcon}`); + copyButtonEl.appendChild(shareIconEl); + inputSuffixEl.prepend(copyButtonEl); + + const clipboard = new window.ClipboardJS(".aa-CopyButton", { + text: function (_trigger) { + const copyUrl = new URL(window.location); + copyUrl.searchParams.set(kQueryArg, lastQuery); + copyUrl.searchParams.set(kResultsArg, "1"); + return copyUrl.toString(); + }, + }); + clipboard.on("success", function (e) { + // Focus the input + + // button target + const button = e.trigger; + const icon = button.querySelector("i.bi"); + + // flash "checked" + icon.classList.add(checkIcon); + icon.classList.remove(linkIcon); + setTimeout(function () { + icon.classList.remove(checkIcon); + icon.classList.add(linkIcon); + }, 1000); + }); + } + + // If there is a query, show the link icon + if (copyButtonEl) { + if (lastQuery && options["copy-button"]) { + copyButtonEl.style.display = "flex"; + } else { + copyButtonEl.style.display = "none"; + } + } + } +} + +/* Search Index Handling */ +// create the index +var fuseIndex = undefined; +async function readSearchData() { + // Initialize the search index on demand + if (fuseIndex === undefined) { + // create fuse index + const options = { + keys: [ + { name: "title", weight: 20 }, + { name: "section", weight: 20 }, + { name: "text", weight: 10 }, + ], + ignoreLocation: true, + threshold: 0.1, + }; + const fuse = new window.Fuse([], options); + + // fetch the main search.json + const response = await fetch(offsetURL("search.json")); + if (response.status == 200) { + return response.json().then(function (searchDocs) { + searchDocs.forEach(function (searchDoc) { + fuse.add(searchDoc); + }); + fuseIndex = fuse; + return fuseIndex; + }); + } else { + return Promise.reject( + new Error( + "Unexpected status from search index request: " + response.status + ) + ); + } + } + return fuseIndex; +} + +function inputElement() { + return window.document.body.querySelector(".aa-Form .aa-Input"); +} + +function focusSearchInput() { + setTimeout(() => { + const inputEl = inputElement(); + if (inputEl) { + inputEl.focus(); + } + }, 50); +} + +/* Panels */ +const kItemTypeDoc = "document"; +const kItemTypeMore = "document-more"; +const kItemTypeItem = "document-item"; +const kItemTypeError = "error"; + +function renderItem( + item, + createElement, + state, + setActiveItemId, + setContext, + refresh +) { + switch (item.type) { + case kItemTypeDoc: + return createDocumentCard( + createElement, + "file-richtext", + item.title, + item.section, + item.text, + item.href + ); + case kItemTypeMore: + return createMoreCard( + createElement, + item, + state, + setActiveItemId, + setContext, + refresh + ); + case kItemTypeItem: + return createSectionCard( + createElement, + item.section, + item.text, + item.href + ); + case kItemTypeError: + return createErrorCard(createElement, item.title, item.text); + default: + return undefined; + } +} + +function createDocumentCard(createElement, icon, title, section, text, href) { + const iconEl = createElement("i", { + class: `bi bi-${icon} search-result-icon`, + }); + const titleEl = createElement("p", { class: "search-result-title" }, title); + const titleContainerEl = createElement( + "div", + { class: "search-result-title-container" }, + [iconEl, titleEl] + ); + + const textEls = []; + if (section) { + const sectionEl = createElement( + "p", + { class: "search-result-section" }, + section + ); + textEls.push(sectionEl); + } + const descEl = createElement("p", { + class: "search-result-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + textEls.push(descEl); + + const textContainerEl = createElement( + "div", + { class: "search-result-text-container" }, + textEls + ); + + const containerEl = createElement( + "div", + { + class: "search-result-container", + }, + [titleContainerEl, textContainerEl] + ); + + const linkEl = createElement( + "a", + { + href: offsetURL(href), + class: "search-result-link", + }, + containerEl + ); + + const classes = ["search-result-doc", "search-item"]; + if (!section) { + classes.push("document-selectable"); + } + + return createElement( + "div", + { + class: classes.join(" "), + }, + linkEl + ); +} + +function createMoreCard( + createElement, + item, + state, + setActiveItemId, + setContext, + refresh +) { + const moreCardEl = createElement( + "div", + { + class: "search-result-more search-item", + onClick: (e) => { + // Handle expanding the sections by adding the expanded + // section to the list of expanded sections + toggleExpanded(item, state, setContext, setActiveItemId, refresh); + e.stopPropagation(); + }, + }, + item.title + ); + + return moreCardEl; +} + +function toggleExpanded(item, state, setContext, setActiveItemId, refresh) { + const expanded = state.context.expanded || []; + if (expanded.includes(item.target)) { + setContext({ + expanded: expanded.filter((target) => target !== item.target), + }); + } else { + setContext({ expanded: [...expanded, item.target] }); + } + + refresh(); + setActiveItemId(item.__autocomplete_id); +} + +function createSectionCard(createElement, section, text, href) { + const sectionEl = createSection(createElement, section, text, href); + return createElement( + "div", + { + class: "search-result-doc-section search-item", + }, + sectionEl + ); +} + +function createSection(createElement, title, text, href) { + const descEl = createElement("p", { + class: "search-result-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + + const titleEl = createElement("p", { class: "search-result-section" }, title); + const linkEl = createElement( + "a", + { + href: offsetURL(href), + class: "search-result-link", + }, + [titleEl, descEl] + ); + return linkEl; +} + +function createErrorCard(createElement, title, text) { + const descEl = createElement("p", { + class: "search-error-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + + const titleEl = createElement("p", { + class: "search-error-title", + dangerouslySetInnerHTML: { + __html: ` ${title}`, + }, + }); + const errorEl = createElement("div", { class: "search-error" }, [ + titleEl, + descEl, + ]); + return errorEl; +} + +function positionPanel(pos) { + const panelEl = window.document.querySelector( + "#quarto-search-results .aa-Panel" + ); + const inputEl = window.document.querySelector( + "#quarto-search .aa-Autocomplete" + ); + + if (panelEl && inputEl) { + panelEl.style.top = `${Math.round(panelEl.offsetTop)}px`; + if (pos === "start") { + panelEl.style.left = `${Math.round(inputEl.left)}px`; + } else { + panelEl.style.right = `${Math.round(inputEl.offsetRight)}px`; + } + } +} + +/* Highlighting */ +// highlighting functions +function highlightMatch(query, text) { + if (text) { + const start = text.toLowerCase().indexOf(query.toLowerCase()); + if (start !== -1) { + const startMark = ""; + const endMark = ""; + + const end = start + query.length; + text = + text.slice(0, start) + + startMark + + text.slice(start, end) + + endMark + + text.slice(end); + const startInfo = clipStart(text, start); + const endInfo = clipEnd( + text, + startInfo.position + startMark.length + endMark.length + ); + text = + startInfo.prefix + + text.slice(startInfo.position, endInfo.position) + + endInfo.suffix; + + return text; + } else { + return text; + } + } else { + return text; + } +} + +function clipStart(text, pos) { + const clipStart = pos - 50; + if (clipStart < 0) { + // This will just return the start of the string + return { + position: 0, + prefix: "", + }; + } else { + // We're clipping before the start of the string, walk backwards to the first space. + const spacePos = findSpace(text, pos, -1); + return { + position: spacePos.position, + prefix: "", + }; + } +} + +function clipEnd(text, pos) { + const clipEnd = pos + 200; + if (clipEnd > text.length) { + return { + position: text.length, + suffix: "", + }; + } else { + const spacePos = findSpace(text, clipEnd, 1); + return { + position: spacePos.position, + suffix: spacePos.clipped ? "…" : "", + }; + } +} + +function findSpace(text, start, step) { + let stepPos = start; + while (stepPos > -1 && stepPos < text.length) { + const char = text[stepPos]; + if (char === " " || char === "," || char === ":") { + return { + position: step === 1 ? stepPos : stepPos - step, + clipped: stepPos > 1 && stepPos < text.length, + }; + } + stepPos = stepPos + step; + } + + return { + position: stepPos - step, + clipped: false, + }; +} + +// removes highlighting as implemented by the mark tag +function clearHighlight(searchterm, el) { + const childNodes = el.childNodes; + for (let i = childNodes.length - 1; i >= 0; i--) { + const node = childNodes[i]; + if (node.nodeType === Node.ELEMENT_NODE) { + if ( + node.tagName === "MARK" && + node.innerText.toLowerCase() === searchterm.toLowerCase() + ) { + el.replaceChild(document.createTextNode(node.innerText), node); + } else { + clearHighlight(searchterm, node); + } + } + } +} + +function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string +} + +// highlight matches +function highlight(term, el) { + const termRegex = new RegExp(term, "ig"); + const childNodes = el.childNodes; + + // walk back to front avoid mutating elements in front of us + for (let i = childNodes.length - 1; i >= 0; i--) { + const node = childNodes[i]; + + if (node.nodeType === Node.TEXT_NODE) { + // Search text nodes for text to highlight + const text = node.nodeValue; + + let startIndex = 0; + let matchIndex = text.search(termRegex); + if (matchIndex > -1) { + const markFragment = document.createDocumentFragment(); + while (matchIndex > -1) { + const prefix = text.slice(startIndex, matchIndex); + markFragment.appendChild(document.createTextNode(prefix)); + + const mark = document.createElement("mark"); + mark.appendChild( + document.createTextNode( + text.slice(matchIndex, matchIndex + term.length) + ) + ); + markFragment.appendChild(mark); + + startIndex = matchIndex + term.length; + matchIndex = text.slice(startIndex).search(new RegExp(term, "ig")); + if (matchIndex > -1) { + matchIndex = startIndex + matchIndex; + } + } + if (startIndex < text.length) { + markFragment.appendChild( + document.createTextNode(text.slice(startIndex, text.length)) + ); + } + + el.replaceChild(markFragment, node); + } + } else if (node.nodeType === Node.ELEMENT_NODE) { + // recurse through elements + highlight(term, node); + } + } +} + +/* Link Handling */ +// get the offset from this page for a given site root relative url +function offsetURL(url) { + var offset = getMeta("quarto:offset"); + return offset ? offset + url : url; +} + +// read a meta tag value +function getMeta(metaName) { + var metas = window.document.getElementsByTagName("meta"); + for (let i = 0; i < metas.length; i++) { + if (metas[i].getAttribute("name") === metaName) { + return metas[i].getAttribute("content"); + } + } + return ""; +} + +function algoliaSearch(query, limit, algoliaOptions) { + const { getAlgoliaResults } = window["@algolia/autocomplete-preset-algolia"]; + + const applicationId = algoliaOptions["application-id"]; + const searchOnlyApiKey = algoliaOptions["search-only-api-key"]; + const indexName = algoliaOptions["index-name"]; + const indexFields = algoliaOptions["index-fields"]; + const searchClient = window.algoliasearch(applicationId, searchOnlyApiKey); + const searchParams = algoliaOptions["params"]; + const searchAnalytics = !!algoliaOptions["analytics-events"]; + + return getAlgoliaResults({ + searchClient, + queries: [ + { + indexName: indexName, + query, + params: { + hitsPerPage: limit, + clickAnalytics: searchAnalytics, + ...searchParams, + }, + }, + ], + transformResponse: (response) => { + if (!indexFields) { + return response.hits.map((hit) => { + return hit.map((item) => { + return { + ...item, + text: highlightMatch(query, item.text), + }; + }); + }); + } else { + const remappedHits = response.hits.map((hit) => { + return hit.map((item) => { + const newItem = { ...item }; + ["href", "section", "title", "text"].forEach((keyName) => { + const mappedName = indexFields[keyName]; + if ( + mappedName && + item[mappedName] !== undefined && + mappedName !== keyName + ) { + newItem[keyName] = item[mappedName]; + delete newItem[mappedName]; + } + }); + newItem.text = highlightMatch(query, newItem.text); + return newItem; + }); + }); + return remappedHits; + } + }, + }); +} + +function fuseSearch(query, fuse, fuseOptions) { + return fuse.search(query, fuseOptions).map((result) => { + const addParam = (url, name, value) => { + const anchorParts = url.split("#"); + const baseUrl = anchorParts[0]; + const sep = baseUrl.search("\\?") > 0 ? "&" : "?"; + anchorParts[0] = baseUrl + sep + name + "=" + value; + return anchorParts.join("#"); + }; + + return { + title: result.item.title, + section: result.item.section, + href: addParam(result.item.href, kQueryArg, query), + text: highlightMatch(query, result.item.text), + }; + }); +} diff --git a/styles.css b/styles.css new file mode 100644 index 0000000..e69de29 diff --git a/unpublished/Binning loss.html b/unpublished/Binning loss.html new file mode 100644 index 0000000..4e985a5 --- /dev/null +++ b/unpublished/Binning loss.html @@ -0,0 +1,562 @@ + + + + + + + + + +Hans Dembinski's blog – binning-loss + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + +
    + +
    + + + + +
    + + + +
    +
    from iminuit import Minuit
    +import numpy as np
    +from scipy.stats import norm, uniform, poisson
    +from matplotlib import pyplot as plt
    +import progressbar
    +
    +
    +
    xrange = (0., 1.)
    +z = 0.2
    +mu = 0.5
    +sigma = 0.05
    +
    +truth = (z, mu, sigma)
    +
    +
    +
    def unbinned(z, mu, sigma):
    +    """Unbinned likelihood"""
    +    probs = (1.0-z) / (xrange[1] - xrange[0]) + z * norm(mu, sigma).pdf(x)
    +    return -np.sum(np.log(probs))
    +
    +
    +def expected(z, mu, sigma):
    +    """Return expected counts per bin in a histogram"""
    +    nx = len(x)
    +    b = uniform(*xrange).cdf(xe)
    +    s = norm(mu, sigma).cdf(xe)
    +    m = (1-z) * b + z * s
    +    return len(x) * (m[1:] - m[:-1])
    +
    +
    +def binned(z, mu, sigma):
    +    """Binned likelihood for Poisson-distributed counts"""
    +    m = expected(z, mu, sigma)
    +    return np.sum(m - w * np.log(m))
    +
    +
    +
    # check expected
    +np.random.seed(1)
    +s = norm(mu, sigma).rvs(int(z * 1000))
    +b = uniform(*xrange).rvs(1000 - len(s))
    +x = np.append(s, b)
    +assert len(x) == 1000
    +
    +fig, ax = plt.subplots(1, 3, figsize=(15, 4), sharex=True)
    +for axi, bins in zip(ax, (5, 50, 100)):
    +    plt.sca(axi)
    +    w, xe, *_ = plt.hist(x, bins=bins, range=xrange, label="data")
    +    m = expected(*truth)
    +    plt.step(xe, np.append(0, m), label="expected")
    +    plt.legend()
    +    plt.xlabel("x")
    +
    +

    +
    +
    +
    +
    np.random.seed(1)
    +
    +bins = (5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 30, 40, 50, 100)
    +ntot = 1000
    +nmc = 100
    +
    +pars_unbinned = []
    +pars_binned = []
    +covs_unbinned = []
    +covs_binned = []
    +
    +minuit_kwargs = {
    +    "z": 0.5, "mu": 0.5, "sigma": 0.05, "pedantic": False, "errordef": 0.5,
    +    "limit_z" : (0, 1), "limit_mu" : xrange, "limit_sigma" : (0, None)
    +}
    +
    +for imc in progressbar.progressbar(range(nmc)):
    +    xs = norm(mu, sigma).rvs(int(z * ntot))
    +    xb = uniform(*xrange).rvs(ntot - len(xs))
    +    x = np.append(xs, xb)
    +    
    +    m_un = Minuit(unbinned, **minuit_kwargs)
    +    m_un.migrad()
    +    m_un.migrad()
    +
    +    valid = m_un.get_fmin().is_valid
    +    if not valid:
    +        continue
    +
    +    pars_b = []
    +    covs_b = []
    +    for b in bins:
    +        w, xe = np.histogram(x, bins=b, range=xrange)
    +
    +        m = Minuit(binned, **minuit_kwargs)
    +        m.migrad()
    +        m.migrad()
    +
    +        valid = m.get_fmin().is_valid
    +        if not valid:
    +            break
    +
    +        pars_b.append(m.np_values())
    +        covs_b.append(m.np_covariance())
    +
    +    if not valid:
    +        continue
    +
    +    pars_unbinned.append(m_un.np_values())
    +    covs_unbinned.append(m_un.np_covariance())
    +    
    +    pars_binned.append(pars_b)
    +    covs_binned.append(covs_b)
    +
    +pars_unbinned = np.array(pars_unbinned)
    +pars_binned = np.array(pars_binned)
    +covs_unbinned = np.array(covs_unbinned)
    +covs_binned = np.array(covs_binned)
    +
    +
    100% (100 of 100) |######################| Elapsed Time: 0:03:22 Time:  0:03:22
    +
    +
    +
    +
    # bias
    +fig, ax = plt.subplots(1, 3, figsize=(16, 4), sharex=True)
    +plt.subplots_adjust(wspace=0.3)
    +for ipar, name in enumerate(("z", "mu", "sigma")):
    +    plt.sca(ax[ipar])
    +    y = pars_unbinned[:, ipar]
    +    m = np.mean(y)
    +    s = np.std(y) / n ** 0.5
    +    n = len(pars_unbinned)
    +    plt.axhline(m, label="unbinned")
    +    plt.axhspan(m-s, m+s, alpha=0.2)
    +    y = pars_binned[:, :, ipar]
    +    m = np.mean(y, axis=0)
    +    s = np.std(y, axis=0) / n ** 0.5
    +    plt.errorbar(bins, m, s, fmt="o", color="C1", label="binned")
    +    plt.legend()
    +    plt.xlabel("bins")
    +    plt.semilogx()
    +    plt.ylabel(name)
    +    plt.axvline((xrange[1] - xrange[0])/sigma, ls="--", color="0.5")
    +    plt.axvline((xrange[1] - xrange[0])/(2 * sigma), ls=":", color="0.5")
    +    plt.axhline(truth[ipar], color="k", zorder=0)
    +
    +

    +
    +
    +
    +
    # coverage probability
    +fig, ax = plt.subplots(1, 3, figsize=(16, 4), sharex=True)
    +plt.subplots_adjust(wspace=0.3)
    +for ipar, name in enumerate(("z", "mu", "sigma")):
    +    plt.sca(ax[ipar])
    +    y = pars_unbinned[:, ipar]
    +    dy = covs_unbinned[:, ipar, ipar] ** 0.5
    +    prob = np.mean((truth[ipar] < y + dy) & (truth[ipar] > y - dy))
    +    plt.axhline(prob, label="unbinned")
    +    y = pars_binned[:, :, ipar]
    +    dy = covs_binned[:, :, ipar, ipar] ** 0.5
    +    prob = np.mean((truth[ipar] < y + dy) & (truth[ipar] > y - dy), axis=0)
    +    plt.plot(bins, prob, "o-", color="C1", label="binned")
    +    plt.ylim(0, 1)
    +    plt.xlabel("bins")
    +    plt.semilogx()
    +    plt.ylabel(name)
    +    plt.axhline(0.68, ls="-", color="k")
    +    plt.axvline((xrange[1] - xrange[0])/sigma, ls="--", color="0.5")
    +    plt.axvline((xrange[1] - xrange[0])/(2 * sigma), ls=":", color="0.5")
    +    plt.legend(loc="lower right")
    +
    +

    +
    +
    + + + +
    + +
    + + + + \ No newline at end of file diff --git a/unpublished/Binning loss_files/figure-html/cell-5-output-1.png b/unpublished/Binning loss_files/figure-html/cell-5-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..f580fb63b106d9d828ec601d4315ba3d373eab4a GIT binary patch literal 12375 zcmb_@2{@GP+xIX<5}rwjLX9nDiLtLe88n2lXG>!2OWAjx789W;3^9?Gku}>W%T$&y z$WnGDGDOzNKFoKGXZyd$`@Y9_9N+PE9CKXep8LM8>%Ok@{GGq^cV6AQWUS9|P~ad0 z0^vYiK$=1zP;Ur?iRAzbxU%LD83vkt{^wEV2f)9G1I`%moz3sU6@LhX{V?Oh)H51I z0~b{Sbgcu-e9-~H*KWE%ysriLdHDu-xjUT*a=Gd6?(2h)k(ZH^I^h-&;HM@l`|s~E zzBgTEpRK43K_Dj}DCFbz6J|xLhlSZfUSpS#Vv>%+p0{=J9B18}&(>%p ze(1|*%e|~{N;|HL+A^4WQJgwSO7>sVavdwF?9420a9#A*^6zYwpbm};tIKnvQbbXv zih>qF5b#Gz{B-ksWZj_oa!f`Vo9l22Q+_CTgVR-`kid zp?syn-C(%VlTTkbQh`e85P`RGzMOIv-(FvI?Ju@}a*89k{2YYl} z8-Ihgt|(^_PKP_8kt2Mt@kGp`Zez6UkjXtvxQLT0(hGy(cWF7DnD0TavgrsJ(@szV zD@zGsA)G~YvG79pbQ_(>lSByw)tuZyv=D1Ph5tipC|}ZTxXOFvMvLHetpysjYu%|h 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z5YSOAf4Ex=$nJ_at_Y4_;tWeo3z5r^rN=LOo9*G|4Loqf5Z0MDgj}_@j&VN|S+IBr zi!LObAYqZ21*~VM!Jmu7Ma{S=L6DCHagb zztoOE)A4ifw6wm-nG}J4sY)%rohmPPe9TJTBcZGI| zWa_yd-U`IG&%Erys2OrY6Lt$%Ljc>#R3*@n9Ci;W&gX>!Tkn#Mmo|bKbY8J;E$`bE%%4{*uL*o3X{}oNjzGa<0`FaQ4h>)vc}u^c05sGs(lHTM~wYW#WC#^ z8OLO=Ei6~lJD&_$RW~$-{u%CdQF?esp8fs9!4B6UJ){}O%G@T?1}g@Vhdd%0BN&n{ z{8~TpUf2F8xUO2Ln0fTPYBRW-3!w!1dThPxTXjfROzT=8=!>aI=Aa8NCy_JH z^zc$sxPnC+JYcUjUb!1=@&;H0=9vNj32BCdFPpZJi~s{W0KJJh;0S zB62lF(9})HA&d!4RYNa%PzacD71G9D3fKvjq15U@0}?pro2n0lzY-ec$7RA+c zY~`3l+W#jk1@v!k&UKRR>};)tw8L78K!1oC*Z9pYz?qpZsyUyJ3xmrHaLK*~@v$u{ zq=Jz4;SbPM{78yi7*5>>&NJqf+nIP@woBYM*8eyQ?&A-w>;JdW#sJ`HVrmTlzc{(i b{mm=KI*)krLej|@I06tH6uz0ik#ziTml-oR literal 0 HcmV?d00001 diff --git a/unpublished/Heilter model, influence of alpha.html b/unpublished/Heilter model, influence of alpha.html new file mode 100644 index 0000000..5c91f4b --- /dev/null +++ b/unpublished/Heilter model, influence of alpha.html @@ -0,0 +1,413 @@ + + + + + + + + + +Hans Dembinski's blog – heilter-model-influence-of-alpha + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + +
    + +
    + + + + +
    + + + +
    +
    from sympy import *
    +
    +
    +
    nmu, energy, xi, alpha, M = symbols("N_mu E xi alpha M", real=True, positive=True)
    +
    +
    +
    nmu1 = (energy/xi)**(1 + log(alpha) / log(M))
    +
    +
    +
    d = diff(nmu1, alpha); d
    +
    +

    \(\displaystyle \frac{\left(\frac{E}{\xi}\right)^{1 + \frac{\log{\left(\alpha \right)}}{\log{\left(M \right)}}} \log{\left(\frac{E}{\xi} \right)}}{\alpha \log{\left(M \right)}}\)

    +
    +
    +
    +
    d.subs(nmu1, nmu)
    +
    +

    \(\displaystyle \frac{N_{\mu} \log{\left(\frac{E}{\xi} \right)}}{\alpha \log{\left(M \right)}}\)

    +
    +
    + + + +
    + +
    + + + + \ No newline at end of file diff --git a/unpublished/discrete profiling.html b/unpublished/discrete profiling.html new file mode 100644 index 0000000..8e06716 --- /dev/null +++ b/unpublished/discrete profiling.html @@ -0,0 +1,1118 @@ + + + + + + + + + +Hans Dembinski's blog – discrete-profiling + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    + +
    + +
    + + + + +
    + + + +
    +
    from iminuit import Minuit
    +from iminuit.cost import ExtendedBinnedNLL
    +from scipy.stats import norm, expon
    +from scipy.interpolate import BPoly
    +import numpy as np
    +from pyik.mplext import plot_hist
    +import matplotlib.pyplot as plt
    +
    +
    +
    def scaled_cdf_bernstein(x, pars):
    +    pars = np.atleast_1d(pars)
    +    c = BPoly(pars[..., np.newaxis], xrange).antiderivative()
    +    return c(x)
    +
    +
    +def scaled_cdf_expon(x, a, b):
    +    return a * (1 - np.exp(-b * x))
    +
    +
    +def model1(x, a, b, c):
    +    sig = a * norm(0.5, 0.05).cdf(x)
    +    bkg = scaled_cdf_expon(x, b, c)
    +    return bkg + sig
    +
    +
    +def model2(x, par):
    +    sig = par[0] * norm(0.5, 0.05).cdf(x)
    +    bkg = scaled_cdf_bernstein(x, par[1:])
    +    return bkg + sig
    +
    +
    +
    rng = np.random.default_rng(1)
    +xrange = np.array((0, 1))
    +x_bkg = rng.exponential(0.4, size=10000)
    +x_sig = norm(0.5, 0.05).rvs(5000)
    +x = np.append(x_bkg, x_sig)
    +x = x[(x > xrange[0]) & (x < xrange[1])]
    +n, xe = np.histogram(x, bins=50, range=xrange)
    +plot_hist(xe, n)
    +
    +

    +
    +
    +
    +
    m1 = Minuit(ExtendedBinnedNLL(n, xe, model1), 500, 1000, 1)
    +m1.limits = (0, None)
    +m1.migrad()
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 50.68Nfcn = 98
    EDM = 4e-06 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0a5.01e30.08e30
    1b10.06e30.13e30
    2c2.460.040
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    abc
    a7.23e+03-3.09e+03 (-0.280)0.703 (0.186)
    b-3.09e+03 (-0.280)1.68e+04-2.61 (-0.454)
    c0.703 (0.186)-2.61 (-0.454)0.00196
    +
    +
    +
    +
    plot_hist(xe, n)
    +plot_hist(xe, np.diff(model1(xe, *m1.values)))
    +
    +

    +
    +
    +
    +
    m2 = Minuit(ExtendedBinnedNLL(n, xe, model2), (500, 2e4, 2e3, 3e3))
    +m2.limits = (0, None)
    +m2.migrad()
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 65.9Nfcn = 126
    EDM = 1.1e-06 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0x04.95e30.09e30
    1x123.0e30.4e30
    2x22.2e30.5e30
    3x32.54e30.19e30
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    x0x1x2x3
    x07.86e+031.98e+03 (0.060)-1.41e+04 (-0.345)3.41e+03 (0.207)
    x11.98e+03 (0.060)1.4e+05-1.02e+05 (-0.590)2.48e+04 (0.356)
    x2-1.41e+04 (-0.345)-1.02e+05 (-0.590)2.13e+05-6.22e+04 (-0.722)
    x33.41e+03 (0.207)2.48e+04 (0.356)-6.22e+04 (-0.722)3.47e+04
    +
    +
    +
    +
    plot_hist(xe, n)
    +plot_hist(xe, np.diff(model2(xe, m2.values)))
    +
    +

    +
    +
    +
    +
    m3 = Minuit(ExtendedBinnedNLL(n, xe, model2), (500, 1000, 1000, 1000, 1000))
    +m3.limits = (0, None)
    +m3.migrad()
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 52.27Nfcn = 336
    EDM = 7.46e-05 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0x05.03e30.09e30
    1x124.5e30.6e30
    2x25.8e30.9e30
    3x34.3e30.6e30
    4x42.07e30.21e30
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    x0x1x2x3x4
    x08.3e+031.06e+04 (0.207)-2.7e+04 (-0.316)2.81e+03 (0.051)558 (0.029)
    x11.06e+04 (0.207)3.16e+05-3.97e+05 (-0.752)1.63e+05 (0.480)-2.66e+04 (-0.223)
    x2-2.7e+04 (-0.316)-3.97e+05 (-0.752)8.82e+05-4.33e+05 (-0.761)7.95e+04 (0.400)
    x32.81e+03 (0.051)1.63e+05 (0.480)-4.33e+05 (-0.761)3.68e+05-9.17e+04 (-0.714)
    x4558 (0.029)-2.66e+04 (-0.223)7.95e+04 (0.400)-9.17e+04 (-0.714)4.48e+04
    +
    +
    +
    +
    plot_hist(xe, n)
    +plot_hist(xe, np.diff(model2(xe, m3.values)))
    +
    +

    +
    +
    +
    +
    m4 = Minuit(ExtendedBinnedNLL(n, xe, model2), (500, 1000, 1000, 1000, 1000, 1000))
    +m4.limits = (0, None)
    +m4.migrad()
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 49.14Nfcn = 456
    EDM = 7.25e-05 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0x04.97e30.10e30
    1x125.2e30.7e30
    2x28.0e31.5e30
    3x38.6e32.0e30
    4x42.1e31.0e30
    5x52.38e30.29e30
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    x0x1x2x3x4x5
    x09.6e+03-4.21e+03 (-0.061)3.4e+04 (0.224)-8.4e+04 (-0.428)3.7e+04 (0.359)-5.97e+03 (-0.213)
    x1-4.21e+03 (-0.061)4.94e+05-8.14e+05 (-0.748)7e+05 (0.497)-2.8e+05 (-0.378)4.65e+04 (0.231)
    x23.4e+04 (0.224)-8.14e+05 (-0.748)2.39e+06-2.67e+06 (-0.863)1.17e+06 (0.717)-2.02e+05 (-0.457)
    x3-8.4e+04 (-0.428)7e+05 (0.497)-2.67e+06 (-0.863)4.01e+06-1.93e+06 (-0.916)3.54e+05 (0.618)
    x43.7e+04 (0.359)-2.8e+05 (-0.378)1.17e+06 (0.717)-1.93e+06 (-0.916)1.11e+06-2.39e+05 (-0.793)
    x5-5.97e+03 (-0.213)4.65e+04 (0.231)-2.02e+05 (-0.457)3.54e+05 (0.618)-2.39e+05 (-0.793)8.19e+04
    +
    +
    +
    +
    plot_hist(xe, n)
    +plot_hist(xe, np.diff(model2(xe, m4.values)))
    +
    +

    +
    +
    +
    +
    sig1, fcn1, _ = m1.mnprofile("a")
    +sig2, fcn2, _ = m2.mnprofile("x0")
    +sig3, fcn3, _ = m3.mnprofile("x0")
    +sig4, fcn4, _ = m4.mnprofile("x0")
    +plt.plot(sig1, fcn1, label="model1")
    +plt.plot(sig2, fcn2, label="model2(2)")
    +plt.plot(sig3, fcn3, label="model2(3)")
    +plt.plot(sig4, fcn4, label="model2(4)")
    +plt.legend()
    +
    +
    <matplotlib.legend.Legend at 0x132c0bf70>
    +
    +
    +

    +
    +
    +
    +
    sig, fcn1, ok = m1.mnprofile("a", size=50, bound=(4500, 5500))
    +assert np.all(ok)
    +_, fcn2, ok = m2.mnprofile("x0", size=50, bound=(4500, 5500))
    +assert np.all(ok)
    +_, fcn3, ok = m3.mnprofile("x0", size=50, bound=(4500, 5500))
    +assert np.all(ok)
    +_, fcn4, ok = m4.mnprofile("x0", size=50, bound=(4500, 5500))
    +assert np.all(ok)
    +
    +
    +
    def akaike(npar, n):
    +    return 2 * npar / (1 - (npar + 1) / n)
    +
    +def simple(npar, n):
    +    return npar
    +
    +
    +
    fcn_min = np.inf
    +for fcn, npar in ((fcn1, m1.nfit), (fcn2, m2.nfit), (fcn3, m3.nfit), (fcn4, m4.nfit)):
    +    fcn_min = np.minimum(fcn_min, fcn + akaike(npar, len(n)))
    +
    +
    +
    plt.plot(sig, fcn1-offset, label="model1")
    +plt.plot(sig, fcn2-offset, label="model2(2)")
    +plt.plot(sig, fcn3-offset, label="model2(3)")
    +plt.plot(sig, fcn4-offset, label="model2(4)")
    +plt.legend()
    +
    +
    <matplotlib.legend.Legend at 0x1331c2a00>
    +
    +
    +

    +
    +
    +
    +
    offset = np.min(fcn_min)
    +
    +plt.plot(sig, fcn1-offset+simple(m1.nfit, len(n)), label="model1")
    +plt.plot(sig, fcn2-offset+simple(m2.nfit, len(n)), label="model2(2)")
    +plt.plot(sig, fcn3-offset+simple(m3.nfit, len(n)), label="model2(3)")
    +plt.plot(sig, fcn4-offset+simple(m4.nfit, len(n)), label="model2(4)")
    +plt.plot(sig, fcn_min-offset, color="k", label="min")
    +plt.legend()
    +plt.ylim(0, 20)
    +
    +
    (0.0, 20.0)
    +
    +
    +

    +
    +
    + + + +
    + +
    + + + + \ No newline at end of file diff --git a/unpublished/discrete profiling_files/figure-html/cell-10-output-1.png b/unpublished/discrete profiling_files/figure-html/cell-10-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..eec390fd30e106274eda2e2e11f7a1198f517f3d GIT binary patch literal 6344 zcma)BXIN9)wheFurOHv7G!I3JglaL*p%NJ3FSno`9?TId8745FYhpmafkRH-5m z1Hp(WP3ap$$6z?pD;Vt_cok&j9*nw%48G>;aWwR5V305Jx|XVz zs;0_OpWt9rke-@az~2*8k%8W7YIg;BfJF|V%r6FkKs<-`UTmO@EKv|h6k=g|8WEPY zFdF;7Wipz!#1csqy2O(xYFj1rCg}*_1Gg1<*b(!j|L&7yuy+~mu5?~#B~;Kt;+4ib zh{X~4TZfM29E4LNB=84?dTp+h-XMtiWL6#gcAArrbg{ZeQ>jw;%!rUVdqr*8Tkgpj zX%^;XBx7J~U@X_0R*^0_8r|Ix*?jJJG(9MHC2PeLdcXqKl-2ixUCv^_yWSGUFGk1V zO%jN5lMg0TEp$&1d-j1;oa@y?LPND1c`f{uJ{q#96ZqbE4^6pZ zg6Z4R$vRyXSqt4O|9hQ(gse0^y7(34fIqz|U#%0G%ZoILo_knAMm^rBYiTj64{NQT zs%jAl^YwWWnX~x;XYGV;^|*d9+}BVUU)YIy{Pf_wxRmtJYAbqp_k~8PVZ5nBypREC!(2D`KP1+VJC)SlcR;7 z9dP*{7yrlah0`-Mq(*CU7f83xE}-vc=aT~LIh1jJZ8=1*dq8kiKoabSL}s?W2@g7A z^AMrWq(VH)#G-Fn1rQy_wB4=xKMPD(_?)+e-QRnts@Zi_J!}5M$ojB-Wsa`9Hw%uG z6irJB=5p=uJ-`^baeQJk;B?+^z!jxOI3X+;ggJ-FWlIL>5b#X9=3w7hW;PK&DK3Q! z7K3$JeF;jzwvwCoHNv5fIchMOaTOreb9{_M7*9PovOR$8ba8WgI0jdsR}$>#rLC?b z2tldn7g#VKpfRGx?Ws>{$hm>g|!&4uc`D-?V$ zarinHhIKuPBpL>eq!R@-&b$AZLZ$jYKH~S`^YM8S>q|cK;nks<RJVkkgWZ`?g+_X-i(dZ0% zocS8!N4f~B+j8wY!K~;?XS6ZO&a~FsbJ`yh<0i5=wpHa6w=n!oa+q~xB=@|M*cc9R z?omP{)K?(>&@-U=f{g&PrSw{B@NBY)-lsilM@2lCyuH8pcT zP$a#b5IwDW^=CqxxJyF6`hTdG<=Z8QtJw-%ZXF7Db;=wCM#X1cA9xkqePbx23fkeV zmoO(cbM8oe<_>1x#E8S8$g`#(0g!*3?YNc_f#E? zLn@k7JB3sxRT{M80QHSFtz}yvBiZ!f&{$DHGzXf)mEDM6rK8E7o%VhQABY@-yb5w< z07xVz&#X4#N@GCRJ5wVXe=LP4**0$#C?)U@eiT$68>V%*zCv%xF6<7zeO znjBdMyF{g;D%^?mdsttN1xk(GvNc!KfH4d)5_QMjE z^SyeS|I>g*ZaLc7@WnzQSFW!O1cz`1OVTG3TsQ6{_YP<@vk6(if_!FB(m%&?1BVvT zKFCZ+jEVTYsKto!S=`4*1+V>QP_7|SBH2sI(m(10w-Rnet1rtf#wYX>^k#s*ua~6 z+^jPduY&Ku+Wq0!2RKBm&S+pdSIDQ9-iF7Ne(+203Wyh2CZ6P25-j%2`x5^yb+FVj;y} z4dE?xC(^W&Jp$8$?^QE>dzI`-jwc6^pTnV^f*M?5zpRUm??=I*zODnuE2_hXsNK;d z|D-RBBaM%k57CbOm<+j@YgUBm9R@}uj%DAn&W(zN&QxL74~=V&ix1}_3HTTTnd{w5 zV)0$b&|9Dg!Qn7}2B`)*54Pwk&qt+TEiu-hSP)fAkZsEVWm(rjAv0|EvqFYFF#+V* zWURRoVK&vpHM!N&50ZL0Z$mmRR$$A7xgAdyfI}HxG-=Qnv2E649D=#FiK^Q@DOjZ; z6Csm7{VC7H+aX#K*uuuuHv(LnP}Re~S#-Je=0uuqu`ilIXBx2UJ})nyB-Dz1=KVA} zI$BHG$X@Ut9&So27e991F6i_@3#&4md%7K2LDmC|l8V=0abb95JVFQcn|}#I5@t`k z9Y7rgc$Y^p%Q01M;6uSg4s63}T!*bXXVXQA5Gy1|V;_V+(Fi(TX&Bj_JiqB(yN#h>?ydkNnRnFT--DvyPCf3AZL=h`i@sE0o5Wr zdg#i`oN-s~8ACrg2uFoYwrRsp27K*L+G*$_vO`O>DI`v4SJLiGp8Mq^#EhL}SBW9I3!D7F z&56rbiPJAQ-G1I?c0b}ML*AN1Z}rqOVJy~Q*rd@K6#`&MyW(gXRD)VtMtvQIJQ%{P zWAXPI2jgEP|NQo4WW=iqHI~I`ClAq~xPbFYJ&l|gPr`m1zWMI3>`M@ZlnlM!iqzB0 z;+w2@G|~NwwA^DV1ynEhJjpLO=3PH@syKYf#nVCL9aUj|Bs}n_lT*!lD5X1J$uT6# zp{}Sk^_!0K>->2c(cbgk^$r&s!k4xso8R*|L?uridK2;8-`@uv&}Ffpk5=l?F&O)y zS&-&l?=Xy*ZY9CdLZdU1r)_n%?9l z5DT=_shRlWrz|q~DWt^I47KK&_v$>;UGBEG{BZq8fSc^DH*Nj+tTaVh%lBNub-za; zVUtrG$fjMrFDZ;bmCL-|1rX6CD0n{rV)xNwW)550ro3Dwa>8s{C+pn8KUf&)DE?(nFe zcD8hx`BJlQqfQ6{987Ms21{|92mOt_tgI0L%DX3`z%HKOhNMUFP4BB)M0}3xbQB7b zHktzD9UQ)(>9Ej`E-qV~^2I6?!|C>H%&5KBRmG?7q>Z7J>~T~hKJwiT1yJ8r&w^b} z=H3q}H}eZfi0n?)XN&B+gkGI|K%1*= z9+NLVL5WV#gF6mx2y^|x6Hte?b)B@8kC|P7u>Krzr*F4lMw9AYqy^@x|G6{gwyyuT zTxiO3r-#!Ol5(^LN|lIuQzCcP)Ak7&yY{Aw(UmnB86yo9=m6WzLRnXphGqviw5tu} z@UWV+0p=NZ%LT5%WR$~XHwtofV#=YFv<;&`{sJ3CUB2)%R8K#FsO&K1*Z#1gJa4XC zBq<>P@D)8VSAKpJ>nA#1s$Iy)JI2StInl8hcXQUt);83IEa<2ZNGp1aF)+4ZdQ0PV0U8RIg=8IW&UK zNWxwihq?(={-`Az?Ps`_H;INV)IOJd|K-?D*?qyDLxP5P_$4#^MsWpNMJK7U9#V_6 zI6(SH8b&(}T3W-y0arSOhoPaxoB2tq8SQX+i$_I@RN1f2+$}qgl?cu48G!6bz81-~ zXf|lHUqyFe=JAyPz_JuysXGr6U{fo^H${9Ylq8pvrM+UW8<%en$$r&vlUJV)Bc2F( z@su!AbR}m~dA#qGtLds&K=aw(`kTcBU$IrN>v)ai{H1)h9Tcr>Ez6%03K1W!r9Y2T ztO>VD+&;88>^msm<@2}qJpVe#D?PQQ;6ULCC$qiE-E*A`uq>n)u$I)-ajkD(v?ay7 zCSL18oY-xDy4PD;6;T(&05IGDye7rvktUUJ3_%~{i)3V{j_U8G=Lf)DW^H~>Jt6j= z5$?`3i5w}*-?ORL7;BeKZ(|M7#qlb7y9NMNU+y{l0cIH9b?tYBJ>OtiKIP`x(|SHL znb+IlWutimH!r$4HOu{B;7o7=u^-y5;#m7)a* z2=7TgSD#a7{p2Vv?Qwgi>x(5#btY0{qSDo{%j9)D|EI0F1cwgW9y-Z zJNX(L#CI2#wr!NoV!Ga(|QAvRq}9ma=ANuw7#(Y^Fh#hr}1h=qbL6ZID`ZpuJAmsp?2d zfDhg?YH7Sk^R!KzzL%Ws!8k$V9dqz7FG>S)sby9?fe$oX6lu>onLV4yc6jVy*qSeM zXU*j7h+oLi1&Rxdj+Y>+>6AXdx11|i$n*lm>)cMPiFTVsFJy58!!OQ<%r{-?HYh`E z@a6Syvr1c0kMYOXGKTx+oxQC8xx?f4D_w92kcpevFkKn?i&lSgTzhFm{kgX)<8$P; zIMwmTGWn%nDr7B`D})VMDIm4ycupA96Hup(M_wn(72msbKHhoHrvaKQ^$_|l_CDDR zW2t6H^tEhLT`#VQ$dQBGy5!|ZnQC~B*XGD%Ya|l;hPUpO#w=KuUV0=q00(Q82~#xp z+*nhyoIen;0KeJZ1$D@Lt;YIrT;~k!9}khUxQd8&JVsc(Zn~wn*c^9}ockQlwBG~< zCI_x`0`lvZMN@E0ot$srF}v7{uOb-)^1SeXNV%QWTRB86Z1b zmWEwl!G_7o?7Zk?VzCuhW2Ixo1*LPCNrm`~ouoJAw_MrM*aOHsr-p7v$MCH`nqTT7 z51I|R-0{53SL>Y;8*)q{sE+oNjE?h;qwmFOE1Kis<89T9UE61KFDM#F&KO7}b>F;y z@MxdGR4#`R@0LsYf`8oZfnQKUmEMgN!^(y%sPO1zE`uz?G0w6?Y?0^v0qq%$=lD

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    KI0+1(=TO!bDPh?q!p=Ox2b(9#k7y{XE>{sBH6}wdekDe3LZD4Ny)DV-?w`(?c zajuPfx-c=GnH|SE2_;R)KfuJOYs>^Ni2WfJ2(~Un^EbYB?2jZ;XP2mZFS5+D+n2A( z^#gb0!qmx7bjBk%DrZ0?o^Kn2FJTO#J!EwR?kJSVNn{lpJi7lhVe@Md{y;b~D#OV> zff@Ey&NUM~W;U;8B1P^EXout-C|< z36=#lp? z;=DpfRsjxv@ma0ALaeA()n+AwrbXQ8ze zL7Lr{;G*jR(qryXV%wHO5gN4z&rfak9cTp|+YWRMCaVd|<;OFLg14AmbUrm`QTlX! zDSfivNvd}H#dk-8t}qXAk_Ax%Oq%o2t{S!Iv-HHS6#Trg*qV8w*c@Z6`L*SQECz9r zEscHWbxH+HV9ifK?pF literal 0 HcmV?d00001 diff --git a/unpublished/finance.html b/unpublished/finance.html new file mode 100644 index 0000000..a5219a2 --- /dev/null +++ b/unpublished/finance.html @@ -0,0 +1,530 @@ + + + + + + + + + +Hans Dembinski’s blog - ETF analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + +

    +
    + +
    + +
    + + + + +
    + +
    +
    +

    ETF analysis

    +
    + + + +
    + + + + +
    + + +
    + +
    +
    import yfinance as yf
    +
    +
    +
    luxor = yf.download("X010.DE", period="max")
    +ishares = yf.download("EUNL.DE", period="max")
    +
    +
    [*********************100%***********************]  1 of 1 completed
    +[*********************100%***********************]  1 of 1 completed
    +
    +
    +
    +
    from matplotlib import pyplot as plt
    +
    +plt.plot(luxor.index, luxor["Open"], label="luxor")
    +plt.plot(ishares.index, ishares["Open"], label="ishares")
    +plt.legend()
    +plt.ylabel("price")
    +plt.xlabel("date")
    +
    +
    Text(0.5, 0, 'date')
    +
    +
    +

    +
    +
    +
    +
    diff = (luxor["Open"] - ishares["Open"]) / luxor["Open"]
    +plt.plot(diff.index, diff)
    +plt.ylim(0, 0.15)
    +plt.xlabel("date")
    +plt.ylabel("(Luxor - iShares) / Luxor")
    +
    +
    Text(0, 0.5, '(Luxor - iShares) / Luxor')
    +
    +
    +

    +
    +
    +
    +

    Monthly delta

    +
    +
    from datetime import date, timedelta
    +
    +start = date(luxor.index[0].year, luxor.index[0].month, 1)
    +end = date(luxor.index[-1].year, luxor.index[-1].month, 31)
    +dt = timedelta(days=1)
    +
    +months = {}
    +
    +for i, v in zip(luxor.index, luxor["Open"]):
    +    months.setdefault((i.year, i.month), []).append(
    +        (i.day, v)
    +    )
    +
    +
    +
    len(months)
    +
    +
    173
    +
    +
    +
    +
    m2 = []
    +for pairs in months.values():
    +    days, values = np.transpose(pairs)
    +    m = np.mean(values)
    +    m2.append(
    +        (days, values / m - 1)
    +    )
    +
    +for days, vals in m2:
    +    plt.plot(days, vals)
    +
    +plt.ylim(-0.1, 0.1)
    +
    +
    (-0.1, 0.1)
    +
    +
    +

    +
    +
    +
    +
    import numpy as np
    +
    +data = {}
    +
    +for days, vals in m2:
    +    for d, v in zip(days.astype(int), vals):
    +        data.setdefault(d, []).append(v)
    +
    +stats = {}
    +for d, vals in data.items():
    +    stats[d] = (np.mean(vals), np.std(vals), np.median(vals))
    +
    +
    +
    plt.errorbar(stats.keys(), [v[0] for v in stats.values()], [v[1] for v in stats.values()], fmt="o", label="mean")
    +plt.plot(stats.keys(), [v[2] for v in stats.values()], "s", mfc="none", label="median", zorder=5)
    +plt.legend()
    +plt.axhline(0, ls="--", color="0.5")
    +plt.xlabel("day in month")
    +plt.ylabel("deviation from average price per month")
    +
    +
    Text(0, 0.5, 'deviation from average price per month')
    +
    +
    +

    +
    +
    +
    +
    luxor_info = yf.Ticker("X010.DE").info
    +ishares_info = yf.Ticker("EUNL.DE").info
    +
    +
    +
    luxor_info["averageDailyVolume10Day"]
    +
    +
    57938
    +
    +
    +
    +
    ishares_info["averageDailyVolume10Day"]
    +
    +
    464867
    +
    +
    + + +
    + +
    + +
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    +
    + +
    + + + + +
    + + + +
    +
    from iminuit import Minuit
    +from iminuit.cost import ExtendedBinnedNLL
    +import numpy as np
    +import numba as nb
    +from matplotlib import pyplot as plt
    +from progressbar import progressbar
    +from pyik.mplext import plot_hist
    +from IPython.display import display
    +from scipy.stats import norm
    +
    +
    +
    # numba currently does not support scipy, so we cannot access
    +# scipy.stats.norm.cdf in a JIT'ed function. As a workaround,
    +# we wrap special functions from scipy to implement the needed
    +# functions here.
    +def get(name, narg):
    +    from numba.extending import get_cython_function_address
    +    import ctypes
    +    addr = get_cython_function_address("scipy.special.cython_special", name)
    +    functype = ctypes.CFUNCTYPE(ctypes.c_double, *([ctypes.c_double] * narg))
    +    return functype(addr)
    +
    +
    +erf = get("__pyx_fuse_1erf", 1)
    +
    +
    +@nb.njit
    +def norm_cdf(x, mu, sigma):
    +    z = (x - mu) / (sigma * np.sqrt(2))
    +    r = np.empty_like(x)
    +    for i, zi in enumerate(z):
    +        r[i] = 0.5 * (1.0 + erf(zi))
    +    return r
    +
    +
    +def model(x, sig, bkg):
    +    return sig * norm_cdf(x, 0.5, 0.1) + bkg * x
    +
    +
    +bkg = 200
    +xe = np.linspace(0, 1, 21)
    +n = np.diff(model(xe, 20, bkg))
    +
    +nll = ExtendedBinnedNLL(n, xe, model)
    +
    +m1 = Minuit(nll, 1, 1)
    +m1.limits = (0, None)
    +display(m1.migrad())
    +m2 = Minuit(nll, 1, 1)
    +m2.limits[1] = (0, None)
    +display(m2.migrad())
    +m3 = Minuit(nll, 0, 1)
    +m3.fixed[0] = True
    +m3.limits = (0, None)
    +display(m3.migrad())
    +exp1 = m3.fval - m1.fval, m1.values[0], m1.errors[0]
    +exp2 = m3.fval - m2.fval, m2.values[0], m2.errors[0]
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 3.861e-06Nfcn = 91
    EDM = 3.86e-06 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0sig20120
    1bkg200180
    + + + + + + + + + + + + + + + + + + +
    sigbkg
    sig138-118 (-0.564)
    bkg-118 (-0.564)318
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 2.091e-06Nfcn = 99
    EDM = 2.09e-06 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0sig2012
    1bkg200180
    + + + + + + + + + + + + + + + + + + +
    sigbkg
    sig138-118 (-0.564)
    bkg-118 (-0.564)318
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    FCN = 3.132Nfcn = 42
    EDM = 2.31e-05 (Goal: 0.0002)
    Valid MinimumValid ParametersNo Parameters at limit
    Below EDM threshold (goal x 10)Below call limit
    CovarianceHesse okAccuratePos. def.Not forced
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NameValueHesse ErrorMinos Error-Minos Error+Limit-Limit+Fixed
    0sig0.00.10yes
    1bkg220150
    + + + + + + + + + + + + + + + + + + +
    sigbkg
    sig00
    bkg0220
    +
    +
    +
    +
    plot_hist(xe, n)
    +
    +

    +
    +
    +
    +
    print(f"with limit   : dQ = {exp1[0]:.3g}     "
    +      f"significance of signal amplitude = {exp1[1] / exp1[2]:.1f}σ")
    +print(f"without limit: dQ = {exp2[0]:.3g}     "
    +      f"significance of signal amplitude = {exp2[1] / exp2[2]:.1f}σ")
    +
    +
    with limit   : dQ = 3.13     significance of signal amplitude = 1.7σ
    +without limit: dQ = 3.13     significance of signal amplitude = 1.7σ
    +
    +
    +
    +
    rng = np.random.default_rng(1)
    +vbkg = np.diff(model(xe, 0, bkg))
    +r1 = []
    +r2 = []
    +for itoy in progressbar(range(50000)):
    +    nll.n = rng.poisson(vbkg)
    +    m1 = Minuit(nll, 0, bkg)
    +    m1.limits = (0, None)
    +    m1.migrad()
    +    m2 = Minuit(nll, 0, bkg)
    +    m2.limits[1] = (0, None)
    +    m2.migrad()
    +    m3 = Minuit(nll, 0, bkg)
    +    m3.fixed[0] = True
    +    m3.limits = (0, None)
    +    m3.migrad()
    +    if m3.valid:
    +        if m2.valid:
    +            r1.append((m3.fval - m1.fval, m1.values[0]))
    +        if m2.valid:
    +            r2.append((m3.fval - m2.fval, m2.values[0]))
    +r1 = np.array(r1)
    +r2 = np.array(r2)
    +
    +
    100% (50000 of 50000) |##################| Elapsed Time: 0:01:40 Time:  0:01:40
    +
    +
    +
    +
    plt.figure()
    +plt.title("dQ distribution")
    +plt.hist(r1[:,0], bins=50, alpha=0.5, density=True, label="with limits")
    +plt.hist(r2[:,0], bins=50, alpha=0.5, density=True, label="without limits")
    +plt.legend()
    +plt.axvline(exp1[0], color="C0")
    +plt.axvline(exp2[0], color="C1")
    +plt.semilogy()
    +
    +plt.figure()
    +plt.title("signal amplitude")
    +plt.hist(r1[:,1], bins=50, alpha=0.5, density=True, label="with limits")
    +plt.hist(r2[:,1], bins=50, alpha=0.5, density=True, label="without limits")
    +plt.legend()
    +plt.axvline(exp1[1], color="C0")
    +plt.axvline(exp2[1], color="C1")
    +plt.semilogy();
    +
    +

    +
    +
    +

    +
    +
    +
    +
    def pvalue2sigma(p):
    +    return norm.ppf(1 - p)
    +
    +
    +
    p1 = np.sum(r1[:,1] > exp1[1])/len(r1)
    +p2 = np.sum(r2[:,1] > exp2[1])/len(r2)
    +print(f"p-value based on amplitude with limit    {p1:.3f}    significance = {pvalue2sigma(p1):.1f}σ")
    +print(f"p-value based on amplitude without limit {p2:.3f}    significance = {pvalue2sigma(p2):.1f}σ")
    +
    +
    p-value based on amplitude with limit    0.031    significance = 1.9σ
    +p-value based on amplitude without limit 0.031    significance = 1.9σ
    +
    +
    +
    +
    p1 = np.sum(r1[:,0] > exp1[0])/len(r1)
    +p2 = np.sum(r2[:,0] > exp1[0])/len(r2)
    +print(f"p-value based on dQ with limit    {p1:.3f}    significance = {pvalue2sigma(p1):.1f}σ")
    +print(f"p-value based on dQ without limit {p2:.3f}    significance = {pvalue2sigma(p2):.1f}σ")
    +
    +
    p-value based on dQ with limit    0.037    significance = 1.8σ
    +p-value based on dQ without limit 0.077    significance = 1.4σ
    +
    +
    +

    The p-value computed from the fitted amplitude should agree with the p-value computed from dQ. This is only the case if the fit imposes a limit on the signal amplitude.

    + + + +
    + +
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    +
    + +
    + +
    + + + + +
    + + + +
    +
    import numpy as np
    +import matplotlib.pyplot as plt
    +
    +
    +
    data = """
    +  method     ntot        4             64           256           1024
    +boyett      10     87.4±0.1μs   96.7±0.08μs    180±0.8μs     472±0.7μs
    +boyett     100      919±2μs       864±2μs       962±3μs       1.26±0ms
    +boyett     1000   9.04±0.01ms   8.12±0.01ms   7.99±0.03ms   8.15±0.03ms
    +boyett    10000    99.0±0.4ms    89.9±0.2ms    86.9±0.3ms    86.4±0.6ms
    + boyett2      10     17.0±0.1ms      failed        failed        failed
    + boyett2     100     22.9±0.2ms   23.6±0.05ms    24.4±0.2ms      failed
    + boyett2     1000   85.4±0.09ms    93.0±0.4ms    97.6±0.5ms    102±0.8ms
    + boyett2    10000     722±1ms       798±2ms       846±4ms       904±4ms
    +patefield     10     70.6±0.2μs     852±10μs      2.22±0ms      5.44±0ms
    +patefield    100    98.8±0.08μs   3.65±0.03ms    13.1±0.2ms    41.9±0.4ms
    +patefield    1000    122±0.2μs    4.53±0.01ms   19.5±0.01ms    71.9±0.3ms
    +patefield   10000    156±0.2μs      5.39±0ms    22.6±0.02ms    90.7±0.3ms
    +patefield2     10      112±2μs       1.17±0ms      2.94±0ms    6.48±0.02ms
    +patefield2    100      157±2μs     6.26±0.01ms   22.0±0.02ms    71.9±0.4ms
    +patefield2    1000    166±0.9μs    7.13±0.05ms    32.1±0.1ms     118±1ms
    +patefield2   10000    201±0.8μs    7.53±0.02ms    32.6±0.2ms    137±0.1ms
    +"""
    +
    +
    +
    d = {}
    +ncell = None
    +ntot = []
    +for line in data.split("\n"):
    +    if not line or line.isspace():
    +        continue
    +    method, ntoti, *times = [x for x in line.split() if x]
    +    if ncell is None:
    +        ncell = [int(x) for x in times]
    +        continue
    +    else:
    +        ntoti = int(ntoti)
    +        if ntoti not in ntot:
    +            ntot.append(ntoti)
    +    times = [(float(x.split("±")[0]) * (1e-3 if x.endswith("μs") else 1)) if x != "failed" else np.nan for x in times]
    +    d.setdefault(method, {})[ntoti] = times
    +
    +times = {
    +    key: np.empty((len(ntot), len(ncell))) for key in d
    +}
    +
    +ntot = np.sort(ntot)
    +ncell = np.array(ncell)
    +
    +for key in d:
    +    v = times[key]
    +    for i, ntoti in enumerate(ntot):
    +        v[i] = d[key][ntoti]
    +
    +
    +
    fig, ax = plt.subplots(1, 2, figsize=(10, 4), sharey=True)
    +
    +plt.sca(ax[0])
    +for key, val in times.items():
    +    plt.plot(ntot, val[:, 0], label=key)
    +plt.loglog()
    +plt.legend(title="ncell = 4")
    +plt.ylabel("run time / ms")
    +plt.xlabel("total number of entries")
    +
    +plt.sca(ax[1])
    +for key, val in times.items():
    +    plt.plot(ncell, val[-1], label=key)
    +plt.loglog()
    +plt.legend(title="ntot = 10000")
    +plt.xlabel("total number of cells")
    +
    +
    Text(0.5, 0, 'total number of cells')
    +
    +
    +

    +
    +
    +
    +
    from scipy.optimize import least_squares
    +
    +def patefield(ntot, ncell, par):
    +    return par[0] * np.outer(np.log(ntot) ** par[1], ncell)
    +
    +def boyett(ntot, ncell, par):
    +    return par * np.outer(ntot, np.ones_like(ncell))
    +
    +model = {}
    +r1 = least_squares(lambda par: np.log(patefield(ntot, ncell, par) / times["patefield"]).ravel(), (1.0, 1.0), bounds=np.transpose([(0, np.inf), (-2, 2)]))
    +assert r1.success
    +print(r1)
    +model["patefield"] = lambda ntot, ncell: patefield(ntot, ncell, r1.x)
    +
    +r2 = least_squares(lambda par: np.log(boyett(ntot, ncell, par) / times["boyett"]).ravel(), 2.0, bounds=(0, np.inf))
    +assert r2.success
    +model["boyett"] = lambda ntot, ncell: boyett(ntot, ncell, r2.x[0])
    +
    +fig, ax = plt.subplots(1, 2, figsize=(10, 4), sharey=True)
    +
    +plt.sca(ax[0])
    +icell = 0
    +for key, val in times.items():
    +    l, = plt.plot(ntot, val[:, icell], "o", label=key)
    +    if key in model:
    +        m = model[key]
    +        plt.plot(ntot, m(ntot, ncell)[:, icell], "-", color=l.get_color())
    +plt.loglog()
    +plt.legend(title=f"ncell = {ncell[icell]}")
    +plt.ylabel("run time / ms")
    +plt.xlabel("total number of entries")
    +
    +plt.sca(ax[1])
    +intot = -1
    +for key, val in times.items():
    +    l, = plt.plot(ncell, val[intot], "o", label=key)
    +    if key in model:
    +        m = model[key]
    +        plt.plot(ncell, m(ntot, ncell)[intot], "-", color=l.get_color())
    +plt.loglog()
    +plt.legend(title=f"ntot = {ntot[intot]}")
    +plt.xlabel("total number of cells")
    +
    +
     active_mask: array([0, 0])
    +        cost: 1.3946620974325732
    +         fun: array([-4.03239230e-01, -1.21206891e-01,  3.07411523e-01,  7.97434019e-01,
    +        2.43799745e-01, -5.92996375e-01, -4.84587076e-01, -2.60966312e-01,
    +        6.07956704e-01, -2.33910748e-01, -3.07308913e-01, -2.25876351e-01,
    +        7.70147778e-01,  2.91843448e-04, -4.68183168e-02, -5.01314065e-02])
    +        grad: array([-1.91174556e-06, -5.01863655e-08])
    +         jac: array([[276.76918167,   0.83403245],
    +       [276.76918168,   0.83403245],
    +       [276.76918167,   0.83403244],
    +       [276.76918168,   0.83403245],
    +       [276.76918167,   1.52717962],
    +       [276.76918168,   1.52717963],
    +       [276.76918167,   1.52717962],
    +       [276.76918168,   1.52717963],
    +       [276.76918166,   1.93264473],
    +       [276.76918167,   1.93264474],
    +       [276.76918168,   1.93264474],
    +       [276.76918167,   1.93264474],
    +       [276.76918167,   2.2203268 ],
    +       [276.76918168,   2.2203268 ],
    +       [276.76918167,   2.2203268 ],
    +       [276.76918167,   2.2203268 ]])
    +     message: '`ftol` termination condition is satisfied.'
    +        nfev: 18
    +        njev: 16
    +  optimality: 1.911745556526512e-06
    +      status: 2
    +     success: True
    +           x: array([0.00361311, 1.41832278])
    +
    +
    +
    Text(0.5, 0, 'total number of cells')
    +
    +
    +

    +
    +
    +
    +
    for ntoti in ntot:
    +    for ncelli in ncell:
    +        print(ntoti, ncelli, model["boyett"](ntoti, ncelli) / ntoti  / (model["patefield"](ntoti, ncelli) / (np.log(ntoti) * ncelli)))
    +
    +
    10 4 [[2.05157943]]
    +10 64 [[2.05157943]]
    +10 256 [[2.05157943]]
    +10 1024 [[2.05157943]]
    +100 4 [[1.53518469]]
    +100 64 [[1.53518469]]
    +100 256 [[1.53518469]]
    +100 1024 [[1.53518469]]
    +1000 4 [[1.29567967]]
    +1000 64 [[1.29567967]]
    +1000 256 [[1.29567967]]
    +1000 1024 [[1.29567967]]
    +10000 4 [[1.14876958]]
    +10000 64 [[1.14876958]]
    +10000 256 [[1.14876958]]
    +10000 1024 [[1.14876958]]
    +
    +
    + + + +
    + +
    + + + + \ No newline at end of file diff --git a/unpublished/plot_random_table_results_files/figure-html/cell-5-output-2.png b/unpublished/plot_random_table_results_files/figure-html/cell-5-output-2.png new file mode 100644 index 0000000000000000000000000000000000000000..d5e7cd946e5708fb7d39abfd720c41a892ae83d0 GIT binary patch literal 48385 zcmcG$1yEH{`!{+N=@2PtL{hpzS|p?;q`O198x%ykyFt23x=TX3ySux&>-fIkf4;dh z_kJ^X<~oBSp0m&1E1u_9%Ye@^VyH;?NDv4FRs55%JOl!x3V}RxLVyE5QRec|0Dru< z7g4cSurjiD(zP{&NbA~Lf3>pzYO43f(a_e;)XI{TfrEjC?v07Py|o=TBcsLtynw;V z)|ioc8&?lp1kw7FsvQLKQWyI5EKeZU6as0q6c_%W=$yR2;H2^8;;Hpys%gLJ)n|07 zz(7{Qn5_Ajxgx7gt!z{IL&7>%I|a4PKld?)X0bW>hM|OYf+9A$4Cvt$ceY124>s5$ z5se5M-FG+6$KG~>TkeNFe{BjuTL3?hZ8q0Mo3R%TGG84B zxEUK%Z2)t)z#Ah2iia(MMW$`M8qNDJZeWmBBGPy@2=_a4-s%$OpOT=lOS@-!={pL#wN+-SHSHutZ4N*<-f1O(48{d?ADp|7M9= zJkTSu+G>fP|KVIAlte(rz<@l7(<(^$;H2|wO}QqZvh(^+|yKVha;WBE7Zqd%n9n4qLb$$H&L^68l!6kU54bL|E>(kWh{2hxm$b%Qn8-}b@R+Pj 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