First implementation of KSB

README.md

@@ -27,9 +27,9 @@ $ conda env create -f environment.yml
 $ conda activate shear
 ```
 
-## Install the `shrbk` library 
+## Install the `shrbk` library
 
-`shrbk` is the library made for this book. 
+`shrbk` is the library made for this book.
 Install it ither in development mode
 
 ```bash

notebooks/taylor_series.ipynb

@@ -0,0 +1,219 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Taylor Series for Approximations\n",
+    "\n",
+    "Taylor series are commonly used in physics to approximate functions making them easier to handle specially when solving equations. In this notebook we give a visual example on how it works and the biases that it introduces.\n",
+    "\n",
+    "## Theoretical Formula\n",
+    "\n",
+    "Consider a function $f$ that is $n$ times differentiable in a point $a$. Then by Taylor's theorem, for any point $x$ in the domain of f, we have the Taylor expansion about the point $a$ is defined as:\n",
+    "\\begin{equation}\n",
+    "f(x) = f(a) + \\sum_{k=1}^n \\frac{f^{k}(a)}{k!}(x-a)^k + o\\left((x-a)^n\\right) \\quad,\n",
+    "\\end{equation}\n",
+    "where $f^{(k)}$ is the derivative of order $k$ of $f$. Usually, we consider $a=0$ which gives:\n",
+    "\\begin{equation}\n",
+    "f(x) = f(0) + \\sum_{k=1}^n \\frac{f^{k}(0)}{k!}(x)^k + o\\left((x)^n\\right) \\quad.\n",
+    "\\end{equation}\n",
+    "\n",
+    "For example, the exponential, $e$ is infinitely differentiable with $e^{(k)}=e$ and $e^0=1$. This gives us the following Taylor expansion:\n",
+    "\\begin{equation}\n",
+    "e(x) = 1 + \\sum_{k=1}^\\infty \\frac{x^k}{k!} \\quad.\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Visualising Taylor Expansion Approximation and its Bias\n",
+    "\n",
+    "Let us see visually how the Taylor expansion approximatees a given function. We start by defining our function below, for example we will consider the exponential function, $e$ again up to order 3."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "code_folding": [
+     0
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "#### FOLDED CELL\n",
+    "%matplotlib inline\n",
+    "import matplotlib.pyplot as plt\n",
+    "from IPython.display import Markdown as md\n",
+    "from sympy import Symbol, series, lambdify, latex\n",
+    "from sympy.functions import *\n",
+    "from ipywidgets import interactive_output\n",
+    "import ipywidgets as widgets\n",
+    "from sympy.parsing.sympy_parser import parse_expr\n",
+    "import numpy as np\n",
+    "\n",
+    "x = Symbol('x')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "order = 3\n",
+    "func = exp(x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "code_folding": [
+     0
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "#### FOLDED CELL\n",
+    "taylor_exp = series(func,x,n=order+1)\n",
+    "approx = lambdify(x, sum(taylor_exp.args[:-1]), \"numpy\")\n",
+    "func_np = lambdify(x, func, \"numpy\")\n",
+    "latex_func = '$'+latex(func)+'$'\n",
+    "latex_taylor = '\\\\begin{equation} '+latex(taylor_exp)+' \\end{equation}'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "variables": {
+     " latex_func ": "$e^{x}$",
+     "latex_taylor": "\\begin{equation} 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + O\\left(x^{4}\\right) \\end{equation}"
+    }
+   },
+   "source": [
+    "The Taylor expansion of  {{ latex_func }} is :\n",
+    "{{latex_taylor}}\n",
+    "\n",
+    "Now let's plot the function and its expansion while considering a point, noted $p$, to study the biais that we introduce when we approximate the function by its expansion:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "code_folding": [
+     0
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "5f5d027e2c284af4869d9e9c6ea083f9",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "Output(layout=Layout(height='650px'))"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "d8b970e851ef4e688880bc1db4a46238",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "HBox(children=(FloatSlider(value=3.0, description='Point Absciss', max=4.0, min=-4.0, step=0.2), IntSlider(val…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#### FOLDED CELL\n",
+    "order = widgets.IntSlider(min=0,max=20,step=1,value=3,description='Order')\n",
+    "x_min = -4\n",
+    "x_max = 4\n",
+    "x1 = widgets.FloatSlider(min=x_min,max=x_max,value=3,step=0.2,description='Point Absciss')\n",
+    "func = widgets.Text('exp(x)',description='Function')\n",
+    "text_offset = np.array([-0.15,2.])\n",
+    "\n",
+    "\n",
+    "ui = widgets.HBox([x1, order, func])\n",
+    "\n",
+    "def f(order=widgets.IntSlider(min=1,max=10,step=1,value=3)\n",
+    "      ,x1=1.5\n",
+    "      ,func='exp(x)'):\n",
+    "    func_sp = parse_expr(func)\n",
+    "    taylor_exp = series(func_sp,x,n=order+1)\n",
+    "    approx = lambdify(x, sum(taylor_exp.args[:-1]), \"numpy\")\n",
+    "    func_np = lambdify(x, func_sp, \"numpy\")\n",
+    "    n_points = 1000\n",
+    "    x_array = np.linspace(x_min,x_max,n_points)\n",
+    "    approx_array = np.array([approx(z) for z in x_array])\n",
+    "    func_array = np.array([func_np(z) for z in x_array])\n",
+    "    func_x1 = func_np(x1)\n",
+    "    approx_x1 = approx(x1)\n",
+    "    plt.figure(42,figsize=(10,10))\n",
+    "    plt.plot(x_array,approx_array,color='blue',label='Taylor Expansion')\n",
+    "    plt.plot(x_array,func_array,color='green',label=func)\n",
+    "    plt.plot(0,approx(0),color='black',marker='o')\n",
+    "    plt.annotate(r'(0,0)',[0,approx(0)],xytext=text_offset)\n",
+    "    plt.plot([x1,x1]\n",
+    "             ,[-np.max(np.abs([np.min(func_array),np.max(func_array)])),min(approx_x1, func_x1)]\n",
+    "             ,'--',color='black',marker='x')\n",
+    "    plt.plot([x1,x1],[approx_x1, func_x1],'r--',marker='x')\n",
+    "    plt.annotate(r'$p_{approx}$',[x1,approx(x1)],xytext=[x1,approx(x1)]-text_offset)\n",
+    "    plt.annotate(r'$p$',[x1,func_np(x1)],xytext=[x1,func_np(x1)]-text_offset)\n",
+    "    plt.xlim([x_min,x_max])\n",
+    "    plt.ylim(-np.max(np.abs([np.min(func_array),np.max(func_array)]))\n",
+    "             ,np.max(np.abs([np.min(func_array),np.max(func_array)])))\n",
+    "    plt.legend()\n",
+    "    plt.show()\n",
+    "    print('Approximation  bias : {}'.format(func_x1-approx_x1))\n",
+    "    return None\n",
+    "interactive_plot = widgets.interactive_output(f, {'order': order, 'x1': x1, 'func': func})\n",
+    "interactive_plot.layout.height = '650px'\n",
+    "display(interactive_plot, ui)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Notice that the further $p$ gets away from the point of the expansion (in that case $0$), the higher the approximation bias gets. Samely, the lower the order of approximation is, the higher the approximation bias gets."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.8.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}

shearbook/_bibliography/z_moments_methods.bib

@@ -18,3 +18,38 @@ @ARTICLE{2011MNRAS.410.2156V
        adsurl = {https://ui.adsabs.harvard.edu/abs/2011MNRAS.410.2156V},
       adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
+
+@ARTICLE{1995ApJ.KSB,
+       author = {{Kaiser}, Nick and {Squires}, Gordon and {Broadhurst}, Tom},
+        title = "{A Method for Weak Lensing Observations}",
+      journal = {\apj},
+     keywords = {COSMOLOGY: OBSERVATIONS, COSMOLOGY: DARK MATTER, GALAXIES: FORMATION, COSMOLOGY: GRAVITATIONAL LENSING, COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE, Astrophysics},
+         year = 1995,
+        month = aug,
+       volume = {449},
+        pages = {460},
+          doi = {10.1086/176071},
+archivePrefix = {arXiv},
+       eprint = {astro-ph/9411005},
+ primaryClass = {astro-ph},
+       adsurl = {https://ui.adsabs.harvard.edu/abs/1995ApJ...449..460K},
+      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@ARTICLE{2003MNRAS.343..459H,
+       author = {{Hirata}, Christopher and {Seljak}, Uro{\v{s}}},
+        title = "{Shear calibration biases in weak-lensing surveys}",
+      journal = {\mnras},
+     keywords = {gravitational lensing, methods: data analysis, Astrophysics},
+         year = 2003,
+        month = aug,
+       volume = {343},
+       number = {2},
+        pages = {459-480},
+          doi = {10.1046/j.1365-8711.2003.06683.x},
+archivePrefix = {arXiv},
+       eprint = {astro-ph/0301054},
+ primaryClass = {astro-ph},
+       adsurl = {https://ui.adsabs.harvard.edu/abs/2003MNRAS.343..459H},
+      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}

shearbook/_toc.yml

@@ -5,6 +5,7 @@
   chapters:
     - file: intro/about
     - file: intro/run_code
+    - file: intro/weak_lensing
 
 - part: Shape Measurement
   chapters:
@@ -17,8 +18,6 @@
     - file: methods/moments-methods
       sections:
         - file: methods/ksb
-        - file: methods/ksb-variants
-        - file: methods/viola2011
         - file: methods/moments-methods-ref
     - file: model/model-intro
       sections:
@@ -35,6 +34,10 @@
       sections:
         - file: calibration/calibration-ref
 
+- part: Appendix
+  chapters:
+    - file: appendix/taylor_series
+
 - part: Contributing
   chapters:
     - file: contrib/contributors