notebooks/ABC.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## MSM.jl and ApproxBayes.jl\n",
    "\n",
    "This notebook shows how one can estimate a model using [MSM.jl](https://github.com/JulienPascal/MSM.jl) and [ApproxBayes.jl](https://github.com/marcjwilliams1/ApproxBayes.jl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using MSM\n",
    "using ApproxBayes\n",
    "using DataStructures\n",
    "using OrderedCollections\n",
    "using Distributions\n",
    "using Random\n",
    "using DataStructures\n",
    "using Statistics\n",
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True intercept = 0.5662374165061859\n",
      "True coefficient beta0 = [0.5908446386657102, 0.7667970365022592]\n",
      "Serial correlation coefficient theta0 = 0.0\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(::MSM.var\"#objective_function_MSM#6\"{MSMProblem}) (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(1234)  #for replicability reasons\n",
    "T = 100000          #number of periods\n",
    "P = 2               #number of dependent variables\n",
    "beta0 = rand(P)     #choose true coefficients by drawing from a uniform distribution on [0,1]\n",
    "alpha0 = rand(1)[]  #intercept\n",
    "theta0 = 0.0        #coefficient to create serial correlation in the error terms\n",
    "println(\"True intercept = $(alpha0)\")\n",
    "println(\"True coefficient beta0 = $(beta0)\")\n",
    "println(\"Serial correlation coefficient theta0 = $(theta0)\")\n",
    "\n",
    "# Generation of error terms\n",
    "# row = individual dimension\n",
    "# column = time dimension \n",
    "U = zeros(T)\n",
    "d = Normal()\n",
    "U[1] = rand(d, 1)[] #first error term\n",
    "# loop over time periods\n",
    "for t = 2:T\n",
    "    U[t] = rand(d, 1)[] + theta0*U[t-1]\n",
    "end\n",
    "\n",
    "# Let's simulate the dependent variables x_t\n",
    "x = zeros(T, P)\n",
    "\n",
    "d = Uniform(0, 5)\n",
    "for p = 1:P  \n",
    "    x[:,p] = rand(d, T)\n",
    "end\n",
    "\n",
    "# Let's calculate the resulting y_t\n",
    "y = zeros(T)\n",
    "\n",
    "for t=1:T\n",
    "    y[t] = alpha0 + x[t,1]*beta0[1] + x[t,2]*beta0[2] + U[t]\n",
    "end\n",
    "\n",
    "optionsSMM = MSMOptions(maxFuncEvals=1000, globalOptimizer = :dxnes, localOptimizer = :NelderMead)\n",
    "myProblem = MSMProblem(options = optionsSMM);\n",
    "\n",
    "# Priors\n",
    "dictPriors = OrderedDict{String,Array{Float64,1}}()\n",
    "dictPriors[\"alpha\"] = [0.5, 0.001, 1.0]\n",
    "dictPriors[\"beta1\"] = [0.5, 0.001, 1.0]\n",
    "dictPriors[\"beta2\"] = [0.5, 0.001, 1.0]\n",
    "set_priors!(myProblem, dictPriors)\n",
    "\n",
    "# Empirical moments\n",
    "dictEmpiricalMoments = OrderedDict{String,Array{Float64,1}}()\n",
    "dictEmpiricalMoments = OrderedDict{String,Array{Float64,1}}()\n",
    "dictEmpiricalMoments[\"mean\"] = [mean(y)] #informative on the intercept\n",
    "dictEmpiricalMoments[\"mean^2\"] = [mean(y.^2)] #informative on the intercept\n",
    "dictEmpiricalMoments[\"mean_x1y\"] = [mean(x[:,1] .* y)] #informative on betas\n",
    "dictEmpiricalMoments[\"mean_x2y\"] = [mean(x[:,2] .* y)] #informative on betas\n",
    "dictEmpiricalMoments[\"mean_x1y^2\"] = [mean((x[:,1] .* y).^2)] #informative on betas\n",
    "dictEmpiricalMoments[\"mean_x2y^2\"] = [mean((x[:,2] .* y).^2)] #informative on betas\n",
    "set_empirical_moments!(myProblem, dictEmpiricalMoments)\n",
    "\n",
    "W = Matrix(1.0 .* I(length(dictEmpiricalMoments)))#initialization\n",
    "#Special case: diagonal matrix\n",
    "#(you may choose something else)\n",
    "for (indexMoment, k) in enumerate(keys(myProblem.empiricalMoments))\n",
    "    W[indexMoment,indexMoment] = 1.0/(myProblem.empiricalMoments[k][1])^2\n",
    "end\n",
    "set_weight_matrix!(myProblem, W)\n",
    "\n",
    "\n",
    "# x[1] corresponds to the intercept, x[2] corresponds to beta1, x[3] corresponds to beta2\n",
    "function functionLinearModel(x; uniform_draws::Array{Float64,1}, simX::Array{Float64,2}, nbDraws::Int64 = length(uniform_draws), burnInPerc::Int64 = 0)\n",
    "    T = nbDraws\n",
    "    P = 2       #number of dependent variables\n",
    "\n",
    "    alpha = x[1]\n",
    "    beta = x[2:end]\n",
    "    theta = 0.0     #coefficient to create serial correlation in the error terms\n",
    "\n",
    "    # Creation of error terms\n",
    "    # row = individual dimension\n",
    "    # column = time dimension\n",
    "    U = zeros(T)\n",
    "    d = Normal()\n",
    "    # Inverse cdf (i.e. quantile)\n",
    "    gaussian_draws = quantile.(d, uniform_draws)\n",
    "    U[1] = gaussian_draws[1] #first error term\n",
    "\n",
    "    # loop over time periods\n",
    "    for t = 2:T\n",
    "        U[t] = gaussian_draws[t] + theta*U[t-1]\n",
    "    end\n",
    "\n",
    "    # Let's calculate the resulting y_t\n",
    "    y = zeros(T)\n",
    "\n",
    "    for t=1:T\n",
    "        y[t] = alpha + simX[t,1]*beta[1] + simX[t,2]*beta[2] + U[t]\n",
    "    end\n",
    "\n",
    "    # Get rid of the burn-in phase:\n",
    "    #------------------------------\n",
    "    startT = max(1, Int(nbDraws * (burnInPerc / 100)))\n",
    "\n",
    "    # Moments:\n",
    "    #---------\n",
    "    output = OrderedDict{String,Float64}()\n",
    "    output[\"mean\"] = mean(y[startT:nbDraws])\n",
    "    output[\"mean^2\"] = mean(y[startT:nbDraws].^2)\n",
    "    output[\"mean_x1y\"] = mean(simX[startT:nbDraws,1] .* y[startT:nbDraws])\n",
    "    output[\"mean_x2y\"] = mean(simX[startT:nbDraws,2] .* y[startT:nbDraws])\n",
    "    output[\"mean_x1y^2\"] = mean((simX[startT:nbDraws,1] .* y[startT:nbDraws]).^2)\n",
    "    output[\"mean_x2y^2\"] = mean((simX[startT:nbDraws,2] .* y[startT:nbDraws]).^2)\n",
    "\n",
    "    return output\n",
    "end\n",
    "\n",
    "# Let's freeze the randomness during the minimization\n",
    "d_Uni = Uniform(0,1)\n",
    "nbDraws = T #number of draws in the simulated data\n",
    "uniform_draws = rand(d_Uni, nbDraws)\n",
    "simX = zeros(length(uniform_draws), 2)\n",
    "d = Uniform(0, 5)\n",
    "for p = 1:2\n",
    "  simX[:,p] = rand(d, length(uniform_draws))\n",
    "end\n",
    "\n",
    "set_simulate_empirical_moments!(myProblem, x -> functionLinearModel(x, uniform_draws = uniform_draws, simX = simX))\n",
    "construct_objective_function!(myProblem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of threads = 1\n"
     ]
    }
   ],
   "source": [
    "n_threads = Threads.nthreads()\n",
    "println(\"Number of threads = $(n_threads)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Vector{Vector{Float64}}:\n",
       " [3.952878050375794]\n",
       " [18.567217663660667]\n",
       " [11.061079621971047]\n",
       " [11.463383169522919]\n",
       " [203.36060647937504]\n",
       " [219.24881943710466]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "targetdata = collect(values(dictEmpiricalMoments))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0006764970573433793"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "initial_guess = myProblem.objective_function([dictPriors[k][1] for k in keys(dictPriors)])\n",
    "ϵ = initial_guess/1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Prior((Uniform{Float64}(a=0.001, b=1.0), Uniform{Float64}(a=0.001, b=1.0), Uniform{Float64}(a=0.001, b=1.0)))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Use dictPriors to create a prior for ApproxBayes\n",
    "priors = \"\"\n",
    "for (i, k) in enumerate(keys(myProblem.priors))\n",
    "  if i==1\n",
    "  priors = string(\"Uniform($(dictPriors[k][2]), $(dictPriors[k][3]))\" )\n",
    "  else\n",
    "      priors = string(priors, \", \", \"Uniform($(dictPriors[k][2]), $(dictPriors[k][3]))\" )\n",
    "  end\n",
    "end\n",
    "\n",
    "priors = eval(Meta.parse(string(\"Prior([\", priors, \"])\")))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "##################################################\n",
      "Use ABC rejection to get first population\n",
      "Preparing to run in parallel on 1 processors\n",
      "Running ABC SMC... \n",
      "\n",
      "Preparing to run in parallel on 1 processors\n",
      "##################################################\n",
      "Total number of simulations: 4.38e+02\n",
      "Cumulative number of simulations = [101, 438]\n",
      "Acceptance ratio: 2.28e-01\n",
      "Tolerance schedule = [0.27, 0.03]\n",
      "\n",
      "Median (95% intervals):\n",
      "Parameter 1: 0.61 (0.09,0.99)\n",
      "Parameter 2: 0.71 (0.25,0.98)\n",
      "Parameter 3: 0.65 (0.26,0.94)\n",
      "##################################################\n",
      "\n",
      "##################################################\n",
      "Total number of simulations: 1.53e+03\n",
      "Cumulative number of simulations = [101, 438, 1532]\n",
      "Acceptance ratio: 6.53e-02\n",
      "Tolerance schedule = [0.27, 0.03, 0.01]\n",
      "\n",
      "Median (95% intervals):\n",
      "Parameter 1: 0.54 (0.06,0.92)\n",
      "Parameter 2: 0.58 (0.39,0.85)\n",
      "Parameter 3: 0.78 (0.54,0.98)\n",
      "##################################################\n",
      "\n",
      "##################################################\n",
      "Total number of simulations: 3.15e+03\n",
      "Cumulative number of simulations = [101, 438, 1532, 3148]\n",
      "Acceptance ratio: 3.18e-02\n",
      "Tolerance schedule = [0.27, 0.03, 0.01, 0.0]\n",
      "\n",
      "Median (95% intervals):\n",
      "Parameter 1: 0.52 (0.07,0.98)\n",
      "Parameter 2: 0.60 (0.43,0.73)\n",
      "Parameter 3: 0.77 (0.63,0.91)\n",
      "##################################################\n",
      "\n",
      "##################################################\n",
      "Total number of simulations: 5.50e+03\n",
      "Cumulative number of simulations = [101, 438, 1532, 3148, 5502]\n",
      "Acceptance ratio: 1.82e-02\n",
      "Tolerance schedule = [0.27, 0.03, 0.01, 0.0, 0.0]\n",
      "\n",
      "Median (95% intervals):\n",
      "Parameter 1: 0.62 (0.06,0.98)\n",
      "Parameter 2: 0.58 (0.49,0.69)\n",
      "Parameter 3: 0.77 (0.65,0.87)\n",
      "##################################################\n",
      "\n",
      " 49.931421 seconds (30.30 M allocations: 162.646 GiB, 7.35% gc time, 14.31% compilation time)\n"
     ]
    },
    {
     "data": {
      "text/plain": []
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Total number of simulations: 1.20e+04\n",
      "Cumulative number of simulations = [101, 438, 1532, 3148, 5502, 12024]\n",
      "Acceptance ratio: 8.32e-03\n",
      "Tolerance schedule = [0.27, 0.03, 0.01, 0.0, 0.0, 0.0]\n",
      "\n",
      "Median (95% intervals):\n",
      "Parameter 1: 0.57 (0.24,0.85)\n",
      "Parameter 2: 0.58 (0.53,0.64)\n",
      "Parameter 3: 0.76 (0.70,0.83)\n"
     ]
    }
   ],
   "source": [
    "# Approximate Bayesian Computation Sequential Monte Carlo\n",
    "setup = ABCSMC((params, constants, targetdata) -> (myProblem.objective_function(params), 1), #simulation function\n",
    "  length(dictPriors), # number of parameters\n",
    "  ϵ, #target ϵ\n",
    "  priors; # Prior for each of the parameters\n",
    "  maxiterations = 10^6, #Maximum number of iterations before the algorithm terminates\n",
    "  )\n",
    "\n",
    "# run ABC inference\n",
    "@time res = runabc(setup, targetdata, verbose=true, progress=true, parallel=true)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True intercept = 0.5662374165061859\n",
      "True coefficient beta0 = [0.5908446386657102, 0.7667970365022592]\n"
     ]
    }
   ],
   "source": [
    "println(\"True intercept = $(alpha0)\")\n",
    "println(\"True coefficient beta0 = $(beta0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Attribute alias `nbins` detected in the user recipe defined for the signature (::ApproxBayes.ABCSMCresults). To ensure expected behavior it is recommended to use the default attribute `bins`.\n",
      "└ @ Plots /home/julien/.julia/packages/Plots/g581z/src/pipeline.jl:26\n"
     ]
    },
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      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(res)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.6.1\n",
      "Commit 6aaedecc44 (2021-04-23 05:59 UTC)\n",
      "Platform Info:\n",
      "  OS: Linux (x86_64-pc-linux-gnu)\n",
      "  CPU: Intel(R) Core(TM) i7-8850H CPU @ 2.60GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-11.0.1 (ORCJIT, skylake)\n"
     ]
    }
   ],
   "source": [
    "versioninfo()"
   ]
  }
 ],
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   "mimetype": "application/julia",
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