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yiyuezhuo/test_mle_cleaned.ipynb

Created February 14, 2020 04:33
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Some Julia Optim.jl MLE example
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test_mle_cleaned.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Normal MLE"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Optim\n",
"using Statistics\n",
"using Distributions\n",
"using Plots\n",
"using LinearAlgebra"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"data = randn(100) .* 0.5 .+ 1. ;"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: success\n",
"\n",
" * Candidate solution\n",
" Minimizer: [9.33e-01, 5.05e-01]\n",
" Minimum: 7.366251e+01\n",
"\n",
" * Found with\n",
" Algorithm: Fminbox with L-BFGS\n",
" Initial Point: [0.00e+00, 1.00e+00]\n",
"\n",
" * Convergence measures\n",
" |x - x'| = 5.43e-10 ≰ 0.0e+00\n",
" |x - x'|/|x'| = 5.12e-10 ≰ 0.0e+00\n",
" |f(x) - f(x')| = 2.84e-14 ≰ 0.0e+00\n",
" |f(x) - f(x')|/|f(x')| = 3.86e-16 ≰ 0.0e+00\n",
" |g(x)| = 2.10e-10 ≤ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 2 (vs limit Inf)\n",
" Iterations: 4\n",
" f(x) calls: 196\n",
" ∇f(x) calls: 196\n"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood = (μ, σ) -> logpdf.( Normal(μ, σ), data ) |> sum\n",
"f = (p) -> -likelihood(p[1], p[2])\n",
"res = optimize(f, [-Inf, 1e-6], [Inf, Inf], [0.0, 1.0], Fminbox(LBFGS()), autodiff = :forward)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.9328596990037146"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mean(data)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.5079926618388317"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"std(data)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" 0.9328596990037143\n",
" 0.5054463166722577"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Optim.minimizer(res)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# MvNormal MLE using over-parameterization"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×100 Array{Float64,2}:\n",
" 0.624657 0.951313 1.04176 1.47361 … 2.76285 -0.320447 0.216333\n",
" 1.03725 1.16691 2.42946 0.387327 2.03342 0.470565 0.794415"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = rand(MvNormal([1.; 1.], [1. 0.5; 0.5 1.]), 100)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#7 (generic function with 1 method)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood = (μ, Σ) -> begin\n",
" dist = MvNormal(μ, Σ)\n",
" sum([logpdf(dist, data[:, i]) for i in 1:size(data, 2)])\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#11 (generic function with 1 method)"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = (p) -> begin\n",
" μ = p[1:2]\n",
" f = reshape(p[3:6], 2, 2)\n",
" Σ = f' * f\n",
" -likelihood(μ, Σ)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: failure (objective increased between iterations) (line search failed)\n",
"\n",
" * Candidate solution\n",
" Minimizer: [9.42e-01, 8.06e-01, 8.68e-01, ...]\n",
" Minimum: 2.486111e+02\n",
"\n",
" * Found with\n",
" Algorithm: L-BFGS\n",
" Initial Point: [0.00e+00, 0.00e+00, 1.00e+00, ...]\n",
"\n",
" * Convergence measures\n",
" |x - x'| = 2.51e-09 ≰ 0.0e+00\n",
" |x - x'|/|x'| = 2.67e-09 ≰ 0.0e+00\n",
" |f(x) - f(x')| = 1.42e-13 ≰ 0.0e+00\n",
" |f(x) - f(x')|/|f(x')| = 5.72e-16 ≰ 0.0e+00\n",
" |g(x)| = 1.19e-08 ≰ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 0 (vs limit Inf)\n",
" Iterations: 14\n",
" f(x) calls: 47\n",
" ∇f(x) calls: 47\n"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x0 = [0., 0., 1., 0., 0., 1.]\n",
"res = optimize(f, x0, LBFGS(), autodiff=:forward)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Array{Float64,2}:\n",
" 0.798341 0.39631 \n",
" 0.39631 0.816564"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_hat = Optim.minimizer(res)[1:2]\n",
"f = reshape(Optim.minimizer(res)[3:6], 2, 2)\n",
"Σ_hat = f' * f"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" 0.9416516365632621\n",
" 0.8059733390223162"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_hat"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×1 Array{Float64,2}:\n",
" 0.9416516369020049\n",
" 0.8059733394231644"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mean(data, dims=2)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Array{Float64,2}:\n",
" 0.806405 0.400313\n",
" 0.400313 0.824812"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cov(data, dims=2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# MvNormal MLE using Cholesky parameterization"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [],
"source": [
"using Distributions.PDMats"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#13 (generic function with 1 method)"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = (p) -> begin\n",
" μ = p[1:2]\n",
" Σ = PDMat(Cholesky([p[3] p[4]; 0 p[5]], 'U', 0))\n",
" dist = MvNormal(μ, Σ)\n",
" -sum([logpdf(dist, data[:, i]) for i in 1:size(data, 2)])\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: success\n",
"\n",
" * Candidate solution\n",
" Minimizer: [9.42e-01, 8.06e-01, 8.93e-01, ...]\n",
" Minimum: 2.486111e+02\n",
"\n",
" * Found with\n",
" Algorithm: Fminbox with L-BFGS\n",
" Initial Point: [0.00e+00, 0.00e+00, 1.00e+00, ...]\n",
"\n",
" * Convergence measures\n",
" |x - x'| = 1.11e-09 ≰ 0.0e+00\n",
" |x - x'|/|x'| = 6.26e-10 ≰ 0.0e+00\n",
" |f(x) - f(x')| = 5.68e-14 ≰ 0.0e+00\n",
" |f(x) - f(x')|/|f(x')| = 2.29e-16 ≰ 0.0e+00\n",
" |g(x)| = 3.69e-09 ≤ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 1 (vs limit Inf)\n",
" Iterations: 4\n",
" f(x) calls: 318\n",
" ∇f(x) calls: 318\n"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x0 = [0., 0., 1., 0., 1.]\n",
"lower = [-Inf, -Inf, 0, -Inf, 0]\n",
"upper = [Inf, Inf, Inf, Inf, Inf]\n",
"res = optimize(f, lower, upper, x0, Fminbox(LBFGS()), autodiff=:forward)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"PDMat{Float64,Array{Float64,2}}(2, [0.7983405442326638 0.39630991515272923; 0.39630991515272923 0.8165635604687043], Cholesky{Float64,Array{Float64,2}}([0.8934990454570524 0.4435482244415879; 0.0 0.7872918982584661], 'U', 0))"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p_fit = Optim.minimizer(res)\n",
"μ_fit = p_fit[1:2]\n",
"Σ_fit = PDMat(Cholesky([p_fit[3] p_fit[4]; 0 p_fit[5]], 'U', 0))"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" 0.9416516369020044\n",
" 0.8059733394231686"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_fit"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×2 Array{Float64,2}:\n",
" 0.798341 0.39631 \n",
" 0.39631 0.816564"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Σ_fit.mat"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mixture model MLE"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"num_arr = [50, 30, 20]\n",
"mu_arr = [-1, 0, 1]\n",
"sigma_arr = [0.2, 0.3, 0.4]\n",
"\n",
"data = zeros(sum(num_arr))\n",
"idx = 1\n",
"for (num, mu, sigma) in zip(num_arr, mu_arr, sigma_arr)\n",
" for i in 1:num\n",
" data[idx] = randn()*sigma+mu\n",
" idx += 1\n",
" end\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [],
"source": [
"theme(:dark)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
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},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"histogram(data, bins=20)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align}\n",
"P_\\theta(X) = \\int_Z P_\\theta(X,Z) = \\int_Z P_\\theta(X|Z)P(Z) \n",
"\\end{align}\n",
"$$\n",
"\n",
"$$\n",
"P(X|Z)P(Z)=P(X_1,\\dots,X_N|Z_1,\\dots,Z_N)P(Z_1,\\dots,Z_N)=(\\prod P(X_i|Z_i)) (\\prod P(Z_i))\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align}\n",
"P(X) = \\int_Z \\prod_i P(X_i|Z_i)P(Z_i) \n",
"&= \\sum_{Z_1} \\dots \\sum_{Z_N} \\prod_i^N P(X_i|Z_i)P(Z_i) \\\\\n",
"&= \\sum_{Z_1} \\dots \\sum_{Z_{N-1}} \\prod_i^{N-1} P(X_i|Z_i)P(Z_i) \\\\\n",
"&= (\\sum_{Z_N} P(X_N|Z_N)P(Z_N)) \\sum_{Z_1} \\dots \\sum_{Z_{N-1}} \\prod_i^{N-1} P(X_i|Z_i)P(Z_i) \\\\\n",
"&= \\dots \\\\\n",
"&= \\prod_{i} \\sum_{Z_i} P(X_i|Z_i)P(Z_i)\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align}\n",
"\\log P(X) &= \\log \\prod_i \\sum_{Z_i} P(X_i|Z_i)P(Z_i) \\\\\n",
"&= \\sum_i \\log \\sum_{Z_i} \\exp \\log (P(X_i|Z_i)P(Z_i)) \\\\\n",
"&= \\sum_i \\mathrm{logsumexp}_{Z_i} \\log(P(X_i|Z_i)P(Z_i))\n",
"\\end{align}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
"using StatsFuns"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.4076059644443806"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"logsumexp([1 2 3])"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.40760596444438"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[1 2 3] .|> exp |> sum |> log"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.40760596444438"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(sum(exp.([1 2 3])))"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#17 (generic function with 1 method)"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood = (μ_arr, σ_arr) -> begin\n",
" dist_arr = [Normal(μ, σ) for (μ, σ) in zip(μ_arr, σ_arr)]\n",
" num_classes = length(dist_arr)\n",
" log_prior = log(1. / num_classes)\n",
" lp = 0\n",
" for d in data\n",
" lp += logsumexp([logpdf(dist, d) + log_prior for dist in dist_arr])\n",
" end\n",
" lp\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-207.59091458221434"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood([1, 1, 1], [1, 1, 1])"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-140.63496716867343"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood([-1, 0, 1], [1, 1, 1])"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-103.40893932627914"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood(mu_arr, sigma_arr)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#23 (generic function with 1 method)"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"encode = (p) -> (p[1:3], p[4:6])"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#25 (generic function with 1 method)"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = (p) -> begin\n",
" μ_arr, σ_arr = encode(p)\n",
" -likelihood(μ_arr, σ_arr)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: success\n",
"\n",
" * Candidate solution\n",
" Minimizer: [-3.70e-01, -1.01e+00, 7.02e-01, ...]\n",
" Minimum: 1.004915e+02\n",
"\n",
" * Found with\n",
" Algorithm: Fminbox with L-BFGS\n",
" Initial Point: [1.14e-01, -3.57e-01, 7.81e-01, ...]\n",
"\n",
" * Convergence measures\n",
" |x - x'| = 8.30e-10 ≰ 0.0e+00\n",
" |x - x'|/|x'| = 5.57e-10 ≰ 0.0e+00\n",
" |f(x) - f(x')| = 4.26e-14 ≰ 0.0e+00\n",
" |f(x) - f(x')|/|f(x')| = 4.24e-16 ≰ 0.0e+00\n",
" |g(x)| = 6.52e-09 ≤ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 1 (vs limit Inf)\n",
" Iterations: 4\n",
" f(x) calls: 344\n",
" ∇f(x) calls: 344\n"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lower = [-Inf, -Inf, -Inf, 0, 0, 0]\n",
"upper = [Inf, Inf, Inf, Inf, Inf, Inf]\n",
"x0 = [randn(3); ones(3)]\n",
"res = optimize(f, lower, upper, x0, autodiff=:forward)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6-element Array{Float64,1}:\n",
" -0.37044298988147994\n",
" -1.01267871166701 \n",
" 0.7019056655903312 \n",
" 0.5083640800869492 \n",
" 0.18825829381585862\n",
" 0.5156193681707262 "
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Optim.minimizer(res)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([-0.37044298988147994, -1.01267871166701, 0.7019056655903312], [0.5083640800869492, 0.18825829381585862, 0.5156193681707262])"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_arr_fit, σ_arr_fit = encode(Optim.minimizer(res))"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-3.0:0.06060606060606061:3.0"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = range(-3, 3, length=100)"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Array{Float64,1},1}:\n",
" [3.8472993133532097e-22, 7.6078125022908e-21, 1.372408845984938e-19, 2.258536424078426e-18, 3.390709409718904e-17, 4.643804481873946e-16, 5.801992505420009e-15, 6.613026802451713e-14, 6.876117424567452e-13, 6.522381065864739e-12 … 3.2706106311613235e-65, 1.665413479987071e-67, 7.736333609161463e-70, 3.2784471370139403e-72, 1.2674214813178594e-74, 4.469859005606432e-77, 1.4380913270912092e-79, 4.220840998109037e-82, 1.130137796458194e-84, 2.7604741810798818e-87]\n",
" [2.56486620890214e-22, 1.894807335009776e-21, 1.3438193856020688e-20, 9.149392306566254e-20, 5.980246707952049e-19, 3.752506515078028e-18, 2.260472939812651e-17, 1.3072320081271332e-16, 7.257407967851866e-16, 3.867995003613286e-15 … 3.867995003613286e-15, 7.257407967851866e-16, 1.3072320081271332e-16, 2.260472939812651e-17, 3.752506515078028e-18, 5.980246707952049e-19, 9.149392306566254e-20, 1.3438193856020688e-20, 1.894807335009776e-21, 2.56486620890214e-22] \n",
" [1.9236496566766049e-22, 8.652924351444221e-22, 3.8039062511454e-21, 1.634280781993768e-20, 6.86204422992469e-20, 2.815855229739687e-19, 1.129268212039237e-18, 4.426024914453011e-18, 1.695354704859488e-17, 6.346543600414966e-17 … 0.0013411631096047344, 0.0007642155636951401, 0.0004255789391686351, 0.00023161908830753624, 0.000123196547883328, 6.404020298784535e-5, 3.25339527571786e-5, 1.6152915981621987e-5, 7.83781497728003e-6, 3.716798786835744e-6] "
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pdf_arr = [pdf.(Normal(μ,σ), x) for (μ, σ) in zip(mu_arr, sigma_arr)]"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1-element Array{Array{Float64,1},1}:\n",
" [2.778605059643985e-22, 3.4559707574816664e-21, 5.149432823521996e-20, 7.887910516546753e-19, 1.1524579736761165e-17, 1.5613818007514888e-16, 1.9419088343433915e-15, 2.2088472416748095e-14, 2.2945147893352633e-13, 2.1754375087681186e-12 … 0.00044705436986953415, 0.0002547385212319553, 0.0001418596463895886, 7.720636276918628e-5, 4.106551596111058e-5, 2.1346734329281984e-5, 1.0844650919059565e-5, 5.384305327207333e-6, 2.612604992426677e-6, 1.238932928945248e-6]"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"true_pdf = sum(pdf_arr, dims=1) ./ 3"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1-element Array{Array{Float64,1},1}:\n",
" [4.052291244805683e-7, 7.454676378589778e-7, 1.3520242306357958e-6, 2.417506901642637e-6, 4.261657817475742e-6, 7.406569170993491e-6, 1.2690633604763932e-5, 2.1437657135663452e-5, 3.5702549198067325e-5, 5.862044982039496e-5 … 0.0007990979070055408, 0.0005322056004138422, 0.00034959103514147627, 0.00022648643261263652, 0.0001447187786272269, 9.120280075658358e-5, 5.668809912152609e-5, 3.475168960847889e-5, 2.101165071373719e-5, 1.2529812347666799e-5]"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pdf_arr_fit = [pdf.(Normal(μ,σ), x) for (μ, σ) in zip(μ_arr_fit, σ_arr_fit)]\n",
"true_pdf_fit = sum(pdf_arr_fit, dims=1) ./ 3"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [],
"source": [
"using KernelDensity"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"UnivariateKDE{StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}}}(-2.5878209898955893:0.002601480157877084:2.737408893278802, [2.322758670636471e-5, 2.362077681969388e-5, 2.4043713184207505e-5, 2.4496864632295168e-5, 2.4980729018020398e-5, 2.5495833607697138e-5, 2.6042735488895907e-5, 2.662202199815722e-5, 2.723431116710695e-5, 2.788025218711243e-5 … 2.0848643890587076e-5, 2.096401617730087e-5, 2.110596934651321e-5, 2.1274699601883315e-5, 2.147042923128853e-5, 2.169340682347913e-5, 2.194390750120609e-5, 2.222223317165789e-5, 2.2528712793300903e-5, 2.286370266073659e-5])"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kde_data = kde(data)"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
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"<text style=\"fill:#adadad; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 208.744)\" x=\"2133.79\" y=\"208.744\">true</text>\n",
"</g>\n",
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"<text style=\"fill:#adadad; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 269.224)\" x=\"2133.79\" y=\"269.224\">hist</text>\n",
"</g>\n",
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" 1965.79,312.204 2109.79,312.204 \n",
" \"/>\n",
"<g clip-path=\"url(#clip0800)\">\n",
"<text style=\"fill:#adadad; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 329.704)\" x=\"2133.79\" y=\"329.704\">MLE</text>\n",
"</g>\n",
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"</svg>\n"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot()\n",
"plot!(x, true_pdf, label=\"true\")\n",
"histogram!(data, bins=20, alpha=0.2, normalize=true, label=\"hist\")\n",
"plot!(x, true_pdf_fit, label=\"MLE\")\n",
"plot!(x, pdf(kde_data, x), label=\"KDE\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mixture multiple dimension model MLE"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Int64,1}:\n",
" 50\n",
" 30\n",
" 20"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_arr = [[2.; 2], [-2; -2], [2; -2]]\n",
"Σ_arr = [[1. 0.5; 0.5 1.], [1. 0; 0 1], [1 -0.5; -0.5 1]]\n",
"num_arr = [50, 30, 20]"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×100 Array{Float64,2}:\n",
" 2.15057 2.64107 2.26895 3.74687 … 0.5344 2.18645 1.53061\n",
" 1.132 0.956469 2.0206 2.56987 -1.40667 -2.18465 -1.174 "
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat_arr = Matrix[]\n",
"for (μ, Σ, num) in zip(μ_arr, Σ_arr, num_arr)\n",
" #println(μ, Σ, num)\n",
" mat = rand(MvNormal(μ, Σ), num) # 2 x num\n",
" push!(mat_arr, mat)\n",
"end\n",
"data = hcat(mat_arr...)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
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"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(data[1,:], data[2,:])"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#31 (generic function with 1 method)"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"likelihood = (μ_arr, σ_arr) -> begin\n",
" dist_arr = [MvNormal(μ, σ) for (μ, σ) in zip(μ_arr, σ_arr)]\n",
" num_classes = length(dist_arr)\n",
" num_samples = size(data, 2)\n",
" log_prior = log(1. / num_classes)\n",
" lp = 0\n",
" for d in 1:num_samples\n",
" lp += logsumexp([logpdf(dist, data[:,d]) + log_prior for dist in dist_arr])\n",
" end\n",
" lp\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#37 (generic function with 1 method)"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"encode = (p) -> begin\n",
" μ_arr = Vector[]\n",
" Σ_arr = Matrix[]\n",
" for i in 1:3\n",
" push!(μ_arr, p[(2*i-1):(2*i)])\n",
" f = reshape(p[6+4*(i-1)+1:6+4*i], 2, 2)\n",
" push!(Σ_arr, f' * f)\n",
" end\n",
" μ_arr, Σ_arr\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"#39 (generic function with 1 method)"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = (p) -> begin\n",
" μ_arr, Σ_arr = encode(p)\n",
" -likelihood(μ_arr, Σ_arr)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"18-element Array{Float64,1}:\n",
" 0.3319091667148046\n",
" -1.6433095219624725\n",
" -0.7458544889135744\n",
" -0.7459243343257457\n",
" -0.6550476182144508\n",
" 1.8289442798916673\n",
" 1.0 \n",
" 0.0 \n",
" 0.0 \n",
" 1.0 \n",
" 1.0 \n",
" 0.0 \n",
" 0.0 \n",
" 1.0 \n",
" 1.0 \n",
" 0.0 \n",
" 0.0 \n",
" 1.0 "
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x0 = [randn(6); [1;0;0;1.]; [1;0;0;1.]; [1;0;0;1.]]"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"615.345389425848"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(x0)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: failure (objective increased between iterations) (line search failed)\n",
"\n",
" * Candidate solution\n",
" Minimizer: [-4.16e-01, -2.01e+00, 2.01e+00, ...]\n",
" Minimum: 3.975467e+02\n",
"\n",
" * Found with\n",
" Algorithm: L-BFGS\n",
" Initial Point: [3.32e-01, -1.64e+00, -7.46e-01, ...]\n",
"\n",
" * Convergence measures\n",
" |x - x'| = 9.89e-08 ≰ 0.0e+00\n",
" |x - x'|/|x'| = 4.01e-08 ≰ 0.0e+00\n",
" |f(x) - f(x')| = 1.71e-13 ≰ 0.0e+00\n",
" |f(x) - f(x')|/|f(x')| = 4.29e-16 ≰ 0.0e+00\n",
" |g(x)| = 4.52e-06 ≰ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 1 (vs limit Inf)\n",
" Iterations: 272\n",
" f(x) calls: 886\n",
" ∇f(x) calls: 886\n"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"res = optimize(f, x0, LBFGS(), autodiff=:forward)"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: failure (reached maximum number of iterations) (line search failed)\n",
"\n",
" * Candidate solution\n",
" Minimizer: [1.28e+00, -2.18e+00, -2.20e+00, ...]\n",
" Minimum: 3.737273e+02\n",
"\n",
" * Found with\n",
" Algorithm: Nelder-Mead\n",
" Initial Point: [3.32e-01, -1.64e+00, -7.46e-01, ...]\n",
"\n",
" * Convergence measures\n",
" √(Σ(yᵢ-ȳ)²)/n ≰ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 1 (vs limit Inf)\n",
" Iterations: 1000\n",
" f(x) calls: 1332\n"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"res = optimize(f, x0, )"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
" * Status: success\n",
"\n",
" * Candidate solution\n",
" Minimizer: [2.15e+00, -2.32e+00, -2.05e+00, ...]\n",
" Minimum: 3.649739e+02\n",
"\n",
" * Found with\n",
" Algorithm: Nelder-Mead\n",
" Initial Point: [3.32e-01, -1.64e+00, -7.46e-01, ...]\n",
"\n",
" * Convergence measures\n",
" √(Σ(yᵢ-ȳ)²)/n ≤ 1.0e-08\n",
"\n",
" * Work counters\n",
" Seconds run: 2 (vs limit Inf)\n",
" Iterations: 3613\n",
" f(x) calls: 4821\n"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"res = optimize(f, x0, iterations=10000)"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{MvNormal{Float64,PDMat{Float64,Array{Float64,2}},Array{Float64,1}},1}:\n",
" FullNormal(\n",
"dim: 2\n",
"μ: [2.151822356603406, -2.315319978017564]\n",
"Σ: [0.6501771077022512 -0.12898110129028423; -0.12898110129028423 0.4968955916966953]\n",
")\n",
" \n",
" FullNormal(\n",
"dim: 2\n",
"μ: [-2.0476516492012187, -1.8681425532978642]\n",
"Σ: [0.8100083035203361 -0.2611628201941864; -0.2611628201941864 0.6368959491108641]\n",
")\n",
"\n",
" FullNormal(\n",
"dim: 2\n",
"μ: [1.9836294394227587, 1.9896007284861126]\n",
"Σ: [1.1255851119177147 0.5623401883267891; 0.5623401883267891 1.042561808182948]\n",
")\n",
" "
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_arr_fit, Σ_arr_fit = encode(Optim.minimizer(res))\n",
"dist_arr_fit = [MvNormal(μ, Σ) for (μ, Σ) in zip(μ_arr_fit, Σ_arr_fit)]"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
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],
"source": [
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{
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