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Plots some figures in Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming
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basis_functions_approach_to_GP_exp.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"using SpecialFunctions\n",
"using LinearAlgebra\n",
"using Distributions\n",
"using Plots\n",
"using Random"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"S_32 (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# α: magnitude (marginal variance)\n",
"# ℓ: length-scale\n",
"\n",
"k_∞(r, α, ℓ) = α * exp(-1/2 * r^2/ℓ^2)\n",
"k_32(r, α, ℓ) = α * exp(1+ sqrt(3) * r / ℓ) * exp(-sqrt(3) * r / ℓ)\n",
"\n",
"S_∞(ω, α, ℓ) = α * sqrt(2π) * ℓ * exp(-ℓ^2 * ω^2 / 2)\n",
"S_32(ω, α, ℓ) = α * 2 * sqrt(π) * gamma(2) * 3^(3/2) / (1/2 * sqrt(π) * ℓ^3) * (3/ℓ^2 + ω^2)^(-2)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"𝜙_j (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# l: boundary condition, half-range of the input space\n",
"# Ω: [-L, L] input space, the interval in which the approximations are valid\n",
"\n",
"λ_j(j, L) = (j * π / 2 / L)^2\n",
"𝜙_j(x, j, L) = sqrt(1 / L) * sin(sqrt(λ_j(j, L)) * (x + L))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# L = c ⋅ S , set L an extension of the desired covariate input domain \n",
"# Ψ = max_i |x_i|\n",
"# Ψ = [−S, S] ⊂ Ω, input domain\n",
"# c: c ≥ 1, proportional extension factor\n",
"# S: half-range of the input space\n",
"\n",
"α = 1\n",
"ℓ = 0.3\n",
"S = 1\n",
"# varying m and L"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Int64,1}:\n",
" -2\n",
" -1\n",
" 2"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sort([-2,-1,2])"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"69"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Random.seed!(89)\n",
"x = sort(rand(80) * 2 .- 1.)\n",
"x = x[.!(-0.4 .< x .< -0.1)]\n",
"length(x)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(69, 69)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"τ = x .- x'\n",
"r = norm.(τ)\n",
"size(r)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ZeroMeanFullNormal(\n",
"dim: 69\n",
"μ: 69-element FillArrays.Zeros{Float64,1,Tuple{Base.OneTo{Int64}}} = 0.0\n",
"Σ: [1.1 0.9910364047490361 … 2.279314715341694e-9 3.989976949412656e-10; 0.9910364047490361 1.1 … 5.266945358365081e-9 9.560620281097424e-10; … ; 2.279314715341694e-9 5.266945358365081e-9 … 1.1 0.9640916555933162; 3.989976949412656e-10 9.560620281097424e-10 … 0.9640916555933162 1.1]\n",
")\n"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"true_dist = MvNormal(k_∞.(r, α, ℓ) + 0.1I)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"y = rand(true_dist);"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
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]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#plot(x, y, legend=false)\n",
"scatter(x, y, markersize=2, color=:white)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"xl = range(-1,1,length=100);"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": [
"#Σ₂₂⁻¹=inv(k_∞.(r, α, ℓ) + 1e-6I);\n",
"Σ₂₂⁻¹=inv(k_∞.(r, α, ℓ) + 0.1I);"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"predict (generic function with 1 method)"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function predict(x0)\n",
" T = eltype(x0)\n",
" \n",
" Σ₁₂ = k_∞.(x0 .- x, α, ℓ)'\n",
" Σ₂₁ = Σ₁₂'\n",
" Σ₁₁ = k_∞(T(0), α, ℓ) + T(0.1)I\n",
" μ = Σ₁₂ * Σ₂₂⁻¹ * y\n",
" Σ = Σ₁₁ - Σ₁₂ * Σ₂₂⁻¹ * Σ₂₁\n",
" return μ, Σ\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-1.1327119258871023, 0.11117543940403352)"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"predict([0.])"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-1.1327119258871023, 0.11117543940403352)"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"predict(0.)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"pl = map(x->predict(x), xl);\n",
"mean_l = map(x->x[1], pl)\n",
"sigma_l = map(x->sqrt(x[2]), pl)\n",
";"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
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],
"source": [
"#plot(xl, mean_l, ribbon=sigma_l, legend=false, linestyle=:dash, color=:red)\n",
"plot(xl, mean_l, legend=false, linestyle=:dash, color=:red)\n",
"scatter!(x, y, markersize=2, color=:white)\n",
"title!(\"Mean posterior predictive functions (GP groundtruth)\")"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
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},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(xl, sigma_l, xlabel=\"x\", ylabel=\"σ\", linestyle=:dash, color=:red)\n",
"title!(\"posterior standard deviations (GP groundtruth)\")"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": [
"cov_x = range(0, 2., length=100)\n",
"cov_f = map(τ->k_∞(τ, α, ℓ), cov_x)\n",
";"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
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" \n",
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"</svg>\n"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(cov_x, cov_f, linestyle=:dash, color=:red, legend=false)\n",
"title!(\"covariance functions (GP groundtruth)\")"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"20"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"m_l = [2, 4, 7, 10, 15, 20]\n",
"max_m = maximum(m_l)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = 3 # for L = c * S and S=1 and c is fixed to a large enough value (c=3)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((20,), (20, 69))"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"λ_l = [λ_j(j, L) for j in 1:max_m]\n",
"𝜙_mat = [𝜙_j(x[i], j, L) for j=1:max_m, i=1:length(x)]\n",
"\n",
"size(λ_l), size(𝜙_mat)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [],
"source": [
"S_l = map(λ->S_∞(sqrt(λ), α, ℓ), λ_l);"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(69, 20)"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"project_mat = 𝜙_mat' * Diagonal(S_l)\n",
"size(project_mat)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(20, 100)"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"𝜙_l_mat = [𝜙_j(xl[i], j, L) for j=1:max_m, i=1:length(xl)]\n",
"\n",
"size(𝜙_l_mat)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(69, 100)"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"k_x_xl = project_mat * 𝜙_l_mat\n",
"size(k_x_xl)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((1, 20), (20, 20), (1, 20))"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"𝜙_cov_x0 = [𝜙_j(0., j, L) for j=1:max_m]\n",
"project_cov_x = 𝜙_cov_x0' * Diagonal(S_l)\n",
"\n",
"size(𝜙_cov_x0'), size(Diagonal(S_l)), size(project_cov_x)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(20, 100)"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"𝜙_cov_x = [𝜙_j(cov_x[i], j, L) for j=1:max_m, i=1:length(cov_x)]\n",
"\n",
"size(𝜙_cov_x)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
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},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(cov_x, vec(project_cov_x * 𝜙_cov_x))"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [],
"source": [
"cov_per_m = [𝜙_cov_x0[1:m]' * Diagonal(S_l[1:m]) * 𝜙_cov_x[1:m,:] for m in m_l] .|> vec\n",
";"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
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]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(cov_x, cov_per_m, palette = palette(:blues, length(cov_per_m)), label=m_l')\n",
"plot!(cov_x, cov_f, linestyle=:dash, color=:red, label=\"GP\")\n",
"title!(\"covariance functions (m approx vs GP GT)\")\n"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(20, 69)"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"𝜙_x_mat = [𝜙_j(x[i], j, L) for j=1:max_m, i=1:length(x)]\n",
"size(𝜙_x_mat)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([(69, 69), (69, 69), (69, 69), (69, 69), (69, 69), (69, 69)], [-149.7797534625447, -145.5238148662229, -140.09528747799033, -136.01689721814333, -131.52435977810003, -129.7262284059348])"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K_per_m = [𝜙_x_mat'[:,1:m] * Diagonal(S_l[1:m]) * 𝜙_x_mat[1:m,:] + 0.1I for m in m_l]\n",
"\n",
"map(size, K_per_m), map(logdet, K_per_m)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"#=\n",
"function predict(x0)\n",
" T = eltype(x0)\n",
" \n",
" Σ₁₂ = k_∞.(x0 .- x, α, ℓ)'\n",
" Σ₂₁ = Σ₁₂'\n",
" Σ₁₁ = k_∞(T(0), α, ℓ) + T(0.1)I\n",
" μ = Σ₁₂ * Σ₂₂⁻¹ * y\n",
" Σ = Σ₁₁ - Σ₁₂ * Σ₂₂⁻¹ * Σ₂₁\n",
" return μ, Σ\n",
"end\n",
"=#"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(69, 69)"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Σ₂₂⁻¹_per_m = map(inv, K_per_m)\n",
"\n",
"size(Σ₂₂⁻¹_per_m[1])"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((100, 69), (69, 100), (100, 100))"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Σ₁₂_per_m = [𝜙_l_mat'[:,1:m] * Diagonal(S_l[1:m]) * 𝜙_x_mat[1:m,:] for m in m_l]\n",
"Σ₂₁_per_m = map(x->x', Σ₁₂_per_m)\n",
"Σ₁₁_per_m = [𝜙_l_mat'[:,1:m] * Diagonal(S_l[1:m]) * 𝜙_l_mat[1:m,:] + 0.1I for m in m_l]\n",
"\n",
"size(Σ₁₂_per_m[1]), size(Σ₂₁_per_m[1]), size(Σ₁₁_per_m[1])"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Array{T,2} where T[]"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"μ_per_m = Vector[]\n",
"Σ_per_m = Matrix[]"
]
},
{
"cell_type": "code",
"execution_count": 40,
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"source": [
"for (Σ₂₂⁻¹, Σ₁₂, Σ₂₁, Σ₁₁) in zip(Σ₂₂⁻¹_per_m, Σ₁₂_per_m, Σ₂₁_per_m, Σ₁₁_per_m)\n",
" μ = Σ₁₂ * Σ₂₂⁻¹ * y\n",
" Σ = Σ₁₁ - Σ₁₂ * Σ₂₂⁻¹ * Σ₂₁\n",
" push!(μ_per_m, μ)\n",
" push!(Σ_per_m, Σ)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
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],
"source": [
"plot(xl, μ_per_m)"
]
},
{
"cell_type": "code",
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},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(xl, μ_per_m, palette = palette(:blues, length(cov_per_m)), label=m_l')\n",
"plot!(xl, mean_l, linestyle=:dash, color=:red, label=\"GP\")\n",
"scatter!(x, y, markersize=2, color=:white, label=\"\")\n",
"title!(\"Mean posterior predictive functions (GP groundtruth)\")"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [],
"source": [
"σ_per_m = map(x->sqrt.(diag(x)), Σ_per_m);"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
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]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(xl, σ_per_m, palette = palette(:blues, length(cov_per_m)), label=m_l')\n",
"plot!(xl, sigma_l, xlabel=\"x\", ylabel=\"σ\", linestyle=:dash, color=:red)\n",
"title!(\"posterior standard deviations (GP groundtruth)\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"@webio": {
"lastCommId": null,
"lastKernelId": null
},
"kernelspec": {
"display_name": "Julia 1.5.0",
"language": "julia",
"name": "julia-1.5"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.5.0"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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