Created
November 14, 2019 10:39
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Example sampling of a Gaussian Process posterior
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| using Distributions | |
| using LinearAlgebra | |
| using Plots | |
| using Random | |
| pyplot(); | |
| # Training points | |
| P = [ | |
| 1.0 1.0 | |
| 2.0 3.0 | |
| 4.0 -2.0 | |
| 7.0 5.0 | |
| 8.0 3.0 | |
| ] | |
| # Dataset size | |
| N₁ = size(P, 1) | |
| N₂ = 500; | |
| # I.i.d. observation noise standard deviation | |
| σₙ = 0.05; | |
| f₁ = P[:, 2] | |
| X₁ = P[:, 1] | |
| X₂ = collect(range(0, stop=10, length=N₂)); | |
| ϵ = 1e-12 * rand() * I; | |
| # Build conditional distribution | |
| k(x, y) = exp(-0.5 * norm(x - y)^2); | |
| K(X, Y) = [k(X[i], Y[j]) for i in 1:size(X, 1), j in 1:size(Y, 1)]; | |
| Σ₁₁¯¹ = inv(K(X₁, X₁) + σₙ^2 * I); | |
| Σ₁₂ = K(X₁, X₂); | |
| Σ₂₂ = K(X₂, X₂); | |
| μ = Σ₁₂' * Σ₁₁¯¹ * f₁; | |
| Σ = Σ₂₂ - Σ₁₂' * Σ₁₁¯¹ * Σ₁₂ + ϵ; | |
| σ = sqrt.(diag(Σ)); | |
| d = MvNormal(μ, Matrix(Hermitian(Σ))); | |
| # Sample some functions | |
| f₂ = rand(d, 5); | |
| p = plot(X₂, f₂); | |
| plot!(X₂, μ, linewidth=2, ribbon=(2σ, 2σ), fillalpha=0.2, c="grey", label="μ ± 2σ") | |
| scatter!(X₁, f₁, label="P") | |
| display(p); | |
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