From 0fdac314c9ba81c0f58a3ce5aba7d5c0f8b46108 Mon Sep 17 00:00:00 2001 From: leachim Date: Mon, 16 Aug 2021 15:00:04 +0100 Subject: [PATCH] basic flow of tutorial; requires clean-up and more text --- ..._gaussian-process-latent-variable-model.md | 495 ++++++++ .../Data.txt | 1000 +++++++++++++++++ .../DataLabels.txt | 1000 +++++++++++++++++ ...ian-process-latent-variable-model_10_1.png | Bin 0 -> 15934 bytes ...ian-process-latent-variable-model_11_1.png | Bin 0 -> 15584 bytes ...ian-process-latent-variable-model_12_1.png | Bin 0 -> 16430 bytes ...ian-process-latent-variable-model_14_1.png | Bin 0 -> 14844 bytes ...ian-process-latent-variable-model_15_1.png | Bin 0 -> 15425 bytes ...ian-process-latent-variable-model_18_1.png | Bin 0 -> 15168 bytes ...ian-process-latent-variable-model_19_1.png | Bin 0 -> 15753 bytes ...sian-process-latent-variable-model_7_1.png | Bin 0 -> 15638 bytes ...sian-process-latent-variable-model_8_1.png | Bin 0 -> 14490 bytes ...gaussian-process-latent-variable-model.jmd | 143 ++- 13 files changed, 2612 insertions(+), 26 deletions(-) create mode 100644 markdown/12-gaussian-process-latent-variable-model/12_gaussian-process-latent-variable-model.md create mode 100644 markdown/12-gaussian-process-latent-variable-model/Data.txt create mode 100644 markdown/12-gaussian-process-latent-variable-model/DataLabels.txt create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_10_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_11_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_12_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_14_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_15_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_18_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_19_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_7_1.png create mode 100644 markdown/12-gaussian-process-latent-variable-model/figures/12_gaussian-process-latent-variable-model_8_1.png diff --git a/markdown/12-gaussian-process-latent-variable-model/12_gaussian-process-latent-variable-model.md b/markdown/12-gaussian-process-latent-variable-model/12_gaussian-process-latent-variable-model.md new file mode 100644 index 000000000..377e41654 --- /dev/null +++ b/markdown/12-gaussian-process-latent-variable-model/12_gaussian-process-latent-variable-model.md @@ -0,0 +1,495 @@ +--- +redirect_from: "tutorials/12-gaussian-process-latent-variable-model/" +title: "Gaussian Process Latent Variable Model" +permalink: "/:collection/:name/" +--- + + +# Gaussian Process Latent Variable Model + +In a previous tutorial, we have discussed latent variable models, in particular probabilistic principal +component analysis (pPCA). Here, we show how we can extend the mapping provided by pPCA to non-linear mappings between input and output. For more details about the Gaussian Process Latent Variable Model (GPLVM), we refer the reader to the [original +publication](https://jmlr.org/papers/v6/lawrence05a.html) and a [further extension](http://proceedings.mlr.press/v9/titsias10a/titsias10a.pdf). + +In short, the GPVLM is a dimensionality reduction technique that allows us to embed a high-dimensional dataset in a lower-dimensional embedding. Importantly, it provides the advantage that the linear mappings from the embedded space can be non-linearised through the use of Gaussian Processes. + +Let's start by loading some dependencies. +```julia +# Load Turing. +using Turing +using AbstractGPs, KernelFunctions, Random, Plots + +# Load other dependencies +using Distributions, LinearAlgebra +using VegaLite, DataFrames, StatsPlots, StatsBase +using DelimitedFiles + +Random.seed!(1789); +``` + + + + +We use a classical data set of synthetic data that is also used in the original publication. +```julia +oil_matrix = readdlm("Data.txt", Float64); +labels = readdlm("DataLabels.txt", Float64); +labels = mapslices(x -> findmax(x)[2], labels, dims=2); +# normalize data +dt = fit(ZScoreTransform, oil_matrix, dims=2); +StatsBase.transform!(dt, oil_matrix); +``` + + + + +We will start out, by demonstrating the basic similarity between pPCA (see the tutorial on this topic) +and the GPLVM model. Indeed, pPCA is basically equivalent to running the GPLVM model with an +automatic relevance determination (ARD) linear kernel. + +First, we re-introduce the pPCA model (see the tutorial on pPCA for details) +```julia +@model function pPCA(x) + # Dimensionality of the problem. + N, D = size(x) + + # latent variable z + z ~ filldist(Normal(), D, N) + + # weights/loadings W + w ~ filldist(Normal(0.0, 1.0), D, D) + + # mean offset + mu = (w * z)' + + for d in 1:D + x[:,d] ~ MvNormal(mu[:,d], 1.) + end +end; +``` + + + + +And here is the GPLVM model. The choice of kernel is determined by the kernel_function parameter. +```julia +@model function GPLVM(Y, kernel_function, ndim=4,::Type{T} = Float64) where {T} + + # Dimensionality of the problem. + N, D = size(Y) + # dimensions of latent space + K = ndim + noise = 1e-3 + + # Priors + α ~ MvLogNormal(MvNormal(K, 1.0)) + σ ~ LogNormal(0.0, 1.0) + Z ~ filldist(Normal(0., 1.), K, N) + + kernel = kernel_function(α, σ) + + ## DENSE GP + gp = GP(kernel) + prior = gp(ColVecs(Z), noise) + + for d in 1:D + Y[:, d] ~ prior + end +end +``` + +``` +GPLVM (generic function with 3 methods) +``` + + + + + +We define two different kernels, a simple linear kernel with an Automatic Relevance Determination transform and a +squared exponential kernel. The latter represents a fully non-linear transform. +```julia +sekernel(α, σ²) = σ² * SqExponentialKernel() ∘ ARDTransform(α) +linear_kernel(α, σ²) = LinearKernel() ∘ ARDTransform(α) + +# note we keep the problem very small for reasons of runtime +# n_features=size(oil_matrix)[2]; +n_features=4 +# latent dimension for GP case +ndim=2 +n_data=40 +``` + +``` +40 +``` + + + +```julia +ppca = pPCA(oil_matrix[1:n_data,1:n_features]) +chain_ppca = sample(ppca, NUTS(), 1000) +``` + +``` +Chains MCMC chain (1000×188×1 Array{Float64, 3}): + +Iterations = 501:1:1500 +Number of chains = 1 +Samples per chain = 1000 +Wall duration = 48.43 seconds +Compute duration = 48.43 seconds +parameters = z[3,9], z[1,6], z[1,3], z[4,40], z[1,31], z[1,32], w[1, +4], z[1,11], z[3,2], z[2,26], z[2,30], z[4,21], z[2,36], z[4,30], z[3,26], +w[4,2], w[3,3], z[4,13], z[1,22], z[4,36], z[4,2], z[4,23], z[2,14], w[1,3] +, z[3,12], z[4,24], z[2,23], z[3,16], z[4,5], z[1,18], w[4,3], z[2,22], z[1 +,10], z[4,31], z[1,28], z[1,8], z[3,17], z[3,4], z[1,14], w[3,4], z[2,39], +w[2,2], z[2,38], z[3,40], z[2,25], w[2,3], z[1,20], z[3,30], z[3,31], z[3,7 +], z[1,17], z[1,16], z[2,18], w[2,4], z[3,1], z[4,28], z[4,4], z[1,29], z[2 +,17], z[2,29], z[1,27], z[1,25], z[3,36], z[4,27], z[2,13], z[2,10], z[1,26 +], z[4,3], z[2,12], z[2,7], z[4,19], z[4,33], w[2,1], z[1,12], z[3,33], z[2 +,5], z[4,26], z[3,29], z[3,22], z[2,15], z[3,14], z[4,39], z[1,21], z[3,24] +, z[2,24], z[1,15], z[3,13], z[1,13], z[4,22], z[4,16], z[4,25], z[3,39], z +[2,31], z[4,12], z[4,29], z[4,11], z[2,16], z[2,11], z[2,6], z[1,24], z[1,1 +9], z[2,1], z[3,3], z[2,34], z[1,30], z[1,5], z[3,6], z[4,7], z[1,34], z[2, +40], z[2,19], z[3,35], z[2,28], z[1,40], z[3,11], z[4,37], z[3,5], z[4,15], + z[4,34], z[4,14], z[4,6], z[1,9], z[4,10], z[1,1], z[2,21], w[1,2], z[1,38 +], z[3,27], z[2,2], z[1,2], w[1,1], z[3,19], w[4,4], z[1,4], z[3,23], z[3,1 +0], z[3,37], z[3,18], z[2,4], z[3,38], z[3,25], w[3,1], z[4,38], z[2,27], z +[2,3], z[2,37], z[4,18], z[2,20], z[2,33], z[4,20], z[3,32], z[1,33], z[4,3 +2], z[2,32], z[1,23], z[2,8], w[4,1], z[3,34], z[3,28], z[1,35], z[3,8], z[ +2,9], z[3,21], z[4,8], z[1,37], w[3,2], z[3,20], z[1,7], z[4,9], z[1,36], z +[1,39], z[4,35], z[4,17], z[4,1], z[3,15], z[2,35] +internals = lp, n_steps, is_accept, acceptance_rate, log_density, h +amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, +tree_depth, numerical_error, step_size, nom_step_size + +Summary Statistics + parameters mean std naive_se mcse ess rhat + ⋯ + Symbol Float64 Float64 Float64 Float64 Float64 Float64 + ⋯ + + z[1,1] 0.0335 0.8742 0.0276 0.0356 555.0660 1.0017 + ⋯ + z[2,1] 0.0125 0.8996 0.0284 0.0474 359.7716 1.0001 + ⋯ + z[3,1] -0.0772 0.8917 0.0282 0.0443 509.0491 0.9991 + ⋯ + z[4,1] -0.0560 0.9174 0.0290 0.0482 604.4676 0.9994 + ⋯ + z[1,2] -0.0947 1.0530 0.0333 0.0640 192.6264 1.0009 + ⋯ + z[2,2] -0.0161 1.0426 0.0330 0.0770 178.0413 0.9990 + ⋯ + z[3,2] 0.1544 1.0234 0.0324 0.0639 234.7909 1.0000 + ⋯ + z[4,2] 0.0977 0.9850 0.0311 0.0876 222.7565 0.9991 + ⋯ + z[1,3] 0.0662 0.9897 0.0313 0.0458 401.9331 1.0020 + ⋯ + z[2,3] 0.0270 1.0025 0.0317 0.0677 281.7341 1.0018 + ⋯ + z[3,3] -0.1329 0.9741 0.0308 0.0373 613.6068 1.0005 + ⋯ + z[4,3] -0.0221 0.9954 0.0315 0.0521 468.7021 0.9999 + ⋯ + z[1,4] -0.0496 0.9509 0.0301 0.0550 414.0184 1.0078 + ⋯ + z[2,4] -0.0050 0.9088 0.0287 0.0603 267.2155 0.9996 + ⋯ + z[3,4] 0.1181 0.9513 0.0301 0.0641 249.7106 1.0003 + ⋯ + z[4,4] 0.0890 0.8975 0.0284 0.0556 458.8122 0.9991 + ⋯ + z[1,5] 0.0579 0.8101 0.0256 0.0255 942.7670 1.0007 + ⋯ + ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ + ⋱ + 1 column and 159 rows om +itted + +Quantiles + parameters 2.5% 25.0% 50.0% 75.0% 97.5% + Symbol Float64 Float64 Float64 Float64 Float64 + + z[1,1] -1.6805 -0.5841 0.0738 0.6067 1.6903 + z[2,1] -1.7307 -0.5979 0.0483 0.6098 1.7872 + z[3,1] -1.8672 -0.6864 -0.1105 0.5493 1.7318 + z[4,1] -1.7930 -0.7319 -0.0378 0.5480 1.7750 + z[1,2] -2.0686 -0.8266 -0.1019 0.6335 1.9847 + z[2,2] -1.9686 -0.7430 -0.0222 0.6723 2.1067 + z[3,2] -1.8533 -0.5669 0.2030 0.8743 2.1369 + z[4,2] -1.8551 -0.5761 0.0965 0.7280 2.0448 + z[1,3] -1.9024 -0.5786 0.0591 0.7587 2.0076 + z[2,3] -2.0057 -0.6549 0.0622 0.7699 1.8922 + z[3,3] -2.0316 -0.7775 -0.1436 0.5121 1.7903 + z[4,3] -1.9309 -0.7007 -0.0555 0.6719 1.8539 + z[1,4] -1.9506 -0.6747 -0.0423 0.6183 1.7704 + z[2,4] -1.7112 -0.6613 -0.0466 0.6564 1.7538 + z[3,4] -1.7941 -0.5449 0.1398 0.7644 1.9194 + z[4,4] -1.7260 -0.5264 0.1218 0.7191 1.7838 + z[1,5] -1.4470 -0.4825 0.0189 0.5895 1.6322 + ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ + 159 rows omitted +``` + + + +```julia +w = permutedims(reshape(mean(group(chain_ppca, :w))[:,2], (n_features,n_features))) +z = permutedims(reshape(mean(group(chain_ppca, :z))[:,2], (n_features, n_data)))' +X = w * z + +#df_rec = DataFrame(X', :auto) +#df_rec[!,:datapoints] = 1:n_data + +df_pre = DataFrame(z', :auto) +rename!(df_pre, Symbol.( ["z"*string(i) for i in collect(1:n_features)])) +df_pre[!,:type] = labels[1:n_data] +df_pre |> @vlplot(:point, x=:z1, y=:z2, color="type:n") +``` + +![](figures/12_gaussian-process-latent-variable-model_7_1.png) + +```julia +df_pre |> @vlplot(:point, x=:z2, y=:z3, color="type:n") +df_pre |> @vlplot(:point, x=:z2, y=:z4, color="type:n") +df_pre |> @vlplot(:point, x=:z3, y=:z4, color="type:n") +``` + +![](figures/12_gaussian-process-latent-variable-model_8_1.png) + +```julia +gplvm_linear = GPLVM(oil_matrix[1:n_data,1:n_features], linear_kernel, n_features) + +chain_linear = sample(gplvm_linear, NUTS(), 800) +z_mean = permutedims(reshape(mean(group(chain_linear, :Z))[:,2], (n_features, n_data)))' +alpha_mean = mean(group(chain_linear, :α))[:,2] +``` + +``` +4-element Vector{Float64}: + 0.6432505968563335 + 0.6678945068525474 + 0.6395027704297788 + 0.6543767042298047 +``` + + + +```julia +df_gplvm_linear = DataFrame(z_mean', :auto) +rename!(df_gplvm_linear, Symbol.( ["z"*string(i) for i in collect(1:n_features)])) +df_gplvm_linear[!,:sample] = 1:n_data +df_gplvm_linear[!,:labels] = labels[1:n_data] +alpha_indices = sortperm(alpha_mean, rev=true)[1:2] +println(alpha_indices) +df_gplvm_linear[!,:ard1] = z_mean[alpha_indices[1], :] +df_gplvm_linear[!,:ard2] = z_mean[alpha_indices[2], :] + +p1 = df_gplvm_linear|> @vlplot(:point, x=:z1, y=:z2, color="labels:n") +``` + +``` +[2, 4] +``` + + +![](figures/12_gaussian-process-latent-variable-model_10_1.png) + +```julia +p2 = df_gplvm_linear |> @vlplot(:point, x=:ard1, y=:ard2, color="labels:n") +``` + +![](figures/12_gaussian-process-latent-variable-model_11_1.png) + +```julia +df_gplvm_linear|> @vlplot(:point, x=:z2, y=:z3, color="labels:n") +df_gplvm_linear|> @vlplot(:point, x=:z2, y=:z4, color="labels:n") +df_gplvm_linear|> @vlplot(:point, x=:z3, y=:z4, color="labels:n") +``` + +![](figures/12_gaussian-process-latent-variable-model_12_1.png) + + + +Finally, we demonstrate that by changing the kernel to a non-linear function, we are better able to separate the data. + +```julia +gplvm = GPLVM(oil_matrix[1:n_data,:], sekernel, ndim) + +chain = sample(gplvm, NUTS(), 800) +z_mean = permutedims(reshape(mean(group(chain, :Z))[:,2], (ndim, n_data)))' +alpha_mean = mean(group(chain, :α))[:,2] +``` + +``` +2-element Vector{Float64}: + 2.2804807885302676 + 2.0226820902393263 +``` + + + +```julia +df_gplvm = DataFrame(z_mean', :auto) +rename!(df_gplvm, Symbol.( ["z"*string(i) for i in collect(1:ndim)])) +df_gplvm[!,:sample] = 1:n_data +df_gplvm[!,:labels] = labels[1:n_data] +alpha_indices = sortperm(alpha_mean, rev=true)[1:2] +println(alpha_indices) +df_gplvm[!,:ard1] = z_mean[alpha_indices[1], :] +df_gplvm[!,:ard2] = z_mean[alpha_indices[2], :] + +p1 = df_gplvm|> @vlplot(:point, x=:z1, y=:z2, color="labels:n") +``` + +``` +[1, 2] +``` + + +![](figures/12_gaussian-process-latent-variable-model_14_1.png) + +```julia +p2 = df_gplvm |> @vlplot(:point, x=:ard1, y=:ard2, color="labels:n") +``` + +![](figures/12_gaussian-process-latent-variable-model_15_1.png) + + + +### Speeding up inference + +Gaussian processes tend to be slow, as they naively require. + +```julia +@model function GPLVM_sparse(Y, kernel_function, ndim=4,::Type{T} = Float64) where {T} + + # Dimensionality of the problem. + N, D = size(Y) + # dimensions of latent space + K = ndim + # number of inducing points + n_inducing = 20 + noise = 1e-3 + + # Priors + α ~ MvLogNormal(MvNormal(K, 1.0)) + σ ~ LogNormal(0.0, 1.0) + Z ~ filldist(Normal(0., 1.), K, N) + # σ_n ~ MvLogNormal(MvNormal(K, 1.0)) + # α = rand( MvLogNormal(MvNormal(K, 1.0))) + # σ = rand( LogNormal(0.0, 1.0)) + # Z = rand( filldist(Normal(0., 1.), K, N)) + # σ_n = rand( MvLogNormal(MvNormal(K, 1.0))) + σ = σ + 1e-12 + + kernel = kernel_function(α, σ) + + ## Standard + # gp = GP(kernel) + # prior = gp(ColVecs(Z), noise) + + ## SPARSE GP + # xu = reshape(repeat(locations, K), :, K) # inducing points + # xu = reshape(repeat(collect(range(-2.0, 2.0; length=20)), K), :, K) # inducing points + lbound = minimum(Y) + 1e-6 + ubound = maximum(Y) - 1e-6 + locations ~ filldist(Uniform(lbound, ubound), n_inducing) + xu = reshape(repeat(locations, K), :, K) # inducing points + gp = Stheno.wrap(GP(kernel), GPC()) + fobs = gp(ColVecs(Z), noise) + finducing = gp(ColVecs(xu'), noise) + prior = SparseFiniteGP(fobs, finducing) + + for d in 1:D + Y[:, d] ~ prior + end +end +``` + +``` +GPLVM_sparse (generic function with 3 methods) +``` + + + +```julia +ndim=2 +n_data=40 +# n_features=size(oil_matrix)[2]; +n_features=4 + +gplvm = GPLVM(oil_matrix[1:n_data,1:n_features], sekernel, ndim) + +chain_gplvm_sparse = sample(gplvm, NUTS(), 500) +z_mean = permutedims(reshape(mean(group(chain_gplvm_sparse, :Z))[:,2], (ndim, n_data)))' +alpha_mean = mean(group(chain_gplvm_sparse, :α))[:,2] +``` + +``` +2-element Vector{Float64}: + 1.3457169236983682 + 1.0759707153432654 +``` + + + +```julia +df_gplvm_sparse = DataFrame(z_mean', :auto) +rename!(df_gplvm_sparse, Symbol.( ["z"*string(i) for i in collect(1:ndim)])) +df_gplvm_sparse[!,:sample] = 1:n_data +df_gplvm_sparse[!,:labels] = labels[1:n_data] +alpha_indices = sortperm(alpha_mean, rev=true)[1:2] +println(alpha_indices) +df_gplvm_sparse[!,:ard1] = z_mean[alpha_indices[1], :] +df_gplvm_sparse[!,:ard2] = z_mean[alpha_indices[2], :] +p1 = df_gplvm_sparse|> @vlplot(:point, x=:z1, y=:z2, color="labels:n") +``` + +``` +[1, 2] +``` + + +![](figures/12_gaussian-process-latent-variable-model_18_1.png) + +```julia +p2 = df_gplvm_sparse |> @vlplot(:point, x=:ard1, y=:ard2, color="labels:n") +``` + +![](figures/12_gaussian-process-latent-variable-model_19_1.png) + + +## Appendix + This tutorial is part of the TuringTutorials repository, found at: . + +To locally run this tutorial, do the following commands: +```julia, eval = false +using TuringTutorials +TuringTutorials.weave_file("12-gaussian-process-latent-variable-model", "12_gaussian-process-latent-variable-model.jmd") +``` + +Computer Information: +``` +Julia Version 1.6.1 +Commit 6aaedecc44 (2021-04-23 05:59 UTC) +Platform Info: + OS: Linux (x86_64-pc-linux-gnu) + CPU: Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz + WORD_SIZE: 64 + LIBM: libopenlibm + LLVM: libLLVM-11.0.1 (ORCJIT, skylake) +Environment: + JULIA_NUM_THREADS = 8 + +``` + +Package Information: + +``` + Status `~/TuringDev/TuringTutorials/tutorials/12-gaussian-process-latent-variable-model/Project.toml` (empty project) + +``` diff --git a/markdown/12-gaussian-process-latent-variable-model/Data.txt b/markdown/12-gaussian-process-latent-variable-model/Data.txt new file mode 100644 index 000000000..d77bf9dc4 --- /dev/null +++ b/markdown/12-gaussian-process-latent-variable-model/Data.txt @@ -0,0 +1,1000 @@ +3.3150000e-01 2.1560000e-01 6.8020000e-01 1.4340000e-01 6.8250000e-01 2.7200000e-01 6.2230000e-01 2.0920000e-01 7.9610000e-01 1.5300000e-01 5.8560000e-01 2.5730000e-01 +9.3900000e-02 1.0089000e+00 3.6500000e-02 6.9440000e-01 9.0800000e-02 4.9610000e-01 7.2200000e-02 6.5210000e-01 -1.3000000e-02 6.0850000e-01 6.3100000e-02 6.5970000e-01 +5.1840000e-01 2.2830000e-01 5.3000000e-01 6.8840000e-01 7.4560000e-01 6.1710000e-01 6.1360000e-01 5.9280000e-01 7.6780000e-01 6.1300000e-01 6.7050000e-01 5.2020000e-01 +4.2080000e-01 6.7400000e-01 1.6510000e-01 7.5920000e-01 1.8100000e-01 5.4480000e-01 1.7070000e-01 7.5540000e-01 1.6350000e-01 5.4920000e-01 2.5980000e-01 6.4550000e-01 +3.1300000e-01 6.4650000e-01 5.9080000e-01 6.9240000e-01 7.6640000e-01 6.2620000e-01 1.7177000e+00 1.5000000e-02 8.5100000e-02 1.9046000e+00 -1.6500000e-02 2.2100000e-02 +1.1458000e+00 -4.6700000e-02 4.0560000e-01 5.6620000e-01 3.1230000e-01 4.5800000e-01 3.6360000e-01 6.1340000e-01 3.3050000e-01 4.1320000e-01 4.1670000e-01 5.5140000e-01 +-1.9000000e-03 1.7320000e-01 5.7000000e-03 4.8820000e-01 2.0760000e-01 3.9100000e-01 8.6000000e-03 1.7198000e+00 2.1500000e-02 -2.5800000e-02 6.7300000e-02 -8.2900000e-02 +7.8000000e-03 4.6150000e-01 1.1810000e-01 6.5900000e-01 2.5870000e-01 6.3520000e-01 -1.9100000e-02 1.7491000e+00 1.0980000e-01 -1.3150000e-01 4.0700000e-02 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oil_matrix); ``` We will start out, by demonstrating the basic similarity between pPCA (see the tutorial on this topic) @@ -60,7 +61,7 @@ First, we re-introduce the pPCA model (see the tutorial on pPCA for details) end; ``` -And here is the GPLVM model. +And here is the GPLVM model. The choice of kernel is determined by the kernel_function parameter. ```julia @model function GPLVM(Y, kernel_function, ndim=4,::Type{T} = Float64) where {T} @@ -81,28 +82,29 @@ And here is the GPLVM model. gp = GP(kernel) prior = gp(ColVecs(Z), noise) - Y ~ filldist(prior, D) + for d in 1:D + Y[:, d] ~ prior + end end ``` We define two different kernels, a simple linear kernel with an Automatic Relevance Determination transform and a squared exponential kernel. The latter represents a fully non-linear transform. - ```julia sekernel(α, σ²) = σ² * SqExponentialKernel() ∘ ARDTransform(α) linear_kernel(α, σ²) = LinearKernel() ∘ ARDTransform(α) -Y = oil_matrix # note we keep the problem very small for reasons of runtime -ndim=12 +# n_features=size(oil_matrix)[2]; +n_features=4 +# latent dimension for GP case +ndim=2 n_data=40 -n_features=size(oil_matrix)[2]; ``` ```julia -ppca = pPCA(oil_matrix[1:n_data,:]) -chain_ppca = sample(ppca, NUTS(), 500) -StatsPlots.plot(group(chain_ppca, :alpha)) +ppca = pPCA(oil_matrix[1:n_data,1:n_features]) +chain_ppca = sample(ppca, NUTS(), 1000) ``` ```julia w = permutedims(reshape(mean(group(chain_ppca, :w))[:,2], (n_features,n_features))) @@ -118,20 +120,28 @@ df_pre[!,:type] = labels[1:n_data] df_pre |> @vlplot(:point, x=:z1, y=:z2, color="type:n") ``` +```julia +df_pre |> @vlplot(:point, x=:z2, y=:z3, color="type:n") +df_pre |> @vlplot(:point, x=:z2, y=:z4, color="type:n") +df_pre |> @vlplot(:point, x=:z3, y=:z4, color="type:n") +``` + ```julia -gplvm_linear = GPLVM(oil_matrix[1:n_data,:], linear_kernel, ndim) +gplvm_linear = GPLVM(oil_matrix[1:n_data,1:n_features], linear_kernel, n_features) -# takes about 4hrs chain_linear = sample(gplvm_linear, NUTS(), 800) -z_mean = permutedims(reshape(mean(group(chain_linear, :Z))[:,2], (ndim, n_data)))' +z_mean = permutedims(reshape(mean(group(chain_linear, :Z))[:,2], (n_features, n_data)))' alpha_mean = mean(group(chain_linear, :α))[:,2] +``` +```julia df_gplvm_linear = DataFrame(z_mean', :auto) -rename!(df_gplvm_linear, Symbol.( ["z"*string(i) for i in collect(1:ndim)])) +rename!(df_gplvm_linear, Symbol.( ["z"*string(i) for i in collect(1:n_features)])) df_gplvm_linear[!,:sample] = 1:n_data df_gplvm_linear[!,:labels] = labels[1:n_data] alpha_indices = sortperm(alpha_mean, rev=true)[1:2] +println(alpha_indices) df_gplvm_linear[!,:ard1] = z_mean[alpha_indices[1], :] df_gplvm_linear[!,:ard2] = z_mean[alpha_indices[2], :] @@ -140,37 +150,118 @@ p1 = df_gplvm_linear|> @vlplot(:point, x=:z1, y=:z2, color="labels:n") ```julia p2 = df_gplvm_linear |> @vlplot(:point, x=:ard1, y=:ard2, color="labels:n") +``` -save("figure1-gplvm-linear.png", p1) -save("figure2-gplvm-linear.png", p2) +```julia +df_gplvm_linear|> @vlplot(:point, x=:z2, y=:z3, color="labels:n") +df_gplvm_linear|> @vlplot(:point, x=:z2, y=:z4, color="labels:n") +df_gplvm_linear|> @vlplot(:point, x=:z3, y=:z4, color="labels:n") ``` +Finally, we demonstrate that by changing the kernel to a non-linear function, we are better able to separate the data. + ```julia gplvm = GPLVM(oil_matrix[1:n_data,:], sekernel, ndim) -# takes about 4hrs -chain = sample(gplvm, NUTS(), 1400) +chain = sample(gplvm, NUTS(), 800) z_mean = permutedims(reshape(mean(group(chain, :Z))[:,2], (ndim, n_data)))' alpha_mean = mean(group(chain, :α))[:,2] +``` +```julia df_gplvm = DataFrame(z_mean', :auto) rename!(df_gplvm, Symbol.( ["z"*string(i) for i in collect(1:ndim)])) df_gplvm[!,:sample] = 1:n_data df_gplvm[!,:labels] = labels[1:n_data] alpha_indices = sortperm(alpha_mean, rev=true)[1:2] +println(alpha_indices) df_gplvm[!,:ard1] = z_mean[alpha_indices[1], :] df_gplvm[!,:ard2] = z_mean[alpha_indices[2], :] p1 = df_gplvm|> @vlplot(:point, x=:z1, y=:z2, color="labels:n") - ``` ```julia p2 = df_gplvm |> @vlplot(:point, x=:ard1, y=:ard2, color="labels:n") +``` + +### Speeding up inference + +Gaussian processes tend to be slow, as they naively require. + +```julia +@model function GPLVM_sparse(Y, kernel_function, ndim=4,::Type{T} = Float64) where {T} -save("figure1-gplvm-exp.png", p1) -save("figure2-gplvm-exp.png", p2) + # Dimensionality of the problem. + N, D = size(Y) + # dimensions of latent space + K = ndim + # number of inducing points + n_inducing = 20 + noise = 1e-3 + + # Priors + α ~ MvLogNormal(MvNormal(K, 1.0)) + σ ~ LogNormal(0.0, 1.0) + Z ~ filldist(Normal(0., 1.), K, N) + # σ_n ~ MvLogNormal(MvNormal(K, 1.0)) + # α = rand( MvLogNormal(MvNormal(K, 1.0))) + # σ = rand( LogNormal(0.0, 1.0)) + # Z = rand( filldist(Normal(0., 1.), K, N)) + # σ_n = rand( MvLogNormal(MvNormal(K, 1.0))) + σ = σ + 1e-12 + + kernel = kernel_function(α, σ) + + ## Standard + # gp = GP(kernel) + # prior = gp(ColVecs(Z), noise) + + ## SPARSE GP + # xu = reshape(repeat(locations, K), :, K) # inducing points + # xu = reshape(repeat(collect(range(-2.0, 2.0; length=20)), K), :, K) # inducing points + lbound = minimum(Y) + 1e-6 + ubound = maximum(Y) - 1e-6 + locations ~ filldist(Uniform(lbound, ubound), n_inducing) + xu = reshape(repeat(locations, K), :, K) # inducing points + gp = Stheno.wrap(GP(kernel), GPC()) + fobs = gp(ColVecs(Z), noise) + finducing = gp(ColVecs(xu'), noise) + prior = SparseFiniteGP(fobs, finducing) + + for d in 1:D + Y[:, d] ~ prior + end +end ``` + +```julia +ndim=2 +n_data=40 +# n_features=size(oil_matrix)[2]; +n_features=4 + +gplvm = GPLVM(oil_matrix[1:n_data,1:n_features], sekernel, ndim) + +chain_gplvm_sparse = sample(gplvm, NUTS(), 500) +z_mean = permutedims(reshape(mean(group(chain_gplvm_sparse, :Z))[:,2], (ndim, n_data)))' +alpha_mean = mean(group(chain_gplvm_sparse, :α))[:,2] +``` + +```julia +df_gplvm_sparse = DataFrame(z_mean', :auto) +rename!(df_gplvm_sparse, Symbol.( ["z"*string(i) for i in collect(1:ndim)])) +df_gplvm_sparse[!,:sample] = 1:n_data +df_gplvm_sparse[!,:labels] = labels[1:n_data] +alpha_indices = sortperm(alpha_mean, rev=true)[1:2] +println(alpha_indices) +df_gplvm_sparse[!,:ard1] = z_mean[alpha_indices[1], :] +df_gplvm_sparse[!,:ard2] = z_mean[alpha_indices[2], :] +p1 = df_gplvm_sparse|> @vlplot(:point, x=:z1, y=:z2, color="labels:n") +``` + +```julia +p2 = df_gplvm_sparse |> @vlplot(:point, x=:ard1, y=:ard2, color="labels:n") ``` ```julia, echo=false, skip="notebook"