### A Pluto.jl notebook ### # v0.19.38 using Markdown using InteractiveUtils # ╔═╡ 5e902e3c-2c19-11ec-39df-3fba9fe5640c begin import MLJLinearModels import MLJDecisionTreeInterface using DataFrames: DataFrame, select, Not using Distributions: Normal using CairoMakie: Axis, Figure, lines, lines!, scatter, scatter!, current_figure, axislegend, help, linkxaxes!, linkyaxes!, xlabel!, density, density!, hidedecorations!, violin!, boxplot!, hidexdecorations!, hideydecorations! using MLJ: CV, evaluate, models, matching, @load, machine, fit!, predict, predict_mode, rms using Random: seed! using Statistics: mean, std, var, median using MLJTuning: TunedModel, Explicit using MLJModelInterface: Probabilistic, Deterministic end # ╔═╡ ea44c5c8-02d6-4bcb-90e7-01fb3db9e44e # hideall md""" Nested cross-validation is said to be an improvement over cross-validation. Unfortunately, I found most explanations quite confusing, so decided to simulate some data and see what happens. In this post, I simulate two models: one linear model which perfectly fits the data and one which overfits the data. Next, cross-validation and nested cross-validation are plotted. To keep the post short, I've hidden the code to produce the plots. """ # ╔═╡ b37e1bc1-2f53-4637-852b-5c8e38912a0e # hideall function compatible_models() X = (a = [1.0], b = [2.0], c = [3.0]) y = [4.0] M = models(matching(X, y)) filter(m -> m.is_pure_julia, M) end; # ╔═╡ b0b55bb3-d445-4f45-bd33-c26f28d074e1 # hideall compatible_models(); # ╔═╡ e4584145-243d-4ba9-853c-82189a9b96df y_true(x) = 2x + 10; # ╔═╡ a998b36b-7183-4657-ae14-ab81212138ab y_real(x) = y_true(x) + rand(Normal(0, 40)); # ╔═╡ fc89a45d-fe73-45aa-a8ef-bf090884dd00 indexes = 1.0:100; # ╔═╡ 3f39b4eb-7aab-4dda-adbf-3be971d4cb99 df = let seed!(0) DataFrame(x = indexes, y = y_real.(indexes)) end # ╔═╡ b641a8d6-624b-4562-92e2-276388f573de LinearModel = @load LinearRegressor pkg=MLJLinearModels verbosity=0; # ╔═╡ d7751696-1bb5-4ee8-a8e7-45da5dd71947 TreeModel = @load DecisionTreeRegressor pkg=DecisionTree verbosity=0; # ╔═╡ 3f99e49e-9e51-4f8f-9c21-9f670d63f3c4 X, y = (select(df, Not(:y)), df.y); # ╔═╡ 1e0ac73d-1faf-4250-a6f5-032c67649326 function linear_model() model = LinearModel(fit_intercept=true) mach = machine(model, X, y) fit!(mach) return mach end; # ╔═╡ 772ec163-7861-4bbe-a90b-d6b6f7c08040 function tree_model() model = TreeModel() mach = machine(model, X, y) fit!(mach) return mach end; # ╔═╡ d07d4596-bc21-4ba7-9f78-9ce00d1d9339 # hideall let linear_predictions = predict(linear_model()) tree_predictions = predict(tree_model()) color = :black markersize = 2 linewidth = 2 fig = Figure() ax1 = Axis(fig[1, 1]; ylabel="y") ax2 = Axis(fig[2, 1]; xlabel="x", ylabel="y") lines!(ax1, df.x, linear_predictions; linewidth, color, label="LinearRegressor") lines!(ax1, df.x, y_true.(df.x); linewidth, linestyle=:dash, color, label="True model") scatter!(ax1, df.x, df.y; color, markersize) axislegend(ax1; position=:lt) lines!(ax2, df.x, tree_predictions; linewidth, color, label="DecisionTreeRegressor") lines!(ax2, df.x, y_true.(df.x); linewidth, linestyle=:dash, color, label="True model") scatter!(ax2, df.x, df.y; color, markersize) axislegend(ax2; position=:lt) fig end # ╔═╡ 86a2c569-d139-467e-948c-18f60d0cd93a # hideall md""" Okay, so which model performs better. I would guess the `LinearRegressor`, but let's see what the root-mean-square error (RMS) is when we fit the models on the training data: """ # ╔═╡ a09b498e-931e-4fd3-b3e3-eefc16d2e65f rms(predict(linear_model()), df.y) # ╔═╡ 662d7494-6248-4adc-ac0d-0f4c68e4418d rms(predict(tree_model()), df.y) # ╔═╡ c0a0bd8b-fe78-4ea1-8619-62a26a968a10 # hideall md""" Clearly, the tree model is overfitting the data. In other words, the model is not expected to perform well on new data. Now the question is whether we can determine that the linear model is the right one via cross-validation. Let's first plot the error for each of our $k$ folds. """ # ╔═╡ 696808d5-3547-4666-bfcc-46e658ebbced # hideall r1(x) = round(x; digits=1); # ╔═╡ 916a78d9-a6f7-4e60-8511-9d654f74f3de # hideall function evaluate_model(model, resampling) e = evaluate(model, X, y; resampling) per_fold = first(e.per_fold) end; # ╔═╡ bfebf58b-daf5-4d74-ba9b-fc853da3b166 # hideall foldsmean(per_fold) = fill(mean(per_fold), length(per_fold)); # ╔═╡ da994d08-9d14-45b9-a80a-7ea6a13edd67 # hideall let nfolds = 10 color = :black fig = Figure() resampling = CV(; nfolds) linear_per_fold = evaluate_model(LinearModel(), resampling) tree_per_fold = evaluate_model(TreeModel(), resampling) foldsmean(per_fold) = fill(mean(per_fold), length(folds)) folds = 1:nfolds ax1 = Axis(fig[1, 1]; xlabel="fold", ylabel="root-mean-square error") lines!(ax1, folds, linear_per_fold; color, label="LinearRegressor") lines!(ax1, folds, foldsmean(linear_per_fold); color, linestyle=:dash, label="Mean") axislegend(ax1; position=:lt) ax2 = Axis(fig[1, 2]; xlabel="fold") lines!(ax2, folds, tree_per_fold; color, label="DecisionTreeRegressor") lines!(ax2, folds, foldsmean(tree_per_fold); color, linestyle=:dash, label="Mean") axislegend(ax2; position=:lt) hideydecorations!(ax2) # ax3 = Axis(fig[1, 4:5]) # density!(ax3, linear_per_fold; strokewidth=1, strokecolor=:black, color=:white, direction=:y) # hidedecorations!(ax3; grid=false) linkyaxes!(ax1, ax2) current_figure() end # ╔═╡ 641ac771-1424-45ba-a313-d6e1e8b7eaf4 # hideall md""" So, basically cross-validation isn't gonna be perfect. If the data or standard deviation would have been different, then another model could have obtained a lower error according to the cross-validation. Let's tryout nested cross-validation. According to Zhang ([2015](https://doi.org/10.1016/j.jeconom.2015.02.006)), repeated 50- and 20- fold CV is best for $n_t$ sample points and the best cross-validation parameters for model selection are not necessarily the same the best cross-validation parameters for performance estimation (p. 104 and p. 105). """ # ╔═╡ d0e5d310-d09d-4e00-b942-07e48d8ab00e function evaluate_inner_folds(nfolds::Int, ntrials::Int) inner_resampling = CV(; nfolds=nfolds) multi_model = TunedModel(; models=[LinearModel(), TreeModel()], resampling=inner_resampling); outer_resampling = CV(; nfolds=ntrials) e = evaluate(multi_model, X, y; measure=rms, resampling=outer_resampling) return e end; # ╔═╡ feaca510-5403-474f-b598-8467569161ce # hideall md""" The problem of cross-validation is that it is still possible to overfit during model selection. Therefore, the only reliable way to estimate model performance is to use nested cross-validation (Krstajic et al., [2014](https://doi.org/10.1186/1758-2946-6-10)). Also, repeated k-fold nested cross-validation is the most promising for prediction error estimation. Let's see how the plots for nested cross-validation look: """ # ╔═╡ 0024dfce-b6ff-4a38-8afc-1092591422cb # hideall function plot_trial_model!(ax, e, trial::Int, model::Int) m = e.report_per_fold[trial].history[model] indexes = 1:length(first(m.per_fold)) color = :black label = string(m.model) per_fold = first(m.per_fold) lines!(ax, indexes, per_fold; label, color) # lines!(ax1, folds, linear_per_fold; color, label="LinearRegressor") lines!(ax, indexes, foldsmean(per_fold); color, linestyle=:dash, label="Mean") axislegend(ax; position=:lt) end; # ╔═╡ dccd8ec8-cc40-43a1-bad7-e1ad699a4a40 # hideall function nested_variance(e) var(e.per_fold[1]) end; # ╔═╡ 72012d1a-fd91-4b40-9137-18dc01205c00 # hideall function plot_trial!(fig, e, trial::Int) ax1 = Axis(fig[trial, 1]; ylabel="trial $trial \n root-mean-square error") ax2 = Axis(fig[trial, 2]) plot_trial_model!(ax1, e, trial, 1) plot_trial_model!(ax2, e, trial, 2) linkyaxes!(ax1, ax2) hideydecorations!(ax2) return ax1, ax2 end; # ╔═╡ e1de9f5d-c062-4068-8652-e0bac41bee9b e = evaluate_inner_folds(20, 20); # ╔═╡ 5f92f025-fd2a-4ed4-84b6-ebadb0c689fe # hideall nested_variance(e); # ╔═╡ 95f8e3cf-c6b9-4d84-bc35-b933008bf112 # hideall let color = :black fig = Figure(; size=(900, 1500)) plot_trial!(fig, e, 1) plot_trial!(fig, e, 2) plot_trial!(fig, e, 3) plot_trial!(fig, e, 4) plot_trial!(fig, e, 5) axa, axb = plot_trial!(fig, e, 6) axa.xlabel = "fold" axb.xlabel = "fold" current_figure() end # ╔═╡ 88544422-e781-43a4-8eae-35713aa59193 # hideall foldrange = 2:2:83; # ╔═╡ 43f00e6a-ad6d-47e9-8074-266fffa3cb82 # ╠═╡ show_logs = false # hideall foldevaluations = [evaluate_inner_folds(i, 6) for i in foldrange]; # ╔═╡ 37c2861b-fed1-4c78-a8e5-0827ab98e477 # hideall function evaluate_function!(ax, range, evaluations, f) results = [f(e.per_fold[1]) for e in evaluations] ax.ylabel = f == var ? "variance" : string(f) lines!(ax, range, results; color=:black) scatter!(ax, range, results; color=:black) end; # ╔═╡ 95877eb0-f7b7-4123-8718-b9d30f3e2c3f # hideall md""" That looks interesting, but what happens with the median, mean and variance if we change the number of folds and trials? """ # ╔═╡ 626e0034-caef-4435-9716-c4449375159e # hideall function plot_evaluations(range, evals; xlabel=nothing) means = [mean(e.per_fold[1]) for e in evals] fig = Figure() ax1 = Axis(fig[1, 1]) evaluate_function!(ax1, range, evals, median) ax2 = Axis(fig[2, 1]) evaluate_function!(ax2, range, evals, mean) linkyaxes!(ax1, ax2) hidexdecorations!(ax1) ax3 = Axis(fig[3, 1]; xlabel) evaluate_function!(ax3, range, evals, var) hidexdecorations!(ax2) linkxaxes!(ax1, ax2, ax3) current_figure() end; # ╔═╡ 40f48800-1aca-4587-a26e-fea92b0ce8ba # hideall plot_evaluations(foldrange, foldevaluations; xlabel="number of folds") # ╔═╡ 0630eb20-01a3-41f8-a7ad-202a56b43954 # hideall trialrange = 2:4:60; # ╔═╡ ad007c31-d39b-4f10-9e95-755f40cb6844 # ╠═╡ show_logs = false # hideall trialevaluations = [evaluate_inner_folds(30, i) for i in trialrange]; # ╔═╡ 135f5d2b-4645-493b-97ae-e4d9e76ab8be # hideall plot_evaluations(trialrange, trialevaluations; xlabel="number of trials") # ╔═╡ 734712fa-0871-4793-9ef4-74f856f2613f # hideall md""" I don't know what should be the take-away here. What kind of makes sense is that the variance increases for many trails. The reason is most likely that the samples become too small and fitting a model is either a complete hit or miss. Why the median and mean go down is unclear to me. Maybe, fitting is more likely to be a hit than a miss. Therefore, if the number of trails is increased, then more fits are a hit which result in a lower error on average. """ # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" MLJ = "add582a8-e3ab-11e8-2d5e-e98b27df1bc7" MLJDecisionTreeInterface = "c6f25543-311c-4c74-83dc-3ea6d1015661" MLJLinearModels = "6ee0df7b-362f-4a72-a706-9e79364fb692" MLJModelInterface = "e80e1ace-859a-464e-9ed9-23947d8ae3ea" MLJTuning = "03970b2e-30c4-11ea-3135-d1576263f10f" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" [compat] CairoMakie = "~0.11.8" DataFrames = "~1.6.1" Distributions = "~0.25.107" MLJ = "~0.20.2" MLJDecisionTreeInterface = "~0.4.0" MLJLinearModels = "~0.10.0" MLJModelInterface = "~1.9.5" MLJTuning = "~0.8.1" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is machine-generated - editing it directly is not advised 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"FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" version = "0.15.1+0" [[deps.libblastrampoline_jll]] deps = ["Artifacts", "Libdl"] uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" version = "5.8.0+1" [[deps.libfdk_aac_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" version = "2.0.2+0" [[deps.libpng_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"] git-tree-sha1 = "93284c28274d9e75218a416c65ec49d0e0fcdf3d" uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" version = "1.6.40+0" [[deps.libsixel_jll]] deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"] git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c" uuid = "075b6546-f08a-558a-be8f-8157d0f608a5" version = "1.10.3+0" 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╠═5f92f025-fd2a-4ed4-84b6-ebadb0c689fe # ╠═72012d1a-fd91-4b40-9137-18dc01205c00 # ╠═e1de9f5d-c062-4068-8652-e0bac41bee9b # ╠═95f8e3cf-c6b9-4d84-bc35-b933008bf112 # ╠═88544422-e781-43a4-8eae-35713aa59193 # ╠═43f00e6a-ad6d-47e9-8074-266fffa3cb82 # ╠═37c2861b-fed1-4c78-a8e5-0827ab98e477 # ╠═95877eb0-f7b7-4123-8718-b9d30f3e2c3f # ╠═626e0034-caef-4435-9716-c4449375159e # ╠═40f48800-1aca-4587-a26e-fea92b0ce8ba # ╠═135f5d2b-4645-493b-97ae-e4d9e76ab8be # ╠═0630eb20-01a3-41f8-a7ad-202a56b43954 # ╠═ad007c31-d39b-4f10-9e95-755f40cb6844 # ╠═734712fa-0871-4793-9ef4-74f856f2613f # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002