### A Pluto.jl notebook ###
# v0.17.1

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(; resolution=(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
# 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
# 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]
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Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[compat]
CairoMakie = "~0.6.5"
DataFrames = "~1.2.2"
Distributions = "~0.25.19"
MLJ = "~0.16.9"
MLJDecisionTreeInterface = "~0.1.3"
MLJLinearModels = "~0.5.6"
MLJModelInterface = "~1.3.2"
MLJTuning = "~0.6.13"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa"
version = "0.7.0"

[[Rmath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "68db32dff12bb6127bac73c209881191bf0efbb7"
uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
version = "0.3.0+0"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[SIMD]]
git-tree-sha1 = "9ba33637b24341aba594a2783a502760aa0bff04"
uuid = "fdea26ae-647d-5447-a871-4b548cad5224"
version = "3.3.1"

[[ScanByte]]
deps = ["Libdl", "SIMD"]
git-tree-sha1 = "9cc2955f2a254b18be655a4ee70bc4031b2b189e"
uuid = "7b38b023-a4d7-4c5e-8d43-3f3097f304eb"
version = "0.3.0"

[[ScientificTypes]]
deps = ["CategoricalArrays", "ColorTypes", "Dates", "Distributions", "PrettyTables", "Reexport", "ScientificTypesBase", "StatisticalTraits", "Tables"]
git-tree-sha1 = "b58ef9adeba0d96ae1add87e8df5a7128342e312"
uuid = "321657f4-b219-11e9-178b-2701a2544e81"
version = "2.3.2"

[[ScientificTypesBase]]
git-tree-sha1 = "185e373beaf6b381c1e7151ce2c2a722351d6637"
uuid = "30f210dd-8aff-4c5f-94ba-8e64358c1161"
version = "2.3.0"

[[ScikitLearnBase]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "7877e55c1523a4b336b433da39c8e8c08d2f221f"
uuid = "6e75b9c4-186b-50bd-896f-2d2496a4843e"
version = "0.5.0"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[SignedDistanceFields]]
deps = ["Random", "Statistics", "Test"]
git-tree-sha1 = "d263a08ec505853a5ff1c1ebde2070419e3f28e9"
uuid = "73760f76-fbc4-59ce-8f25-708e95d2df96"
version = "0.4.0"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "793793f1df98e3d7d554b65a107e9c9a6399a6ed"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "1.7.0"

[[StableRNGs]]
deps = ["Random", "Test"]
git-tree-sha1 = "3be7d49667040add7ee151fefaf1f8c04c8c8276"
uuid = "860ef19b-820b-49d6-a774-d7a799459cd3"
version = "1.0.0"

[[StackViews]]
deps = ["OffsetArrays"]
git-tree-sha1 = "46e589465204cd0c08b4bd97385e4fa79a0c770c"
uuid = "cae243ae-269e-4f55-b966-ac2d0dc13c15"
version = "0.1.1"

[[Static]]
deps = ["IfElse"]
git-tree-sha1 = "a8f30abc7c64a39d389680b74e749cf33f872a70"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.3.3"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

[[StatisticalTraits]]
deps = ["ScientificTypesBase"]
git-tree-sha1 = "730732cae4d3135e2f2182bd47f8d8b795ea4439"
uuid = "64bff920-2084-43da-a3e6-9bb72801c0c9"
version = "2.1.0"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "65fb73045d0e9aaa39ea9a29a5e7506d9ef6511f"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.11"

[[StatsFuns]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "95072ef1a22b057b1e80f73c2a89ad238ae4cfff"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.12"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

[[SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[Syslogs]]
deps = ["Printf", "Sockets"]
git-tree-sha1 = "46badfcc7c6e74535cc7d833a91f4ac4f805f86d"
uuid = "cea106d9-e007-5e6c-ad93-58fe2094e9c4"
version = "0.3.0"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[TiffImages]]
deps = ["ColorTypes", "DocStringExtensions", "FileIO", "FixedPointNumbers", "IndirectArrays", "Inflate", "OffsetArrays", "OrderedCollections", "PkgVersion", "ProgressMeter"]
git-tree-sha1 = "945b8d87c5e8d5e34e6207ee15edb9d11ae44716"
uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69"
version = "0.4.3"

[[TimeZones]]
deps = ["Dates", "Downloads", "InlineStrings", "LazyArtifacts", "Mocking", "Pkg", "Printf", "RecipesBase", "Serialization", "Unicode"]
git-tree-sha1 = "9408d0773ed2dbfba68ebc0e2d5dd388a5e668a9"
uuid = "f269a46b-ccf7-5d73-abea-4c690281aa53"
version = "1.6.0"

[[TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "216b95ea110b5972db65aa90f88d8d89dcb8851c"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.6"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "9398e8fefd83bde121d5127114bd3b6762c764a6"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.4"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[isoband_jll]]
deps = ["Libdl", "Pkg"]
git-tree-sha1 = "a1ac99674715995a536bbce674b068ec1b7d893d"
uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4"
version = "0.2.2+0"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "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"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"
"""

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