regularizacao.jl

### A Pluto.jl notebook ###
# v0.16.2

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing
        el
    end
end

# ╔═╡ 39b4ddbe-31dd-11ec-0381-f56db7928454
using ADNLPModels, Lasso, LinearAlgebra, Logging, NLPModels, Percival, Plots, PlutoUI, Random, Statistics

# ╔═╡ d0a48dfe-e305-49c3-a74e-fedc09d4c4e2
md"""
# Regularização Ridge/L2

- Um problema comum em modelos que dependem de otimização (regressões, redes neurais) é mal-condicionamento;
- Considere regressão múltipla: $\min_{\beta} \tfrac{1}{2}\Vert X\beta - y\Vert^2$
- A solução tem que satisfazer $X^T X \beta = X^T y$
- Se as colunas de $X$ forem múltiplas, a matriz $X^T X$ é singular.
"""

# ╔═╡ cdff3790-2382-4c11-baa3-32e14461faa6
md"""
θ = $(@bind θ Slider(0:90, default=76))
ϕ = $(@bind ϕ Slider(0:90, default=0))
"""

# ╔═╡ bb5fb7e0-933d-4416-9e44-2e59bfded3ec
let
	Random.seed!(0)
	n = 30
	X = rand(n, 2)
	X[:,2] = X[:,1] * 4
	y = X * [1.0; 2.0] + randn(n) * 0.2
	L(β) = norm(y - X * β)^2 / 2
	
	plt = plot(layout=grid(1,2), leg=false, size=(800,300))
	xg = range(0, 4, length=50)
	yg = range(0, 4, length=50)
	contour!(plt[1], xg, yg, (x,y)->L([x;y]), levels=30)
	xlabel!(plt[1], "β₁")
	ylabel!(plt[1], "β₂")
	wireframe!(plt[2], xg, yg, (x,y)->L([x;y]), camera=(θ,ϕ))
	xlabel!(plt[2], "β₁")
	ylabel!(plt[2], "β₂")
end

# ╔═╡ 1a6ebaab-049e-4b0a-9ad2-1201be5a710b
md"""
- Uma solução é adicionar um múltiplo da identidade:
$$(X^T X + \lambda I) \beta = X^T y.$$
- Isso equivale a mudar o problema para
$$\min_{\beta} \tfrac{1}{2}\Vert X\beta - y\Vert^2 + \tfrac{1}{2}\lambda \Vert\beta\Vert^2.$$
- O termo adicional é a **regularização L2**.
"""

# ╔═╡ 0017f273-e6c4-435d-8a12-5158b38f2b5c
begin
	λ₁max = 100.0
	nothing
end

# ╔═╡ 1045f959-ddc9-4c37-bf1a-82e9ce747dfd
md"""
λ = $(@bind λ₁ Slider(range(0, λ₁max, length=10)))
"""

# ╔═╡ 8b19d7bd-dc0e-4351-96b5-f1f792339e4b
let
	Random.seed!(0)
	n = 30
	X = rand(n, 2)
	X[:,2] = X[:,1] * 4
	y = X * [1.0; 2.0] + randn(n) * 0.2
	L(β) = norm(y - X * β)^2 / 2 + λ₁ * norm(β)^2 / 2
	sol(λ) = (X' * X + λ * I) \ (X' * y)
	
	plt = plot(layout=grid(1,2), leg=false, size=(800,300))
	xg = range(0, 4, length=50)
	yg = range(0, 4, length=50)
	contour!(plt[1], xg, yg, (x,y)->L([x;y]), levels=30)
	if λ₁ > 0
		βsol = sol(λ₁)
		scatter!(plt[1], [βsol[1]], [βsol[2]], c=:red, ms=3)
	end
	xlabel!(plt[1], "β₁")
	ylabel!(plt[1], "β₂")
	wireframe!(plt[2], xg, yg, (x,y)->L([x;y]), camera=(θ,ϕ))
	xlabel!(plt[2], "β₁")
	ylabel!(plt[2], "β₂")
end

# ╔═╡ b9eb9583-2985-4c68-ba7b-1dc38db9db6c
md"""
- De uma maneira geral, a regularização é feita adicionando o termo $\lambda\Vert\beta\Vert^2$.
"""

# ╔═╡ 5c1d8fbd-4dae-45a8-81af-6437c01cb11d
md"""
## Exemplo polinomial
"""

# ╔═╡ 66130249-4fd8-4e87-a277-9e8db9bef387
md"""
λ = $(@bind λp₁ Slider(-10:2:6))
"""

# ╔═╡ c59c9496-defa-4af5-8256-9848e740efa9
let
	λ = 3.0^λp₁
	n = 50
	x = range(-2, 2, length=n)
	Random.seed!(0)
	y = (x .- 1) .* (x .+ 0.2) .+ 3 + randn(n)
	p = 10

	plt = plot(layout=grid(1, 2, widths=[0.6, 0.4]), leg=false, size=(800,400))
	
	scatter!(plt[1], x, y, m=(3,:lightblue,stroke(1,:blue)))
	X = [x[i]^j for i = 1:n, j = 0:p]
	β = (X' * X + λ * I) \ (X' * y)
	h(x) = β[1] + sum(β[j+1] * x^j for j = 1:p)
	plot!(plt[1], h, extrema(x)..., c=:red, lw=2)

	anns = Tuple[]
	for i = 1:p
		xpos = 0.5
		ypos = 0.9 - 0.8 * (i - 1) / (p - 1)
		text = "β[$i] = $(round(β[i], digits=2))"
		push!(anns, (xpos, ypos, text))
	end
	annotate!(plt[2], anns)
end

# ╔═╡ 0bdf27fa-4b22-4ee3-907a-ce37973714f0
md"""
# Regularização LASSO/L1

- Outra maneira de regularizar é adicionar um termo $\lambda\Vert\beta\Vert_1 = \lambda \sum_{i=1}^p |\beta_i|$.
$$\min_{\beta} \tfrac{1}{2}\Vert X\beta - y\Vert^2 + \lambda \Vert \beta\Vert_1,$$
- Essa maneira é mais difícil de resolver o problema pois não é diferenciável.
- No entanto oferece outra vantagem, pois manda parâmetros pra 0.
- Uma maneira de resolver o problema é alternar para otimização com restrições
$$\min_{\beta} \tfrac{1}{2}\Vert X\beta - y\Vert^2 \qquad \text{suj. to} \qquad \Vert\beta\Vert_1 \leq \Delta,$$
para algum $\Delta \geq 0$.
- A relação entre $\lambda$ e $\Delta$ não é trivial.
"""

# ╔═╡ e88e6dd4-dc3f-4ceb-85c9-2934191faaa8
let
	# Menti, esse é o Δ, mas fiz parecer que era um λ 		8]
	md"""
	λ = $(@bind logΔ Slider(-2:0.5:2))
	"""
end

# ╔═╡ d065a217-f8b7-465c-a598-d85aeb8a60c3
let
	Δ = 2.0 ^ (-logΔ)
	Random.seed!(0)
	n = 30
	X = rand(n, 2)
	X[:,2] = X[:,1] * 4
	y = X * [1.0; 2.0] + randn(n) * 0.2
	nlp = ADNLPModel(
		β -> norm(y - X * β)^2 / 2,
		zeros(2),
		β -> [β[1] + β[2]; β[1] - β[2]; -β[1] + β[2]; -β[1] - β[2]],
		fill(-Inf, 4),
		Δ * ones(4),
	)
	output = with_logger(NullLogger()) do
		percival(nlp, atol=1e-3, rtol=1e-3)
	end
	β = output.solution
	λ₂ = 0.0
	if norm(β) ≈ 0
		λ₂ = 1e2
	elseif abs(norm(β, 1) - Δ) < 1e-3
		λ₂ = -( X' * (X * β - y) )[2]
	end
	L(β) = norm(y - X * β)^2 / 2 + λ₂ * norm(β, 1)
	
	plt = plot(layout=grid(1,2), leg=false, size=(800,300))
	xg = range(0, 4, length=50)
	yg = range(0, 4, length=50)
	contour!(plt[1], xg, yg, (x,y)->L([x;y]), levels=30)
	plot!([Δ; 0], [0; Δ], c=:green, l=:dash)
	scatter!(plt[1], [β[1]], [β[2]], c=:red, ms=3)
	xlabel!(plt[1], "β₁")
	ylabel!(plt[1], "β₂")
	wireframe!(plt[2], xg, yg, (x,y)->L([x;y]), camera=(θ,ϕ))
	xlabel!(plt[2], "β₁")
	ylabel!(plt[2], "β₂")
end

# ╔═╡ 19565c67-eb15-4040-9646-ff11da00113f
md"""
## Exemplo polinomial
"""

# ╔═╡ 135989e8-6346-417c-bce9-8ead951889c0
md"""
λ = $(@bind λp₂ Slider(-5:0.5:0))
"""

# ╔═╡ 7fa11a8a-b0f2-42fd-8a15-11a5ee6b181c
let
	λ = 3.0^λp₂
	n = 50
	x = range(-2, 2, length=n)
	Random.seed!(0)
	y = (x .- 1) .* (x .+ 0.2) .+ 3 + randn(n)
	p = 10

	plt = plot(layout=grid(1, 2, widths=[0.6, 0.4]), leg=false, size=(800,400))
	
	scatter!(plt[1], x, y, m=(3,:lightblue,stroke(1,:blue)))
	X = [x[i]^j for i = 1:n, j = 1:p]
	f = fit(LassoPath, X, y, α=1.0, λ=[λ])
	β = [f.b0; f.coefs][:]
	h(x) = β[1] + sum(β[j+1] * x^j for j = 1:p)
	plot!(plt[1], h, extrema(x)..., c=:red, lw=2)

	anns = Tuple[]
	for i = 1:p
		xpos = 0.5
		ypos = 0.9 - 0.8 * (i - 1) / (p - 1)
		text = "β[$i] = $(round(β[i], digits=2))"
		push!(anns, (xpos, ypos, text))
	end
	annotate!(plt[2], anns)
end

# ╔═╡ 0a144a63-025b-4819-91f6-35c894766218
md"""
# Como escolher $\lambda$?

- Validação cruzada.
"""

# ╔═╡ 4320809b-2915-42df-ad5d-586e9f87a861
let
	Random.seed!(0)
	n = 100
	p = 10
	X = rand(n, p)
	β = randn(p)
	y = X * β + randn(n)
	
	nfolds = 5
	λset = 2.0 .^ (-10:13)
	nλ = length(λset)
	
	sqr_tr = zeros(nfolds, nλ)
	sqr_te = zeros(nfolds, nλ)
	
	I = shuffle(1:n)
	foldsize = ceil(Int, n / nfolds)
	for k = 1:nfolds
		Ite = (k-1)*foldsize + 1:k*foldsize
		Itr = setdiff(1:n, Ite)
		for (j,λ) in enumerate(λset)
			β = (X[Itr,:]' * X[Itr,:] + λ * LinearAlgebra.I) \ (X[Itr,:]' * y[Itr])
			ŷ = X * β
			sqr_tr[k,j] = norm(y[Itr] - ŷ[Itr])^2 / length(Itr)
			sqr_te[k,j] = norm(y[Ite] - ŷ[Ite])^2 / length(Ite)
		end
	end
	
	plot(leg=false, xaxis=:log)
	for k = 1:nfolds
		plot!(λset, sqr_tr[k,:], c=:blue, opacity=0.2)
		plot!(λset, sqr_te[k,:], c=:red, opacity=0.2)
	end
	sqr_tr_avg = sum(sqr_tr, dims=1)[:] / nfolds
	sqr_te_avg = sum(sqr_te, dims=1)[:] / nfolds
	plot!(λset, sqr_tr_avg, c=:blue, m=(2))
	plot!(λset, sqr_te_avg, c=:red, m=(3))
	xlabel!("λ")
	ylabel!("SQR")
end

# ╔═╡ 49e08968-1ccc-453a-acff-2aa6db75d9ae
md"""
# Elastic Net

$$\min_{\beta} \tfrac{1}{2}\Vert X\beta - y\Vert^2 + \lambda \bigg((1 - \alpha)\tfrac{1}{2}\Vert\beta\Vert_2^2 + \alpha\Vert\beta\Vert_1\bigg)$$
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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Percival = "01435c0c-c90d-11e9-3788-63660f8fbccc"
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Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[compat]
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Lasso = "~0.6.2"
NLPModels = "~0.17.2"
Percival = "~0.4.1"
Plots = "~1.22.6"
PlutoUI = "~0.7.1"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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[[FastClosures]]
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[[FillArrays]]
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[[FixedPointNumbers]]
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[[Fontconfig_jll]]
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[[Formatting]]
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[[ForwardDiff]]
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[[FreeType2_jll]]
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[[FriBidi_jll]]
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[[GLFW_jll]]
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[[GLM]]
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[[GR]]
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[[GR_jll]]
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git-tree-sha1 = "cafe0823979a5c9bff86224b3b8de29ea5a44b2e"
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[[GeometryBasics]]
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[[Gettext_jll]]
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[[Glib_jll]]
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[[Graphite2_jll]]
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[[Grisu]]
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[[HTTP]]
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[[HarfBuzz_jll]]
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git-tree-sha1 = "8a954fed8ac097d5be04921d595f741115c1b2ad"
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[[IniFile]]
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[[InlineStrings]]
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[[IntelOpenMP_jll]]
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[[InteractiveUtils]]
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[[Intervals]]
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[[InverseFunctions]]
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[[IrrationalConstants]]
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[[IteratorInterfaceExtensions]]
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[[JLLWrappers]]
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[[JSON]]
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[[JSOSolvers]]
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[[JpegTurbo_jll]]
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git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943"
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[[Krylov]]
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[[LAME_jll]]
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git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
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[[LZO_jll]]
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git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
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[[LaTeXStrings]]
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[[Lasso]]
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[[Latexify]]
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[[LazyArtifacts]]
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[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
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[[LibCURL_jll]]
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uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"

[[LibGit2]]
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[[LibSSH2_jll]]
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uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[Libdl]]
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[[Libffi_jll]]
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git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b"
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[[Libgcrypt_jll]]
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git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
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[[Libglvnd_jll]]
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[[Libgpg_error_jll]]
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uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
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[[Libiconv_jll]]
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git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
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[[Libmount_jll]]
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git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
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[[Libtiff_jll]]
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[[Libuuid_jll]]
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[[LinearAlgebra]]
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[[LinearOperators]]
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[[LogExpFunctions]]
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[[Logging]]
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[[MKL_jll]]
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[[MLBase]]
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[[MacroTools]]
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[[Markdown]]
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[[MbedTLS]]
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[[MbedTLS_jll]]
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[[Measures]]
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[[Missings]]
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[[Mmap]]
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[[Mocking]]
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[[MozillaCACerts_jll]]
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[[MutableArithmetics]]
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[[NLPModels]]
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[[NLPModelsModifiers]]
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[[NaNMath]]
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[[NetworkOptions]]
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[[Ogg_jll]]
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[[OpenLibm_jll]]
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uuid = "05823500-19ac-5b8b-9628-191a04bc5112"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
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[[OpenSpecFun_jll]]
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version = "0.5.5+0"

[[Opus_jll]]
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git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
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[[OrderedCollections]]
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[[PCRE_jll]]
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git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
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[[PDMats]]
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[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "f19e978f81eca5fd7620650d7dbea58f825802ee"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.1.0"

[[Percival]]
deps = ["JSOSolvers", "Krylov", "LinearAlgebra", "LinearOperators", "Logging", "NLPModels", "NLPModelsModifiers", "SolverCore", "SolverTools"]
git-tree-sha1 = "687af63786661de19503c340d950cffd9471ba7c"
uuid = "01435c0c-c90d-11e9-3788-63660f8fbccc"
version = "0.4.1"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.15"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "ba43b248a1f04a9667ca4a9f782321d9211aa68e"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.22.6"

[[PlutoUI]]
deps = ["Base64", "Dates", "InteractiveUtils", "Logging", "Markdown", "Random", "Suppressor"]
git-tree-sha1 = "45ce174d36d3931cd4e37a47f93e07d1455f038d"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.1"

[[Polynomials]]
deps = ["Intervals", "LinearAlgebra", "MutableArithmetics", "RecipesBase"]
git-tree-sha1 = "7499556d31417baeabaa55d266a449ffe4ec5a3e"
uuid = "f27b6e38-b328-58d1-80ce-0feddd5e7a45"
version = "2.0.17"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[QuadGK]]
deps = ["DataStructures", "LinearAlgebra"]
git-tree-sha1 = "78aadffb3efd2155af139781b8a8df1ef279ea39"
uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
version = "2.4.2"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RecipesBase]]
git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.2"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.1"

[[Reexport]]
deps = ["Pkg"]
git-tree-sha1 = "7b1d07f411bc8ddb7977ec7f377b97b158514fe0"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "0.2.0"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[Rmath]]
deps = ["Random", "Rmath_jll"]
git-tree-sha1 = "bf3188feca147ce108c76ad82c2792c57abe7b1f"
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"

[[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"

[[ShiftedArrays]]
git-tree-sha1 = "22395afdcf37d6709a5a0766cc4a5ca52cb85ea0"
uuid = "1277b4bf-5013-50f5-be3d-901d8477a67a"
version = "1.0.0"

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

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

[[SolverCore]]
deps = ["NLPModels", "Printf"]
git-tree-sha1 = "1e1494bc9bae276d78444970fab820b8a111b2d5"
uuid = "ff4d7338-4cf1-434d-91df-b86cb86fb843"
version = "0.2.1"

[[SolverTools]]
deps = ["LinearAlgebra", "LinearOperators", "NLPModels", "Printf"]
git-tree-sha1 = "6f41ab4df5dd522689078a870168c953cfa86971"
uuid = "b5612192-2639-5dc1-abfe-fbedd65fab29"
version = "0.6.2"

[[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 = "2d57e14cd614083f132b6224874296287bfa3979"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "1.8.0"

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

[[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 = "eb35dcc66558b2dda84079b9a1be17557d32091a"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.12"

[[StatsFuns]]
deps = ["Rmath", "SpecialFunctions"]
git-tree-sha1 = "ced55fd4bae008a8ea12508314e725df61f0ba45"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.7"

[[StatsModels]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Printf", "REPL", "ShiftedArrays", "SparseArrays", "StatsBase", "StatsFuns", "Tables"]
git-tree-sha1 = "5cfe2d754634d9f11ae19e7b45dad3f8e4883f54"
uuid = "3eaba693-59b7-5ba5-a881-562e759f1c8d"
version = "0.6.27"

[[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"

[[Suppressor]]
git-tree-sha1 = "a819d77f31f83e5792a76081eee1ea6342ab8787"
uuid = "fd094767-a336-5f1f-9728-57cf17d0bbfb"
version = "0.2.0"

[[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"

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

[[TimeZones]]
deps = ["Dates", "Downloads", "InlineStrings", "LazyArtifacts", "Mocking", "Pkg", "Printf", "RecipesBase", "Serialization", "Unicode"]
git-tree-sha1 = "b4c6460412b1db0b4f1679ab2d5ef72568a14a57"
uuid = "f269a46b-ccf7-5d73-abea-4c690281aa53"
version = "1.6.1"

[[TimerOutputs]]
deps = ["ExprTools", "Printf"]
git-tree-sha1 = "7cb456f358e8f9d102a8b25e8dfedf58fa5689bc"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.13"

[[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"

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

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+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_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+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_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+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_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[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"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+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"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
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# ╟─00000000-0000-0000-0000-000000000001
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