explicação/regressao-logistica.jl

### A Pluto.jl notebook ###
# v0.16.1

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

# ╔═╡ 992de666-e8cf-11eb-3f19-53e69f07a376
using ADNLPModels, JSOSolvers, NLPModels, Logging, LinearAlgebra, Plots, PlutoUI, Random

# ╔═╡ da9f5da6-b61b-4483-bab6-bfce4c19ec5c
md"""
# Regressão Logística

- Apesar do nome, é um método para classificação!
- A ideia vem de cara ou coroa _desbalanceada_.
- Uma moeda desbalanceada tem probabidade $p \in (0,1)$ de cair Cara (1), e portanto $1 - p$ de cair Coroa (0). (Bernoulli)
- Podemos escrever esta probabilidade como
$$P = p^k (1 - p)^{1 - k},$$
com $k = 0$ ou $k = 1$.
- Pra cada $x_i$, existe uma moeda diferente.
"""

# ╔═╡ df9e884f-bf4a-4fd9-8140-f21c82134207
begin
	Random.seed!(1)
	n = 50
	x = range(-3, 3, length=n) + 1e-1 * randn(n)
	y = [x[i] < 0.2 + randn() * 0.8 ? 0 : 1 for i = 1:n]
	I = findall(y .== 0)
	scatter(x[I], y[I], m=(3,:pink,:square,stroke(1,:red)), leg=false)
	I = findall(y .== 1)
	scatter!(x[I], y[I], m=(3,:lightblue,:circle,stroke(1,:blue)))
end

# ╔═╡ f2b17cd7-8f56-4d9b-8599-6f2da53a13f8
md"""
- O modelo $h(x)$ será usado para $p$, e para uma observação de cara ou coroa $y_i$, a probabilidade de ter caído $y_i$ pra moeda de $x_i$ é
$$P_i = h(x_i)^{y_i} \left(1 - h(x_i)\right)^{1 - y_i}.$$
- Como $h(x_i)$ deve estar entre $0$ e $1$, o modelo linear $h(x) = \beta_0 + \beta_1 x$ não é válido, em geral.
- Uma alternativa é usar a função logística $\sigma(t) = (1 + e^{-t})^{-1}$
$$h(x_i) = \sigma(\beta_0 + \beta_1 x_i) = \frac{1}{1 + e^{-\beta_0 - \beta_1 x_i}}.$$
"""

# ╔═╡ 7d5639d5-d490-400d-b4f7-2dadfee9223c
md"""
β₀ = $(@bind ex1β₀ Slider(-8:0.1:8, default=0))
β₁ = $(@bind ex1β₁ Slider(-8:0.1:8, default=0))

mostrar contorno = $(@bind ex1_contorno CheckBox())
"""

# ╔═╡ 3b002053-d892-463c-ab68-0cd48948b27c
md"""
- A probabilidade de acontecer todos as observações $(x_1,y_1),\dots,(x_n,y_n)$, supondo independência, é
$$L(\beta) = \prod_{i=1}^n P_i = \prod_{i=1}^n h(x_i)^{y_i}(1 - h(x_i))^{1 - y_i}.$$
- Para que a probabilidade seja máxima, devemos maximizar essa função, em termo dos parâmetros do modelo.
- Maximizar isso é complicado, então passamos o log:
$$\ell(\beta) = \ln L(\beta) = \sum_{i=1}^n y_i \ln(h(x_i)) + (1 - y_i)\ln(1 - h(x_i)).$$
"""

# ╔═╡ 57c64d6f-8e3b-46b8-b35b-a210294fe191
md"""
- Não existe maneira direta simples de resolver o problema, porque é um problema de otimização não linear.
- Além disso, em geral o problema tem condicionamento ruim, então o uso de regularização é comum - tanto é que o padrão implementado no Scikit-Learn é regularizado.
- Finalmente, a classificação só acontece no final, usando um limiar.
- Tudo acima do limiar é considerado 1, e tudo abaixo é considerado 0.
"""

# ╔═╡ a59a0ab3-04bb-4896-8e1e-c76be3b9c8e1
md"""
mostrar limiar = $(@bind ex1_mostra_limiar CheckBox()), 
limiar = $(@bind ex1limiar Slider(0.05:0.05:0.95, default=0.5))
"""

# ╔═╡ 8816c27a-4ce3-4c97-baf6-77cc3616670c
let
	p1 = plot(leg=false)
	h(x) = 1 / (1 + exp(-ex1β₀ - ex1β₁ * x))
	xg = range(extrema(x)..., length=50)
	plot!(xg, h.(xg), c=:black, lw=2)
	if ex1_mostra_limiar && ex1β₁ != 0
		plot!([xg[1], xg[end]], ex1limiar * ones(2), c=:green, l=:dash)
		ŷ = sign.(h.(x) .- ex1limiar)
		ŷ[findall(ŷ .== -1)] .= 0
		μ = (-log((1 - ex1limiar) / ex1limiar)-ex1β₀) / ex1β₁
		I = findall(h.(xg) .< ex1limiar)
		if length(I) > 0
			plot!([xg[I[1]], xg[I[end]]], zeros(2), c=:red, lw=2)
		end
		I = findall(h.(xg) .> ex1limiar)
		plot!(μ * ones(2), [0, 1], c=:green, l=:dash)
		if length(I) > 0
			plot!([xg[I[1]], xg[I[end]]], ones(2), c=:blue, lw=2)
		end
		M = [
			sum((y .== 0) .& (ŷ .== 0))  sum((y .== 0) .& (ŷ .== 1));
			sum((y .== 1) .& (ŷ .== 0))  sum((y .== 1) .& (ŷ .== 1))
		]
		annotate!([
				(-2.4, 0.8, "R0"),
				(-2.4, 0.7, "R1"),
				(-2.0, 0.9, "P0"),
				(-1.6, 0.9, "P1"),
				(-2.0, 0.8, M[1,1]),
				(-1.6, 0.8, M[1,2]),
				(-2.0, 0.7, M[2,1]),
				(-1.6, 0.7, M[2,2]),
		])
	end
	I = findall(y .== 0)
	scatter!(x[I], y[I], m=(3,:pink,:square,stroke(1,:red)), leg=false)
	I = findall(y .== 1)
	scatter!(x[I], y[I], m=(3,:lightblue,:circle,stroke(1,:blue)))
	if ex1_contorno
		βr = range(-10, 10, length=20)
		h2(β) = 1 ./ (1 .+ exp.(-β[1] .- β[2] * x))
		ℓ(β) = sum(y .* log.(h2(β)) + (1 .- y) .* log.(1 .- h2(β)))
		p2 = contourf(βr, βr, (x,y)->ℓ([x;y]), c=:heat, levels=20, lw=0, leg=false)
		#contour!(βr, βr, (x,y)->ℓ([x;y]), c=:white, levels= .- range(0, 400, length=100), lw=0.2)
		scatter!([ex1β₀], [ex1β₁], c=:lightblue, ms=4, leg=false)
		xlims!(extrema(βr)...)
		ylims!(extrema(βr)...)
		plot(p1, p2, size=(1200,400))
	end
	plot!()
end

# ╔═╡ 2fb9b14b-55bc-4c79-91a0-f1302d21e7cc
md"""
**Solução:**
"""

# ╔═╡ c716490b-3106-4722-a1fe-caa811e3520e
let
	σ(t) = t ≥ 0 ? 1 / (1 + exp(-t)) : exp(t) / (1 + exp(t))
	h(β, x) = σ(β[1] + β[2] * x)
	ℓ(β) = begin
		hβ = h.(Ref(β), x)
		-sum(y[i] * log(hβ[i] + 1e-8) + (1 - y[i]) * log(1 - hβ[i] + 1e-8) for i = 1:n) + 1e-4 * norm(β)
	end
	β = zeros(2)
	nlp = ADNLPModel(ℓ, zeros(2))
	anim = Animation()
	evals = 0
	ks = 1:50
	Ls = Float64[ℓ(β)]
	∇Ls = Float64[norm(grad(nlp, β))]
	ŷ = round.(Int, h.(Ref(β), x))
	Accs = Float64[sum(ŷ .== y) / n]
	for k = ks
		Hβ = hess(nlp, β) + 1e-8 * LinearAlgebra.I
		if minimum(eigen(Hβ).values) < 0
			error(string(eigen(Hβ).values))
		end
		β -= 0.3 * (Hβ \ grad(nlp, β))
		I = findall(y .== 0)

		plt = plot(layout=grid(2,2), size=(800,600))
		scatter!(plt[1,1], x[I], y[I], m=(3,:pink,:square,stroke(1,:red)), leg=false)
		I = findall(y .== 1)
		scatter!(plt[1,1], x[I], y[I], m=(3,:lightblue,:circle,stroke(1,:blue)))
		xg = range(extrema(x)..., length=100)
		plot!(plt[1,1], xg, h.(Ref(β), xg), c=:black, lw=2)
		title!(plt[1,1], "Fit")

		push!(Ls, ℓ(β))
		plot!(plt[1,2], 0:k, Ls, xlims=(0,ks[end]), ylims=(5,Ls[1]), leg=false)
		title!(plt[1,2], "ℓ(β)")

		push!(∇Ls, norm(grad(nlp, β)))
		plot!(plt[2,1], 0:k, ∇Ls, xlims=(0,ks[end]), ylims=(1e-8,∇Ls[1]), yaxis=:log, leg=false)
		title!(plt[2,1], "‖∇ℓ(β)‖")
		
		ŷ = round.(Int, h.(Ref(β), x))
		push!(Accs, sum(ŷ .== y) / n)
		plot!(plt[2,2], 0:k, Accs, xlims=(0,ks[end]), ylims=(0, 1), leg=false)
		title!(plt[2,2], "Acurácia")
		
		frame(anim)
	end
	gif(anim, fps=10)
end

# ╔═╡ ea195c09-f5b6-4854-9a0b-06d4f5def64f
begin
	σ(t) = 1 / (1 + exp(-t))
	h(β, x) = σ(β[1] + β[2] * x)
	ℓ(β) = begin
		hβ = h.(Ref(β), x)
		-sum(y[i] * log(hβ[i]) + (1 - y[i]) * log(1 - hβ[i]) for i = 1:n)
	end
	nlp = ADNLPModel(ℓ, zeros(2))
	output = with_logger(NullLogger()) do
		lbfgs(nlp)
	end
	global β = output.solution

	let
		I = findall(y .== 0)
		scatter(x[I], y[I], m=(3,:pink,:square,stroke(1,:red)), leg=false)
		I = findall(y .== 1)
		scatter!(x[I], y[I], m=(3,:lightblue,:circle,stroke(1,:blue)))
		xg = range(extrema(x)..., length=100)
		plot!(xg, h.(Ref(β), xg), c=:black, lw=2)
	end
end

# ╔═╡ aed83a36-eedf-46c7-bd91-cc3e01289405
β

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deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "9f473cdf6e2eb360c576f9822e7c765dd9d26dbc"
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version = "0.58.0"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "eaf96e05a880f3db5ded5a5a8a7817ecba3c7392"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
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[[GeometryBasics]]
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git-tree-sha1 = "15ff9a14b9e1218958d3530cc288cf31465d9ae2"
uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
version = "0.3.13"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
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[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "47ce50b742921377301e15005c96e979574e130b"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
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[[Grisu]]
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uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
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[[HTTP]]
deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"]
git-tree-sha1 = "c6a1fff2fd4b1da29d3dccaffb1e1001244d844e"
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version = "0.9.12"

[[IniFile]]
deps = ["Test"]
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[[InteractiveUtils]]
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[[IterTools]]
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[[IteratorInterfaceExtensions]]
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[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "642a199af8b68253517b80bd3bfd17eb4e84df6e"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
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[[JSON]]
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[[JSOSolvers]]
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[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943"
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[[Krylov]]
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[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
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[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
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[[LaTeXStrings]]
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[[Latexify]]
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[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[LibVPX_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "12ee7e23fa4d18361e7c2cde8f8337d4c3101bc7"
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version = "1.10.0+0"

[[Libdl]]
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[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
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[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
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[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
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[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
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[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
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[[Libtiff_jll]]
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git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
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[[LinearAlgebra]]
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[[LinearOperators]]
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[[LogExpFunctions]]
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[[Logging]]
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[[MacroTools]]
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[[Markdown]]
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[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
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[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
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[[Missings]]
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[[Mmap]]
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[[MozillaCACerts_jll]]
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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]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
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[[OpenSSL_jll]]
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git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
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[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
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[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
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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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[[Parsers]]
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[[Pixman_jll]]
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git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
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[[Pkg]]
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uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
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[[PlotUtils]]
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[[Plots]]
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[[PlutoUI]]
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[[Preferences]]
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[[Printf]]
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[[Qt5Base_jll]]
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[[REPL]]
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[[Random]]
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[[RecipesBase]]
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[[RecipesPipeline]]
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[[Reexport]]
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[[Requires]]
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[[SHA]]
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[[Scratch]]
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[[Serialization]]
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[[SharedArrays]]
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[[Showoff]]
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[[Sockets]]
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[[SolverCore]]
deps = ["NLPModels", "Printf"]
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[[SolverTools]]
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[[SortingAlgorithms]]
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[[SparseArrays]]
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[[SpecialFunctions]]
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[[StaticArrays]]
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[[Statistics]]
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[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "2f6792d523d7448bbe2fec99eca9218f06cc746d"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.8"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "000e168f5cc9aded17b6999a560b7c11dda69095"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.0"

[[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 = "8ed4a3ea724dac32670b062be3ef1c1de6773ae8"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.4.4"

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

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

[[TimerOutputs]]
deps = ["ExprTools", "Printf"]
git-tree-sha1 = "209a8326c4f955e2442c07b56029e88bb48299c7"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.12"

[[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", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "acc685bcf777b2202a904cdcb49ad34c2fa1880c"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.14.0+4"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7a5780a0d9c6864184b3a2eeeb833a0c871f00ab"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "0.1.6+4"

[[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 = "d713c1ce4deac133e3334ee12f4adff07f81778f"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2020.7.14+2"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "487da2f8f2f0c8ee0e83f39d13037d6bbf0a45ab"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.0.0+3"

[[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:
# ╟─992de666-e8cf-11eb-3f19-53e69f07a376
# ╟─da9f5da6-b61b-4483-bab6-bfce4c19ec5c
# ╟─df9e884f-bf4a-4fd9-8140-f21c82134207
# ╟─f2b17cd7-8f56-4d9b-8599-6f2da53a13f8
# ╟─7d5639d5-d490-400d-b4f7-2dadfee9223c
# ╟─a59a0ab3-04bb-4896-8e1e-c76be3b9c8e1
# ╟─8816c27a-4ce3-4c97-baf6-77cc3616670c
# ╟─3b002053-d892-463c-ab68-0cd48948b27c
# ╟─57c64d6f-8e3b-46b8-b35b-a210294fe191
# ╟─2fb9b14b-55bc-4c79-91a0-f1302d21e7cc
# ╟─c716490b-3106-4722-a1fe-caa811e3520e
# ╟─ea195c09-f5b6-4854-9a0b-06d4f5def64f
# ╟─aed83a36-eedf-46c7-bd91-cc3e01289405
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002