hiperparametros.jl

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

using Markdown
using InteractiveUtils

# ╔═╡ aff83927-6b50-44ef-b6a2-fa6338a32109
using Distributions, Pluto, PlutoUI, Plots, Random

# ╔═╡ 0da8104c-7be1-420f-a99f-37d85d953b2f
begin
	args = Dict(
		:C1 => (3,:pink,stroke(1,:red),:square),
		:C2 => (3,:lightblue,stroke(1,:blue),:circle),
		:C3 => (3,:lightgreen,stroke(1,:green),:diamond),
	)
	clr_range = Dict(
		:C1 => :pink,
		:C2 => :lightblue,
		:C3 => :lightgreen
	)
	nothing
end

# ╔═╡ d3e7a29c-68a0-4f8b-a21f-8358b88f6080
begin
	mutable struct Arvore
		left
		right
		set
		idx
		valor
		y_values
		X
		y
	end
end

# ╔═╡ 60000d11-368c-4524-bfd6-f4391fa71c25
begin
	function set_value(set, y, yvalues)
		D = Dict(yi => count(yi .== y[set]) for yi in yvalues)
		yi = argmax(D)
		return yi
	end
	
	function predict(arvore::Arvore, x)
		if x[arvore.idx] < arvore.valor
			if arvore.left isa Arvore
				return predict(arvore.left, x)
			else
				return set_value(arvore.left, arvore.y, arvore.y_values)
			end
		else
			if arvore.right isa Arvore
				return predict(arvore.right, x)
			else
				return set_value(arvore.right, arvore.y, arvore.y_values)
			end
		end
	end

	function tree_depth(arvore::Arvore)
		d = 1
		if arvore.left isa Arvore
			d = max(d, tree_depth(arvore.left) + 1)
		end
		if arvore.right isa Arvore
			d = max(d, tree_depth(arvore.right) + 1)
		end
		return d
	end

	function plot_noh!(plt, set, y, uy, posx, posy, depth)
		Δx = 1.0
		Δy = 1.0
		δ = 0.3
		T = Dict(yi => count(yi .== y[set]) for yi in uy)
		this = argmax(T)
		perc = count(this .== y[set]) / length(set)
		c = clr_range[this]
		Tann = join(["$k:$v" for (k, v) in T] |> sort, ",")
		plot!(
			plt,
			posx .+ 2δ * [-1, 1, 1, -1, -1] * depth,
			posy .+ δ * [-1, -1, 1, 1, -1] * 2,
			c=c,
			fill=true,
			ann=(posx, posy, text("$this\n$(round(Int, 100 * perc))%", 10))
		)
	end
	
	function plot_arvore!(plt, arvore, x=0.0, y=0.0, depth=length(arvore))
		δx = 2^tree_depth(arvore)
		plot!(plt, [x, x - δx], [y, y - 2], c=:black, l=:arrow, lw=2)
		plot!(plt, [x, x + δx], [y, y - 2], c=:black, l=:arrow, lw=2)
		v = round(arvore.valor, digits=2)
		annotate!(plt, [
			(x - δx, y - 1, text("x$(arvore.idx) < $v", 10)),
			# (x + δx, y - 1, "x$(arvore.idx) ≥ $v"),
		])
		plot_noh!(plt, arvore.set, arvore.y, arvore.y_values, x, y, depth)
		if arvore.left isa Arvore
			plot_arvore!(plt, arvore.left, x - δx, y - 2, depth)
		else
			plot_noh!(plt, arvore.left, arvore.y, arvore.y_values, x - δx, y - 2, depth)
		end
		if arvore.right isa Arvore
			plot_arvore!(plt, arvore.right, x + δx, y - 2, depth)
		else
			plot_noh!(plt, arvore.right, arvore.y, arvore.y_values, x + δx, y - 2, depth)
		end
	end

	function entropy(set, y, y_values)
		if length(set) == 1
			return 1.0
		end
		sum(
			begin
				p = sum(y[set] .== yi) / length(set)
				p == 0 ? 0.0 : -p * log(p)
			end for yi in y_values
		)
	end
			
	function entropy(arvore)
		left_ent = if arvore.left isa Arvore
			entropy(arvore.left)
		else
			entropy(arvore.left, arvore.y, arvore.y_values)
		end
		right_ent = if arvore.right isa Arvore
			entropy(arvore.right)
		else
			entropy(arvore.right, arvore.y, arvore.y_values)
		end
		n = length(arvore.set)
		m = count(arvore.X[arvore.idx,:] .< arvore.valor)
		return (left_ent * m + right_ent * (n - m)) / n
	end
	
	function split(set, X, y, y_values)
		length(set) == 0 && error("Empty set")
		V = []
		ρ0 = entropy(set, y, y_values)
		t_opt, j_opt, ρ_opt = 0, 0, 1.0
		for j = 1:size(X, 2)
			v = sort(unique(X[set,j]))
			v = [(v[i] + v[i+1]) / 2 for i = 1:length(v)-1]
			for t in v
				left = set[findall(X[set,j] .< t)]
				right = setdiff(set, left)
				(length(left) == 0 || length(right) == 0) && continue
				ρ = (entropy(left, y, y_values) * length(left) +
					entropy(right, y, y_values) * length(right)) / length(set)
				if ρ ≤ ρ_opt
					t_opt, j_opt, ρ_opt = t, j, ρ
				end
				push!(V, (t, j, ρ0, ρ, ρ_opt, left, right))
			end
		end
		if j_opt == 0
			error("set = $set, V = $V")
		end
		
		left = set[findall(X[set,j_opt] .< t_opt)]
		right = setdiff(set, left)
		Arvore(left, right, set, j_opt, t_opt, y_values, X, y)
	end
	
	function cria_arvore(X, y; max_depth=Inf)
		@assert max_depth > 0
		y_values = unique(y)
		n = length(y)
		arvore = split(1:n, X, y, y_values)
		next = [(arvore, 1)]
		while length(next) != 0
			(noh, this_depth) = popfirst!(next)
			if this_depth == max_depth
				continue
			end
			if length(unique(y[noh.left])) > 1
				noh.left = split(noh.left, X, y, y_values)
				push!(next, (noh.left, this_depth + 1))
			end
			if length(unique(y[noh.right])) > 1
				noh.right = split(noh.right, X, y, y_values)
				push!(next, (noh.right, this_depth + 1))
			end
		end
		return arvore
	end

	function plot_regions!(
		plt,
		arvore::Arvore,
		lower=minimum(arvore.X[arvore.set,:], dims=1),
		upper=maximum(arvore.X[arvore.set,:], dims=1)
	)
		lower_left = copy(lower)
		lower_right = copy(lower)
		upper_left = copy(upper)
		upper_right = copy(upper)
		lower_right[arvore.idx] = arvore.valor
		upper_left[arvore.idx] = arvore.valor
		if arvore.idx == 1
			plot!(plt, fill(arvore.valor, 2), [lower[2], upper[2]], c=:black)
		else
			plot!(plt, [lower[1], upper[1]], fill(arvore.valor, 2), c=:black)
		end
		if arvore.left isa Arvore
			plot_regions!(plt, arvore.left, lower_left, upper_left)
		else
			set = arvore.left
			if length(set) > 0
				plot_regions!(
					plt,
					arvore.y[set], 
					lower_left,
					upper_left,
				)
			end
		end
		if arvore.right isa Arvore
			plot_regions!(plt, arvore.right, lower_right, upper_right)
		else
			set = arvore.right
			if length(set) > 0
				plot_regions!(
					plt,
					arvore.y[set],
					lower_right,
					upper_right,
				)
			end
		end
	end

	function plot_regions!(
		plt,
		y::Vector,
		lower,
		upper
	)
		if length(y) == 0
			return
		end
		uy = unique(y)
		counts = Dict(yi => count(yi .== y) for yi in unique(y))
		plot!(plt,
			[lower[1], upper[1], upper[1], lower[1], lower[1]],
			[lower[2], lower[2], upper[2], upper[2], lower[2]],
			fill=true,
			c=clr_range[argmax(counts)],
			opacity=0.3,
		)
	end

	function plot_dataset!(plt, X, y)
		for (i,yi) in enumerate(unique(y))
			I = findall(y .== yi)
			scatter!(plt, X[I,1], X[I,2], m=args[yi])
		end
		plt
	end
end

# ╔═╡ 76122c2e-6b91-4f12-a358-e528ec50b106
begin
	mutable struct Floresta
		arvores :: Vector{Arvore}
		y_values
		X
		y
	end

	function cria_floresta(
		X,
		y;
		max_depth=2,
		num_arvores=10,
	)
		@assert max_depth > 0
		y_values = unique(y)
		n = length(y)

		arvores = Arvore[]
		for i = 1:num_arvores
			set = sample(1:n, n)
			push!(arvores, cria_arvore(X[set,:], y[set], max_depth=max_depth))
		end
		return Floresta(arvores, y_values, X, y)
	end

	function predict(floresta::Floresta, x)
		C = Dict(yi => 0 for yi in floresta.y_values) # contador
		for arv in floresta.arvores
			yi = predict(arv, x)
			C[yi] += 1
		end
		return argmax(C)
	end
end

# ╔═╡ 01648df8-cf1b-4020-b6af-fe9bcf3325d6
n = 250

# ╔═╡ ef9fa330-362f-489a-ac9f-01873468a680
md"""
# Otimização de Hiper-parâmetros (Grid search)

- Para escolher os hiper-parâmetros dos modelos de ML, usamos validação cruzada.
- Quantos temos muitos hiper-parâmetros, podemos fazer a combinação de todos para **minimizar o erro médio no teste**.
- Isso é chamado de **grid-search**.
"""

# ╔═╡ 189f6680-a06b-474d-a438-a87caab0d63d
begin
	Random.seed!(0)

	dataset = 4
	X = [range(-1, 1, length=n) shuffle(range(-1, 1, length=n))]
	y = zeros(n)
	if dataset == 4
		y = [
			(X[i,2] + randn() * 0.2)^2 - X[i,1]^2 > randn() * 0.1 - 0.3^2 ? :C2 :
				(X[i,1] > 0 ? :C1 : :C3)
			for i = 1:n
		]
	elseif dataset == 1
		y = [X[i,1]^2+log(1 .+X[i,2].^2)*X[i,1] > 0.2 + randn()*0.1 ? :C1 : :C2 for i = 1:n]
		y[findall(X[:,1] + 0.5X[:,2] + randn(n) * 0.1 .> 0.3)] .= :C3
	elseif dataset == 2
		y = Symbol.("C" .* string.([
			floor(Int, (atan(X[i,2], X[i,1]) + π) * 3 / 2π + 1)
			for i = 1:n
		]))
	elseif dataset == 3
		y = Symbol.("C" .* string.([
			floor(Int, 1 .+ 3*exp(2 * (-X[i,1]^2 - X[i,2]^2))) for i = 1:n
		]))
	end
	uy = sort(unique(y))
	nuy = length(uy)

	plot_dataset!(plot(leg=false, size=(800,600)), X, y)
end

# ╔═╡ 8f69c940-5b52-4305-bbac-991d5d00c912
begin
	Random.seed!(0)
	num_arvores_set = 5:5:70
	max_depth_set = 1:8
	grid_search_tr = zeros(length(max_depth_set), length(num_arvores_set))
	grid_search_te = zeros(length(max_depth_set), length(num_arvores_set))
	n_folds = 5
	folds = reshape(shuffle(1:n), n_folds, :)
	for (i,dpt) = enumerate(max_depth_set),
		(j,n_arv) = enumerate(num_arvores_set)
		
		avg_tr_acc = 0.0
		avg_te_acc = 0.0
		for fold = 1:n_folds
			Ite = folds[fold,:]
			Itr = setdiff(1:n, Ite)
			floresta = cria_floresta(
				X[Itr,:], y[Itr],
				max_depth=dpt,
				num_arvores=n_arv,
			)
			ŷ = [predict(floresta, X[i,:]) for i = 1:n]
			
			avg_tr_acc += 100 * sum(y[Itr] .== ŷ[Itr]) / length(y[Itr])
			avg_te_acc += 100 * sum(y[Ite] .== ŷ[Ite]) / length(y[Ite])
		end
		grid_search_tr[i,j] = avg_tr_acc / n_folds
		grid_search_te[i,j] = avg_te_acc / n_folds
	end
end

# ╔═╡ 3da5347b-fba1-460b-b545-13e5b9439db8
md"""
## Exemplo

- Considere os hiper-parâmetros número de árvores e profundidade máxima para a Floresta.
"""

# ╔═╡ 9a57c966-beb3-4d3c-bcd3-2e8bd0bff772
let
	plt = plot(layout=grid(2,1), size=(1000,1000), leg=false)
	cmap = cgrad([:white,:red,:purple,:orange,:yellow,:green])

	function foo(p, M)
		heatmap!(p, M, c=cmap)
		xticks!(p, 1:length(num_arvores_set), string.(num_arvores_set))
		annotate!(p, [
			(j, i, (string(round(M[i,j], digits=1)) * "%", 9))
			for i = 1:length(max_depth_set), j = 1:length(num_arvores_set)
		][:])
	end

	foo(plt[1], grid_search_tr)
	title!(plt[1], "Acurácia média nos treinos")
	foo(plt[2], grid_search_te)
	title!(plt[2], "Acurácia média nos teste")
	xlabel!(plt[2], "Número de árvores")
	ylabel!("Profundidade máxima")
end

# ╔═╡ 09aaf6f5-7878-443a-a8cf-91a84957440d
let
	idx = argmax(grid_search_te)
	i, j = idx[1], idx[2]

	Random.seed!(0)
	floresta = floresta = cria_floresta(
		X, y,
		max_depth=max_depth_set[i],
		num_arvores=num_arvores_set[j],
	)
	
	plt = plot(size=(800, 800), leg=false)
	value_of_y = Dict(Symbol("C$i") => i for i = 1:3)
	contourf!(
		range(-1, 1, length=100),
		range(-1, 1, length=100),
		(x,y) -> value_of_y[predict(floresta, [x; y])],
		lw=0,
		c=cgrad([clr_range[:C1], clr_range[:C2], clr_range[:C3]]),
	)
	title!(plt, "Resultado no conjunto total")
	plot_dataset!(plt, X, y)
end

# ╔═╡ c4e7edb9-dcb1-464d-90f1-c694501cbc6c
md"""
- Podemos fazer o grid-search independente do número de hiper-parâmetros.
"""

# ╔═╡ 2844f090-5477-4ed1-82c5-3b6acbee38a2
md"""
## Otimização de Hiper-parâmetros

- Em algumas situações o conjunto de hiper-parâmetros é muito grande.
- No lugar, podemos utilizar métodos de otimização sem derivada.
- Esses métodos não são simples o suficiente para explicá-los aqui.
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

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

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

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

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

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

[[deps.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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# ╟─0da8104c-7be1-420f-a99f-37d85d953b2f
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# ╠═01648df8-cf1b-4020-b6af-fe9bcf3325d6
# ╟─ef9fa330-362f-489a-ac9f-01873468a680
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# ╟─8f69c940-5b52-4305-bbac-991d5d00c912
# ╟─3da5347b-fba1-460b-b545-13e5b9439db8
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# ╟─c4e7edb9-dcb1-464d-90f1-c694501cbc6c
# ╟─2844f090-5477-4ed1-82c5-3b6acbee38a2
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002