floresta.jl

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

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 iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ eebc9824-4d61-11ec-1bef-27363898fb1b
using Distributions, Pluto, PlutoUI, Plots, Random

# ╔═╡ 1d6206c7-a3c2-4c72-946a-c15254fcdc5c
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

# ╔═╡ 4485955d-e277-4510-b639-27a9585fd190
begin
	mutable struct Arvore
		left
		right
		set
		idx
		valor
		y_values
		X
		y
	end
end

# ╔═╡ 3f270205-9f9e-41e1-ae5e-5d56a81cce57
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

# ╔═╡ a48d6c64-a048-4b12-ab44-88a7015e0c62
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

# ╔═╡ 543ce27f-391a-42ee-89ab-e66c5a36d0fd
n = 1000

# ╔═╡ a8863bae-b8ba-47f4-a6a8-004d313053b8
md"""
# Floresta aleatória

- Um método que combina várias árvores de decisão
- A árvore não é montada com o conjunto todo
- Cada árvore dá uma resposta
- A floresta usa cada resposta como votação para a previsão
"""

# ╔═╡ df6b443a-ca52-45b6-af83-0d44c3c2e3dc
md"""
- Vamos separar em treino e teste para melhor entender a necessidade e o ganho da floresta
"""

# ╔═╡ b7a454fe-5955-4ef0-a56b-bca6d35a2c87
md"""
### Exemplo
- 4 árvores
- Cada árvore numa amostra de tamanho $n$
- A amostra é feita com repetição (senão era o conjunto original todo)
"""

# ╔═╡ a8691588-514d-44c6-80bd-fcc79f46fb3f
md"""
- Abaixo, as regiões de previsão sobrepostas com transparência
"""

# ╔═╡ 74197ef0-71e1-4745-bbd9-38acd2851d81
md"""
- Abaixo, a região de previsão da floresta
"""

# ╔═╡ 113e7c47-e3f2-4009-9ce0-839d58347bc5
md"""
max\_depth = $(@bind max_depth_slider Slider(2:9, show_value=true))

num\_arvores = $(@bind num_arvores_slider Slider(1:25, show_value=true))

dataset = $(@bind dataset Slider(1:3))
"""

# ╔═╡ b755bcb2-7851-45c4-96fc-3f6d2e5afa18
begin
	Random.seed!(0)
	
	X = [range(-1, 1, length=n) shuffle(range(-1, 1, length=n))]
	y = zeros(n)
	if 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) * 0.4 .> 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

# ╔═╡ 9e40686b-4d79-4a58-a4c2-c173201236d7
begin
	Random.seed!(0)
	m = round(Int, 0.6n)
	Itr = sample(1:n, m, replace=false)
	Ite = setdiff(1:n, Itr)
	Xtr, Xte = X[Itr,:], X[Ite,:]
	ytr, yte = y[Itr], y[Ite]

	plt = plot(layout=grid(2,1), size=(600,1200), leg=false)
	plot_dataset!(plt[1], Xtr, ytr)
	title!(plt[1], "Treino")
	plot_dataset!(plt[2], Xte, yte)
	title!(plt[2], "Teste")
	plt
end

# ╔═╡ 4c381107-28c9-44b3-bd12-2f64f1cd7bbd
let
	Random.seed!(0)
	floresta = cria_floresta(X, y, num_arvores=4, max_depth=4)

	plt = plot(layout=grid(2, 2), size=(800, 800), leg=false)
	for i = 1:4
		arv = floresta.arvores[i]
		set = arv.set
		plot_dataset!(plt[i], X[set,:], y[set])
		plot_regions!(plt[i], floresta.arvores[i])
		ŷtr = [predict(arv, Xtr[i,:]) for i = 1:length(ytr)]
		acctr = round(100 * sum(ytr .== ŷtr) / length(ytr), digits=2)
		ŷte = [predict(arv, Xte[i,:]) for i = 1:length(yte)]
		accte = round(100 * sum(yte .== ŷte) / length(yte), digits=2)
		title!(plt[i], "Acu TR = $acctr%, TE = $accte%")
	end
	plt
end

# ╔═╡ 7b94b4b8-9383-4234-bc4b-96b6a8085d9b
let
	Random.seed!(0)
	floresta = cria_floresta(X, y, num_arvores=4, max_depth=4)
	
	plt = plot(size=(800, 800), leg=false)
	value_of_y = Dict(Symbol("C$i") => i for i = 1:3)
	for arv ∈ floresta.arvores
		contourf!(
			range(-1, 1, length=100),
			range(-1, 1, length=100),
			(x,y) -> value_of_y[predict(arv, [x; y])],
			lw=0,
			c=cgrad([clr_range[:C1], clr_range[:C2], clr_range[:C3]]),
			opacity=0.3,
		)
	end
	ŷtr = [predict(floresta, Xtr[i,:]) for i = 1:length(ytr)]
	acctr = round(100 * sum(ytr .== ŷtr) / length(ytr), digits=2)
	ŷte = [predict(floresta, Xte[i,:]) for i = 1:length(yte)]
	accte = round(100 * sum(yte .== ŷte) / length(yte), digits=2)
	title!(plt, "Previsões sobrepostas - Acu TR = $acctr%, TE = $accte%")
	plot_dataset!(plt, X, y)
end

# ╔═╡ 60200689-1aa5-4c43-8217-5ecb01aa90aa
let
	Random.seed!(0)
	floresta = cria_floresta(X, y, num_arvores=4, max_depth=4)
	
	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]]),
	)
	ŷtr = [predict(floresta, Xtr[i,:]) for i = 1:length(ytr)]
	acctr = round(100 * sum(ytr .== ŷtr) / length(ytr), digits=2)
	ŷte = [predict(floresta, Xte[i,:]) for i = 1:length(yte)]
	accte = round(100 * sum(yte .== ŷte) / length(yte), digits=2)
	title!(plt, "Previsões sobrepostas - Acu TR = $acctr%, TE = $accte%")
	plot_dataset!(plt, X, y)
end

# ╔═╡ 11c11440-bbdd-42ce-b2b0-6a668dc041a7
let
	Random.seed!(0)
	floresta = cria_floresta(Xtr, ytr, num_arvores=num_arvores_slider, max_depth=max_depth_slider)
	
	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]]),
	)
	ŷtr = [predict(floresta, Xtr[i,:]) for i = 1:length(ytr)]
	acctr = round(100 * sum(ytr .== ŷtr) / length(ytr), digits=2)
	ŷte = [predict(floresta, Xte[i,:]) for i = 1:length(yte)]
	accte = round(100 * sum(yte .== ŷte) / length(yte), digits=2)
	title!(plt, "Previsões sobrepostas - Acu TR = $acctr%, TE = $accte%")
	plot_dataset!(plt, Xte, yte)
end

# ╔═╡ b0c2caf5-2c53-43f2-9f82-008fe5967df1
md"""
- A escolha dos parâmetros é essencial.
- Além dos dois acima, podemos ter parâmetros que propagam para as árvores.
- Por fim, a maneira de escolher os parâmetros é validação cruzada, novamente.
- Mas agora temos muitos parâmetros e muitas escolhas. Como fazer?
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

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Distributions = "~0.25.32"
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PlutoUI = "~0.7.20"
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.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"

[[deps.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"

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

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

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[deps.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"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

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

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "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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# ╟─7b94b4b8-9383-4234-bc4b-96b6a8085d9b
# ╟─74197ef0-71e1-4745-bbd9-38acd2851d81
# ╟─60200689-1aa5-4c43-8217-5ecb01aa90aa
# ╟─113e7c47-e3f2-4009-9ce0-839d58347bc5
# ╟─11c11440-bbdd-42ce-b2b0-6a668dc041a7
# ╟─b0c2caf5-2c53-43f2-9f82-008fe5967df1
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002