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kmeans.jl
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kmeans.jl
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### A Pluto.jl notebook ###
# v0.19.9
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 $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ c60f11a2-0924-11ed-12d1-6d57fe2e2ced
using Colors, LinearAlgebra, Plots, PlutoUI, Random, StatsPlots
# ╔═╡ 467b5aa8-dbe0-4cc0-87fa-ea7544f5d9b3
md"""
# K-Médias
- Agrupamento
- O número de grupos (e centros) é pré-definido ($k$)
- Algoritmo:
1. Os $k$ centros são gerados
2. Os dados são atribuídos aos centros
3. Os centros são recalculados como o centro de massa dos dados atribuídos
4. Repete a partir do 2
"""
# ╔═╡ 8fb1f439-ed38-4302-bc7c-01119fb1b0c8
md"""
## Execução do método:
"""
# ╔═╡ 82db353a-cece-4811-abbd-e82217a6038b
md"""
número de clusters = $(@bind n_clusters_ui Slider(2:10, show_value=true))
número de iterações = $(@bind iteracoes_ui Slider(0:30, show_value=true))
dataset = $(@bind dataset_ui Slider(0:13, show_value=true))
seed = $(@bind seed_ui Slider(0:500, show_value=true))
"""
# ╔═╡ 6273ab11-5bba-49e6-8572-389eaab23e71
begin
if dataset_ui ≤ 3
Random.seed!(dataset_ui)
n = 200
X = zeros(0, 2)
while size(X, 1) < n
m = ceil(Int, (0.1 + 0.1 * rand()) * n)
m = min(m, n - size(X, 1))
μ = 10 * randn(2)
Σ = randn(2, 2)
Σ = Σ * Σ' + I
X = [
X;
randn(m, 2) * Σ .+ μ'
]
end
elseif dataset_ui in 4:11
Random.seed!(0)
n = 300
N = div(dataset_ui, 2)
σ = (dataset_ui % 2 + 1) * 3.0
X = zeros(0, 2)
m = div(n, N)
println("N = $N, m = $m")
for θ in (1:N) * 2π / N
μ = [cos(θ); sin(θ)] * (10 + 3N)
X = [
X;
randn(m, 2) * σ .+ μ'
]
end
elseif dataset_ui == 12
X = zeros(0, 2)
N = 200
θ = (1:N) * 2π / N
for i = 1:2
X = [
X;
i * [cos.(θ) sin.(θ)] + randn(N, 2) * 0.05
]
end
n = size(X, 1)
elseif dataset_ui == 13
X = zeros(0, 2)
N = 100
for i = 1:2
θ = (1:N) * π / N
x = cos.(θ)
σ = (i % 2 == 0 ? 1 : -1)
y = σ * sin.(θ) .- 0.3σ
X = [
X;
[x .- 0.5σ y] + randn(N, 2) * 0.02
]
end
n = size(X, 1)
end
scatter(X[:,1], X[:,2], m=(3,:lightblue,stroke(1,:blue)))
end
# ╔═╡ 93ffbd3d-3534-4741-a135-683824a17633
begin
Random.seed!(seed_ui)
idx = shuffle(1:n)
centros = [X[idx[i],:] for i = 1:n_clusters_ui]
colors = [HSV(360i / n_clusters_ui, 0.5, 1.0) for i = 1:n_clusters_ui]
colors_dark = [HSV(360i / n_clusters_ui, 1.0, 0.8) for i = 1:n_clusters_ui]
classes = [argmin(norm(centro - X[i,:]) for centro in centros) for i = 1:n]
atualizados = Int[]
plt_viz = plot(leg=false)
for iter = 1:iteracoes_ui
centros_novos = [
begin
idx = findall(classes .== k)
sum(X[i,:] for i in idx) / length(idx)
end
for k = 1:n_clusters_ui
]
for (centro, centro_novo, c) in zip(centros, centros_novos, colors_dark)
plot!(
plt_viz,
[centro[1], centro_novo[1]],
[centro[2], centro_novo[2]],
c=:black,
l=:dash,
)
scatter!([centro[1]], [centro[2]], m=(4,c,:diamond))
end
centros .= centros_novos
classes_novas = [argmin(norm(centro - X[i,:]) for centro in centros) for i = 1:n]
push!(atualizados, count(classes .!= classes_novas))
classes .= classes_novas
end
for k = 1:n_clusters_ui
idx = findall(classes .== k)
scatter!(
plt_viz,
X[idx,1],
X[idx,2],
m=(3,colors[k],stroke(1,colors_dark[k])),
opacity=0.5
)
end
for (centro, c) in zip(centros, colors_dark)
scatter!(plt_viz, [centro[1]], [centro[2]], m=(4,c,:diamond))
end
title!(plt_viz, "Atualizados / iteração : " * join(atualizados, ", "))
end
# ╔═╡ 4fd9798c-c9a2-4bb0-bc01-96a30a86f67f
md"""
## Medida de qualidade: silhueta
- Baseada na distância **intra**-cluster e **inter**-cluster
- Notações auxiliares:
- Not: $x_i$ como o $i$-ésimo dado
- Not: $\mathcal{C} = \{1,2,\dots,|\mathcal{C}|\}$ é o conjunto das classes
- Not: $I_k$ é o conjunto dos índices dos dados que são da classe $k$
- Not: $c_i$ é a classe de $i$
- Not: Dois dados de mesma classe são "colegas"
- Not: $C_i = I_{c_i}$ é o conjunto dos índices dos colegas de $x_i$
- **Intra**: $a_i$ é a distância média de $x_i$ a todo os seus "colegas"
$$a_i = \frac{1}{|C_i| - 1}\sum_{j \in C_i\setminus\{i\}} d(x_i, x_j)$$
- **Inter**: $b_i$ é a menor distância média do dado $x_i$ aos outros clusters
$$b_i = \min_{k \neq c_i}\frac{1}{|I_k|} \sum_{j \in I_k} d(x_i, x_j)$$
- Pra entender melhor, primeiro considere a distância média entre $x_i$ e o cluster $k$ como
$$D_{ik} = \frac{1}{|I_k|} \sum_{j \in I_k} d(x_i, x_j), \qquad k \neq c_i$$
- Então, $b_i$ é o menor desses valores $\displaystyle b_i = \min_{k \neq c_i} D_{ik}$
- Finalmente, o coeficiente de silhueta do dado $i$ é dado por
$$s_i = \frac{b_i - a_i}{\max\{a_i, b_i\}}$$
- O coeficiente médio $\bar{s}$ pode ser usado como uma medida numérica da qualidade do agrupamento
"""
# ╔═╡ 26ebe8f1-9bb7-4e7e-893e-2f476dfdfa7e
begin
function silhueta(X, classes)
n = size(X, 1)
Nc = length(classes)
s = ones(n)
for i = 1:n
cᵢ = classes[i]
Cᵢ = findall(classes .== cᵢ)
if length(Cᵢ) == 1
s[i] = 0.0
continue
end
a = sum(norm(X[i,:] - X[j,:]) for j in Cᵢ if i != j) / (length(Cᵢ) - 1)
b = Inf
for k = 1:Nc
if k == cᵢ
continue
end
Iₖ = findall(classes .== k)
if length(Iₖ) == 0
continue
end
Dᵢₖ = sum(norm(X[i,:] - X[j,:]) for j in Iₖ) / length(Iₖ)
b = min(b, Dᵢₖ)
end
s[i] = (b - a) / max.(a, b)
end
ordenado = sort(1:n, lt=(i,j) -> classes[i] < classes[j] || (classes[i] == classes[j] && s[i] > s[j]))::Vector{Int}
return s, ordenado
end
s, ordenado = silhueta(X, classes)
plt_bar = bar(s[ordenado], c=colors[classes[ordenado]], lw=0, bar_width=1.0, leg=false)
s̄ = sum(s) / n
plot!(plt_bar, [1, n], [s̄, s̄], c=:black, l=:dash)
title!(plt_bar, "s̄ = $(round(s̄, digits=4))")
end
# ╔═╡ 11b5c31d-6cee-4e45-95cf-9b92a047720b
plot(plt_viz, plt_bar, layout=grid(1, 2), size=(600,300))
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
[compat]
Colors = "~0.12.8"
Plots = "~1.31.3"
PlutoUI = "~0.7.39"
StatsPlots = "~0.15.0"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
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