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dbscan.jl
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dbscan.jl
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### A Pluto.jl notebook ###
# v0.19.11
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
# ╔═╡ 6a63f9c8-1f25-11ed-3b17-fb9ff6469048
using Colors, LinearAlgebra, Plots, PlutoUI, Random
# ╔═╡ 2e0d4b00-0e9b-4426-bdac-2494967d38c3
md"""
# DBSCAN
- Clusterização
- Baseado em distância
- Um ponto _core_ é um ponto com outros $M-1$ pontos à distância $\epsilon$ dele mesmo
- Um ponto é diretamente alcançavel a partir de um ponto _core_ se a distância entre eles é menos que $\epsilon$
- Um ponto $Q$ é alcançavel a partir de um ponto _core_ $P_0$ se existe uma sequência de pontos _core_ $P_1$, $P_2$, $\dots$, $P_N$, tal que $P_i$ é diretamente alcançavel a partir de $P_{i+1}$ e $Q$ é diretamente alcançavel a partir de $P_N$
- Um ponto _core_ e todos os pontos alcançáveis a partir dele formam um cluster
- Pontos não alcançáveis são considerados _outliers_.
"""
# ╔═╡ de61d57d-edc8-4854-ab8f-4533a9886c69
md"""
ϵ = $(@bind ϵ_slider Slider(2.0 .^ (-2:0.1:5), show_value=true))
min pontos no core = $(@bind min_pontos_slider Slider(2:10, show_value=true))
"""
# ╔═╡ 08f25b21-a76b-41d3-a6a4-82d72a6e9ca0
md"""
dataset = $(@bind dataset_ui Slider(0:13, show_value=true))
"""
# ╔═╡ 7edf836f-5c94-482b-b555-19218a1f6876
begin
if dataset_ui ≤ 3
Random.seed!(dataset_ui)
n = 5 * (dataset_ui + 1)^2 + 10
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 = 100
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 = 50
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)
X = 10X
elseif dataset_ui == 13
X = zeros(0, 2)
N = 50
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)
X = 3X
end
"dataset"
end
# ╔═╡ 4eb98ea7-8c86-4d66-bcda-11e9dff92240
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
if b == Inf
s[i] = a
else
s[i] = (b - a) / max.(a, b)
end
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
end
# ╔═╡ a46dbcfb-2c2a-496e-90c9-1f76072c8bb2
begin
function dbscan(
X;
min_pontos_no_core = 3,
ϵ = 7.0,
)
np = size(X, 1)
num_clusters = 0
classes = fill(0, np)
D = [
norm(X[i,:] - X[j,:]) for i = 1:np, j = 1:np
]
distancia(i, j) = D[i, j]
# Encontre o primeiro core point
for i = 1:np
if classes[i] > 0
continue
end
# A partir do ponto core, encontre todos os pontos diretamente conectados a ele
vizinhos = findall(distancia(i, j) ≤ ϵ for j = 1:np)
if length(vizinhos) < min_pontos_no_core
continue
end
# Começo das iterações
num_clusters += 1
pontos_a_explorar = [i]
while length(pontos_a_explorar) > 0
i = popfirst!(pontos_a_explorar)
classes[i] = num_clusters
vizinhos = findall(distancia(i, j) ≤ ϵ for j = 1:np)
for j in vizinhos
if classes[j] == 0
push!(pontos_a_explorar, j)
end
end
end
end
return classes
end
end
# ╔═╡ 1ef0ad25-4149-4a30-b738-6ced59170b76
let
classes = dbscan(X, ϵ=ϵ_slider, min_pontos_no_core=min_pontos_slider)
num_clusters = maximum(classes)
cores = [HSV(360i / num_clusters, 1.0, 0.8) for i = 1:num_clusters]
plot(leg=false, axis_ratio=:equal)
θ = range(0, 2π, length=60)
c, s = cos.(θ), sin.(θ)
for i = 1:n
plot!(
X[i,1] .+ ϵ_slider * c,
X[i,2] .+ ϵ_slider * s,
c=:gray,
)
end
if num_clusters > 0
for k = 1:num_clusters
idx = findall(classes .== k)
scatter!(X[idx,1], X[idx,2], ms=4, opacity=1.0, c=cores[k])
for i in idx
plot!(
X[i,1] .+ ϵ_slider * c,
X[i,2] .+ ϵ_slider * s,
c=cores[k],
)
for j in idx[findall(norm(X[i,:] - X[j,:]) <= ϵ_slider for j = idx)]
plot!(
X[[j;i],1],
X[[j;i],2],
c=cores[k],
)
end
end
end
end
idx = findall(classes .== 0)
scatter!(X[idx,1], X[idx,2], c=:black, ms=3)
xlims!(minimum(X[:,1]) - 1, maximum(X[:,1]) + 1)
ylims!(minimum(X[:,2]) - 1, maximum(X[:,2]) + 1)
title!("Número de clusters = $num_clusters")
end
# ╔═╡ 5be8560f-3657-437e-b761-6692ca1431a8
let
classes = dbscan(X, ϵ=ϵ_slider)
num_clusters = maximum(classes)
idx0 = findall(classes .== 0)
if length(idx0) > 0
num_clusters += 1
classes[idx0] .= num_clusters
end
s, ordenado = silhueta(X, classes)
cores = [HSV(360i / num_clusters, 0.5, 0.6) for i = 1:num_clusters]
plt_bar = bar(s[ordenado], c=cores[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 = "s̄ = $(round(s̄, digits=4))"
if num_clusters == 1
title = "Distância intra-cluster: $title"
else
title = "Coef. de silhueta: $title"
end
title!(plt_bar, title)
end
# ╔═╡ 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"
[compat]
Colors = "~0.12.8"
Plots = "~1.31.7"
PlutoUI = "~0.7.39"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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