Machine Learning Academy · Aula

K-Means: centroides, atribuição e etapas de atualização

K-Means: centroides, atribuição e etapas de atualização é uma aula grátis de Machine Learning Academy no CoddyKit. Esta é a aula 1 de 4. Você pode ler a aula completa abaixo gratuitamente — depois pratica ao vivo no navegador com um editor de código integrado e um tutor de IA 24/7. Faz parte do caminho de aprendizado de Machine Learning Academy, e seu progresso é sincronizado entre a web e o app CoddyKit. O curso de Machine Learning Academy inclui 4 aulas no total.

Partes desta aula ainda não foram traduzidas e aparecem em inglês.

What Is K-Means Clustering?

K-Means is an unsupervised algorithm that partitions n data points into k non-overlapping clusters. Unlike supervised learning, there are no labels — the algorithm discovers structure purely from the feature values. K-Means is fast, scalable, and widely used for customer segmentation, image compression, and anomaly detection.

The Three-Step Algorithm

K-Means repeats three steps until convergence: 1) Initialise — randomly place k centroids in feature space. 2) Assignment — assign every point to the nearest centroid. 3) Update — move each centroid to the mean of its assigned points. The loop stops when assignments no longer change.

Computing Distance to Centroids

In each assignment step, the Euclidean distance from every point to every centroid is computed. A point is assigned to the centroid with the smallest distance. For a point x and centroid c, the squared distance is sum((x_i - c_i)^2). Using squared distance avoids the expensive square-root and gives the same ordering.

import numpy as np

def assign_clusters(X, centroids):
    # X: (n, d), centroids: (k, d)
    distances = np.linalg.norm(X[:, np.newaxis] - centroids, axis=2)  # (n, k)
    return np.argmin(distances, axis=1)  # label for each point

X = np.array([[1, 2], [3, 4], [5, 6], [8, 8]])
centroids = np.array([[2, 2], [7, 7]])
labels = assign_clusters(X, centroids)
print(labels)  # [0, 0, 0, 1]

The Update Step: Recomputing Centroids

After assignment, each centroid is relocated to the arithmetic mean of all points currently in its cluster. If a cluster becomes empty (no points assigned), the centroid is usually re-initialised randomly or removed. This mean-shift minimises the total within-cluster sum of squares (WCSS) — also called inertia.

import numpy as np

def update_centroids(X, labels, k):
    d = X.shape[1]
    new_centroids = np.zeros((k, d))
    for c in range(k):
        points = X[labels == c]
        if len(points) > 0:
            new_centroids[c] = points.mean(axis=0)
    return new_centroids

X = np.array([[1, 2], [3, 4], [5, 6], [8, 8]])
labels = np.array([0, 0, 0, 1])
print(update_centroids(X, labels, k=2))

Tracing Convergence by Hand

Consider four 1D points: 1, 2, 8, 9 and k=2. Init: centroids = [1, 8]. Iter 1 assignment: 1→C0, 2→C0, 8→C1, 9→C1. Iter 1 update: C0=1.5, C1=8.5. Iter 2 assignment: unchanged. Converged in 2 iterations! In higher dimensions convergence may take more steps, but the logic is identical.

Inertia: Measuring Cluster Compactness

Inertia (WCSS) is the sum of squared distances between each point and its cluster centroid. Lower inertia means tighter, more compact clusters. K-Means minimises inertia at each update step, but the algorithm is not guaranteed to find the global minimum — it can get stuck in local optima depending on initialisation.

from sklearn.cluster import KMeans
import numpy as np

X = np.array([[1, 2], [1, 4], [1, 0],
              [10, 2], [10, 4], [10, 0]])

km = KMeans(n_clusters=2, random_state=42)
km.fit(X)

print('Inertia:', km.inertia_)
print('Labels:', km.labels_)
print('Centroids:', km.cluster_centers_)

K-Means++ Initialisation

Random centroid initialisation often leads to slow convergence or poor local optima. K-Means++ (the scikit-learn default via init='k-means++') seeds centroids more intelligently: the first centroid is chosen randomly, and each subsequent centroid is selected with probability proportional to its squared distance from the nearest already-chosen centroid. This spreads starting points and consistently finds better solutions.

from sklearn.cluster import KMeans
import numpy as np

X = np.random.randn(300, 2)

# Default: k-means++ initialisation
km = KMeans(n_clusters=3, init='k-means++', n_init=10, random_state=0)
km.fit(X)
print('Inertia with k-means++:', round(km.inertia_, 2))

# Compare with random init
km_rand = KMeans(n_clusters=3, init='random', n_init=10, random_state=0)
km_rand.fit(X)
print('Inertia with random init:', round(km_rand.inertia_, 2))

Visualising Cluster Assignments

Plotting cluster assignments on a 2D scatter shows the Voronoi partition — the decision boundaries where each point's nearest centroid changes. Plotting centroids as large stars and colouring points by cluster label makes convergence intuitive. This visualisation also reveals when clusters overlap or have unequal sizes.

import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs

X, _ = make_blobs(n_samples=300, centers=3, cluster_std=0.6, random_state=0)
km = KMeans(n_clusters=3, random_state=0)
labels = km.fit_predict(X)

plt.scatter(X[:, 0], X[:, 1], c=labels, cmap='tab10', s=30)
plt.scatter(km.cluster_centers_[:, 0], km.cluster_centers_[:, 1],
            c='black', s=200, marker='*', label='Centroids')
plt.legend()
plt.title('K-Means Clusters')
plt.show()

Multiple Restarts and n_init

Because K-Means can converge to local optima, scikit-learn runs the algorithm n_init times with different random seeds and keeps the result with the lowest inertia. The default is n_init=10. For small datasets 10 is usually sufficient; for large or tricky datasets you may raise it to 20 or 50. Always check the final inertia against the best-run inertia to diagnose poor convergence.

from sklearn.cluster import KMeans
import numpy as np

X = np.random.randn(500, 5)

km = KMeans(n_clusters=4, n_init=20, random_state=0)
km.fit(X)

print('Best inertia over 20 runs:', round(km.inertia_, 2))
print('Number of iterations until convergence:', km.n_iter_)

Limitations of K-Means

K-Means has several well-known weaknesses: 1) Assumes spherical clusters — it struggles with elongated or crescent shapes. 2) Sensitive to outliers — a distant outlier pulls the centroid away from the true cluster mean. 3) Requires k upfront — you must know or estimate the number of clusters before fitting. 4) Feature scale matters — always standardise features before running K-Means so large-scale variables do not dominate distances.

Running K-Means with scikit-learn

In practice, using sklearn.cluster.KMeans is the standard approach. Key parameters: n_clusters (k), init (default 'k-means++'), n_init, max_iter (default 300), and random_state. After fitting, km.labels_ contains cluster assignments, km.cluster_centers_ holds centroid positions, and km.inertia_ reports WCSS.

from sklearn.cluster import KMeans
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_iris

X, _ = load_iris(return_X_y=True)

scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

km = KMeans(n_clusters=3, random_state=42)
km.fit(X_scaled)

print('Cluster sizes:', {i: (km.labels_ == i).sum() for i in range(3)})
print('Inertia:', round(km.inertia_, 2))

Quick Check

Test your understanding of K-Means clustering concepts from this lesson.

Lesson Recap

In this lesson you learned: K-Means iterates assignment and update steps until cluster memberships stabilise, inertia (WCSS) measures compactness and is minimised by each update, and K-Means++ initialisation and multiple restarts help avoid poor local optima. Next up we explore how to choose the right value of k using the elbow method and silhouette score.

Perguntas Frequentes

A aula “K-Means: centroides, atribuição e etapas de atualização” é grátis?

Sim — o texto completo de “K-Means: centroides, atribuição e etapas de atualização” é grátis para ler aqui na web. Para praticá-la interativamente (um editor de código integrado e um tutor de IA 24/7) e desbloquear o restante do curso de Machine Learning Academy, atualize para CoddyKit PRO. O curso de Machine Learning Academy inclui 4 aulas no total.

O que vou aprender em “K-Means: centroides, atribuição e etapas de atualização”?

Os alunos acompanharão manualmente três iterações de K-Means, atribuirão pontos aos centroides mais próximos, recalcularão os centroides e observarão a convergência em um gráfico de dispersão 2D. Você pratica Machine Learning Academy com código prático que executa diretamente no navegador, e um tutor de IA 24/7 responde suas dúvidas enquanto trabalha na aula.

Preciso ter experiência prévia para começar Machine Learning Academy?

Nenhuma experiência prévia é necessária. Machine Learning Academy no CoddyKit é estruturado para alunos iniciantes até avançados, então você pode começar aqui ou desde o início e aprender no seu ritmo. Esta é a aula 1 de 4.

Quanto tempo leva a aula “K-Means: centroides, atribuição e etapas de atualização”?

A maioria das aulas CoddyKit leva cerca de 5–10 minutos. Cada uma é compacta e interativa, então você faz progresso constante e retoma exatamente de onde parou entre web e app.

Posso escrever e executar código nesta aula de Machine Learning Academy?

Sim. Cada aula de Machine Learning Academy inclui um editor de código integrado, então você escreve e executa código real direto no navegador e recebe feedback de IA instantaneamente — nenhuma configuração local necessária.

Todas as aulas deste curso

K-Means: centroides, atribuição e etapas de atualização
Escolha de K: método do cotovelo e pontuação de silhueta
DBSCAN: pontos centrais, pontos de borda e ruído
Agrupamento para segmentação de clientes: exemplo completo

← Voltar para Machine Learning Academy