K-Means：重心、割り当て、更新ステップ
K-Meansの3回の反復を手作業で追い、各点を最も近い重心に割り当て、重心を再計算しながら、2D散布図上で収束を確認します。
「K-Means：重心、割り当て、更新ステップ」はCoddyKit上の無料Machine Learning Academyレッスンです。 これはレッスン1/4です。 下記で完全なレッスンを無料で読むことができます。その後、ブラウザ内の組み込みコードエディタと24時間対応のAIチューターでハンズオン演習できます。 これはMachine Learning Academy学習パスの一部であり、ウェブとCoddyKitアプリ全体で進捗が同期されます。 Machine Learning Academyコースには全4レッスンが含まれています。
このレッスンの一部はまだ翻訳されておらず、英語で表示されています。
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.
よくある質問
「K-Means：重心、割り当て、更新ステップ」レッスンは無料ですか？
はい。「K-Means：重心、割り当て、更新ステップ」の完全なテキストはこのウェブで無料で読めます。インタラクティブに演習し（組み込みコードエディタと24時間対応のAIチューター）、Machine Learning Academyコースの残りをアンロックするには、CoddyKit PROにアップグレードしてください。 Machine Learning Academyコースには全4レッスンが含まれています。
「K-Means：重心、割り当て、更新ステップ」で何を学びますか？
K-Meansの3回の反復を手作業で追い、各点を最も近い重心に割り当て、重心を再計算しながら、2D散布図上で収束を確認します。 ブラウザで直接実行するハンズオンコードでMachine Learning Academyを演習し、24時間対応のAIチューターがレッスンを進める中での質問に答えます。
Machine Learning Academyを始めるのに経験は必要ですか？
事前経験は必要ありません。CoddyKitのMachine Learning Academyは初級者から上級者向けに構成されているため、ここから始めるか最初から始めて、自分のペースで進むことができます。 これはレッスン1/4です。
「K-Means：重心、割り当て、更新ステップ」レッスンにはどのくらい時間がかかりますか？
ほとんどのCoddyKitレッスンは約5～10分かかります。各レッスンはコンパクトでインタラクティブなので、着実に進歩し、ウェブとアプリ全体で正確に前回の場所から再開できます。
このMachine Learning Academyレッスンでコードを書いて実行できますか？
はい。すべてのMachine Learning Academyレッスンに組み込みコードエディタが含まれているため、ブラウザでリアルコードを書いて実行し、即座のAIフィードバックを取得できます。ローカル設定は不要です。
このコースのすべてのレッスン
- K-Means：重心、割り当て、更新ステップ
- Kの選択：エルボー法とシルエットスコア
- DBSCAN：コア点、境界点、ノイズ
- 顧客セグメンテーションのためのクラスタリング：一連の実例