K-means：沒有標籤，能把資料分成什麼？

目標 J = Σᵢ ||xᵢ − μ[zᵢ]||²

初始 J = (0−0)² + (2−0)² + (8−8)² + (10−8)² = 8
更新中心：左群(0+2)/2=1，右群(8+10)/2=9
更新 J = (0−1)² + (2−1)² + (8−9)² + (10−9)² = 4

python -m pip install numpy==1.26.3 scikit-learn==1.5.1
python kmeans_steps.py

import numpy as np
import sklearn
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs
from sklearn.metrics import silhouette_score
from sklearn.model_selection import train_test_split
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
from threadpoolctl import threadpool_limits

points = np.array([0.0, 2.0, 8.0, 10.0]).reshape(-1, 1)
initial = np.array([0.0, 8.0]).reshape(-1, 1)
distance_sq = ((points[:, None, :] - initial[None, :, :]) ** 2).sum(axis=2)
assignment = distance_sq.argmin(axis=1)
centers = np.vstack([points[assignment == k].mean(axis=0) for k in range(2)])
initial_sse = distance_sq[np.arange(len(points)), assignment].sum()
updated_sse = ((points - centers[assignment]) ** 2).sum()

with threadpool_limits(limits=1):
    toy = KMeans(n_clusters=2, init=initial, n_init=1,
                 algorithm="lloyd", random_state=42).fit(points)
    print(f"numpy={np.__version__} sklearn={sklearn.__version__}")
    print(f"toy initial_assignment={assignment.tolist()}")
    print(f"toy initial_SSE={initial_sse:.6f} updated_SSE={updated_sse:.6f}")
    print(f"toy updated_centers={centers.ravel().tolist()}")
    print(f"toy fitted_centers={toy.cluster_centers_.ravel().tolist()} "
          f"inertia={toy.inertia_:.6f}")
    print(f"toy predict_3_and_8={toy.predict([[3.0], [8.0]]).tolist()}")

    # Fictional independent points. Generator labels are intentionally unused.
    X, _ = make_blobs(n_samples=300, centers=3, n_features=2,
                      cluster_std=0.8, random_state=42)
    train, future = train_test_split(X, test_size=100, random_state=42)
    pipe = Pipeline([
        ("scale", StandardScaler()),
        ("cluster", KMeans(n_clusters=3, n_init=10, init="k-means++",
                           algorithm="lloyd", random_state=42)),
    ])
    pipe.fit(train)
    train_scaled = pipe["scale"].transform(train)
    train_labels = pipe["cluster"].labels_
    future_labels = pipe.predict(future)
    print(f"synthetic train={len(train)} future={len(future)} "
          f"scaler_seen={pipe['scale'].n_samples_seen_}")
    print(f"synthetic train_inertia_scaled={pipe['cluster'].inertia_:.6f}")
    print(f"synthetic train_silhouette_scaled="
          f"{silhouette_score(train_scaled, train_labels):.6f}")
    print(f"synthetic train_cluster_sizes={np.bincount(train_labels).tolist()}")
    print(f"synthetic future_cluster_sizes={np.bincount(future_labels, minlength=3).tolist()}")

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