Find the Optimal Number of Clusters¶

Use the Elbow Method and Silhouette Score together to select the right number of clusters for k-Means.

The Elbow Method¶

Plot the inertia (within-cluster sum of squares) for increasing values of \(k\). The "elbow" — where the rate of decrease sharply levels off — suggests the optimal cluster count.

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

X, _ = make_blobs(n_samples=300, centers=4, cluster_std=0.60, random_state=42)

inertias = []
K_range = range(1, 11)

for k in K_range:
    km = KMeans(n_clusters=k, random_state=42, n_init="auto")
    km.fit(X)
    inertias.append(km.inertia_)

plt.figure(figsize=(8, 4))
plt.plot(K_range, inertias, "bo-")
plt.xlabel("Number of Clusters (k)")
plt.ylabel("Inertia")
plt.title("Elbow Method")
plt.tight_layout()
plt.show()

The Silhouette Score¶

The Silhouette Score measures how similar each point is to its own cluster compared to neighbouring clusters. Values range from -1 (wrong cluster) to +1 (well-clustered).

from sklearn.metrics import silhouette_score

scores = []
K_range = range(2, 11)  # Silhouette needs at least 2 clusters

for k in K_range:
    km = KMeans(n_clusters=k, random_state=42, n_init="auto")
    labels = km.fit_predict(X)
    scores.append(silhouette_score(X, labels))

plt.figure(figsize=(8, 4))
plt.plot(K_range, scores, "ro-")
plt.xlabel("Number of Clusters (k)")
plt.ylabel("Silhouette Score")
plt.title("Silhouette Analysis")
plt.tight_layout()
plt.show()

Common Pitfall

The Elbow Method is subjective — the "elbow" is not always obvious. Always validate with the Silhouette Score and domain knowledge.

KSB Mapping¶

KSB	Description	How This Addresses It
K4.2	Advanced analytics and ML techniques	Unsupervised learning algorithms for pattern discovery
K4.4	Trade-offs in selecting algorithms	Choosing between clustering approaches based on data characteristics
S1	Scientific methods and hypothesis testing	Validating cluster quality without ground truth labels
S4	Analysis and models to inform outcomes	Using clustering to derive actionable segments
B1	Inquisitive approach	Exploring hidden structure in unlabelled data