Scegliere K: metodo del gomito e silhouette score
Imparerete a tracciare l’inerzia rispetto a k (metodo del gomito) e calcolare i coefficienti di silhouette per scegliere il numero di cluster che produce gruppi compatti e ben separati.
Scegliere K: metodo del gomito e silhouette score è una lezione Machine Learning Academy gratuita su CoddyKit. Questa è la lezione 2 di 4. Puoi leggere la lezione completa qui gratuitamente — poi esercitati direttamente nel browser con un editor di codice integrato e un tutor IA disponibile 24/7. Fa parte del percorso di apprendimento Machine Learning Academy, e i tuoi progressi si sincronizzano tra il web e l'app CoddyKit. Il corso Machine Learning Academy include 4 lezioni in totale.
Parti di questa lezione non sono ancora state tradotte e vengono mostrate in inglese.
Why Choosing k Matters
K-Means requires you to specify k — the number of clusters — before training. Too few clusters and you lump distinct groups together; too many and you split natural groups artificially. There is no universally correct k, but two diagnostic tools — the elbow method and the silhouette score — give principled guidance.
Inertia Decreases as k Grows
As you increase k, inertia always decreases because points are assigned to closer centroids. At k=n (one cluster per point), inertia is zero. This means you cannot simply minimise inertia — you need to find where additional clusters stop providing meaningful reductions. That point of diminishing returns is the elbow.
from sklearn.cluster import KMeans
import numpy as np
X = np.random.randn(200, 2)
inertias = []
for k in range(1, 11):
km = KMeans(n_clusters=k, random_state=42, n_init=10)
km.fit(X)
inertias.append(km.inertia_)
print('Inertia per k:')
for k, inr in enumerate(inertias, start=1):
print(f' k={k}: {inr:.1f}')The Elbow Method Explained
Plot inertia on the y-axis against k on the x-axis. The curve typically drops steeply for the first few k values then flattens. The elbow — the kink where the rate of decrease sharply slows — is your estimate of the true cluster count. If the true k is 3, the drop from k=1 to k=3 is large, but from k=3 to k=4 is much smaller.
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=0.7, random_state=0)
inertias = []
for k in range(1, 11):
km = KMeans(n_clusters=k, random_state=0, n_init=10)
km.fit(X)
inertias.append(km.inertia_)
plt.plot(range(1, 11), inertias, marker='o')
plt.xlabel('Number of clusters k')
plt.ylabel('Inertia')
plt.title('Elbow Method')
plt.axvline(x=4, color='red', linestyle='--', label='True k=4')
plt.legend()
plt.show()Limitations of the Elbow Method
The elbow method works well when clusters are clearly separated, but real-world data often produces a smooth curve with no obvious kink. In such cases the elbow is ambiguous and different people may pick different k. That is where the silhouette score provides a more objective, mathematically grounded alternative.
Silhouette Score: The Formula
For each point i, compute two values: a(i) = mean distance to other points in the same cluster (cohesion), and b(i) = mean distance to the nearest different cluster (separation). The silhouette for point i is s(i) = (b(i) - a(i)) / max(a(i), b(i)). Values range from −1 (wrong cluster) through 0 (on border) to +1 (tight, well-separated cluster).
Computing Silhouette Score in sklearn
sklearn.metrics.silhouette_score returns the mean silhouette over all points. A score above 0.5 typically indicates reasonable clustering; above 0.7 is strong. Because you cannot compute the silhouette for k=1 (no second cluster), sweep k from 2 to some maximum and pick the k with the highest mean score.
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score
from sklearn.datasets import make_blobs
X, _ = make_blobs(n_samples=300, centers=4, cluster_std=0.7, random_state=0)
scores = {}
for k in range(2, 9):
km = KMeans(n_clusters=k, random_state=0, n_init=10)
labels = km.fit_predict(X)
scores[k] = silhouette_score(X, labels)
print(f'k={k} silhouette={scores[k]:.3f}')
best_k = max(scores, key=scores.get)
print(f'Best k: {best_k}')Silhouette Plots for Per-Point Analysis
A silhouette plot shows the silhouette coefficient of every individual point, sorted by cluster and width. Wide, uniform bars indicate all points are well-placed. Thin bars or points with negative scores reveal misassigned outliers. scikit-learn's silhouette_samples returns per-point scores that you can visualise this way.
from sklearn.metrics import silhouette_samples
import numpy as np
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs
X, _ = make_blobs(n_samples=100, centers=3, cluster_std=0.6, random_state=0)
km = KMeans(n_clusters=3, random_state=0, n_init=10)
labels = km.fit_predict(X)
samples = silhouette_samples(X, labels)
print('Per-cluster mean silhouettes:')
for c in range(3):
print(f' Cluster {c}: {samples[labels == c].mean():.3f}')Combining Elbow and Silhouette
In practice, use both methods together. If the elbow suggests k=4 and the silhouette score is also highest at k=4, you have strong convergent evidence. When they disagree — e.g., elbow at k=3 but silhouette peaks at k=5 — examine the silhouette plot for each candidate k and apply domain knowledge to make the final call.
Gap Statistic: A Statistical Test for k
The gap statistic compares the observed inertia against the expected inertia under a null reference distribution (data sampled uniformly in the feature space). Choose the smallest k where gap(k) >= gap(k+1) - stddev. It is more statistically rigorous than the elbow method but computationally expensive because it requires generating many random reference datasets.
Practical Guidelines for k Selection
Start with domain knowledge — if you know there are 5 product categories, start with k=5. Use the elbow as a quick visual sanity check. Confirm with silhouette for objectivity. Evaluate downstream — for business use cases, test whether the segments are actionable and interpretable. The numerically optimal k is not always the most useful business segmentation.
Elbow and Silhouette Together: Full Example
Here is a compact pipeline that runs both diagnostics side by side, giving you a summary table to help pick k efficiently.
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score
from sklearn.datasets import make_blobs
from sklearn.preprocessing import StandardScaler
X, _ = make_blobs(n_samples=400, centers=5, cluster_std=0.8, random_state=7)
X = StandardScaler().fit_transform(X)
print(f'{'k':>3} {'Inertia':>10} {'Silhouette':>10}')
for k in range(2, 10):
km = KMeans(n_clusters=k, n_init=10, random_state=0)
labels = km.fit_predict(X)
sil = silhouette_score(X, labels)
print(f'{k:>3} {km.inertia_:>10.1f} {sil:>10.3f}')Quick Check
Test your understanding of k selection methods from this lesson.
Lesson Recap
In this lesson you learned: the elbow method plots inertia vs k and looks for the kink where improvement slows, silhouette score ranges from -1 to +1 and measures both cohesion and separation, and combining both methods with domain knowledge gives the most reliable k selection. Next up we explore DBSCAN — a density-based algorithm that discovers clusters of arbitrary shape and handles noise.
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Imparerete a tracciare l’inerzia rispetto a k (metodo del gomito) e calcolare i coefficienti di silhouette per scegliere il numero di cluster che produce gruppi compatti e ben separati. Eserciti Machine Learning Academy con codice pratico che esegui direttamente nel browser, e un tutor IA 24/7 risponde alle tue domande mentre lavori sulla lezione.
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Tutte le lezioni di questo corso
- K-Means: centroidi, assegnazione e aggiornamento
- Scegliere K: metodo del gomito e silhouette score
- DBSCAN: punti core, punti di bordo e rumore
- Clustering per la segmentazione dei clienti: esempio completo