Determining Optimal K using Silhouette Method

 

Experiment

Determining Optimal K using Silhouette Method

---

🎯 Objective

To determine the optimal number of clusters (K) using the Silhouette Method.

---

📘 Theory

🔹 Why Silhouette Method?

Unlike the Elbow Method, which may be ambiguous, the Silhouette Method provides a quantitative measure of clustering quality.

---

🔹 Silhouette Score

It measures how well each data point fits within its cluster.

• a(i) = average distance to points in same cluster
• b(i) = average distance to points in nearest cluster

---

🔹 Silhouette Formula

$s(i) = \frac{b(i) - a(i)}{\max(a(i), b(i))}$

---

🔹 Interpretation

Score	Meaning
≈ 1	Well clustered
≈ 0	Overlapping clusters
< 0	Misclassified

---

🔹 Working Principle

1. Run K-Means for different values of K
2. Compute average silhouette score
3. Choose K with highest score

---

🧾 Sample Dataset

Point	X	Y
P1	2	3
P2	3	4
P3	3	3
P4	8	7
P5	7	8
P6	8	8
P7	15	16

P8       16    15

P9       15    15

💻 Program (Python Code)

import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score

# Step 1: Dataset
X = np.array([
    [2, 3], [3, 4], [3, 3],
    [8, 7], [7, 8], [8, 8],
    [15, 16], [16, 15], [15, 15]
])

# Step 2: Compute Silhouette Scores
sil_scores = []

K_range = range(2, 7)  # silhouette requires at least 2 clusters

for k in K_range:
    kmeans = KMeans(n_clusters=k, init='k-means++', random_state=0)
    labels = kmeans.fit_predict(X)
    
    score = silhouette_score(X, labels)
    sil_scores.append(score)

# Step 3: Plot graph
plt.plot(K_range, sil_scores, marker='o')
plt.title("Silhouette Method")
plt.xlabel("Number of Clusters (K)")
plt.ylabel("Silhouette Score")
plt.show()

# Step 4: Print scores
for k, score in zip(K_range, sil_scores):
    print(f"K = {k}, Silhouette Score = {score}")

---

📊Output

K = 2, Silhouette Score = 0.7519172781342643 K = 3, Silhouette Score = 0.8503109587689267 K = 4, Silhouette Score = 0.5843521134750496 K = 5, Silhouette Score = 0.33839040555236954 K = 6, Silhouette Score = 0.30584671457309703

---

📈 Graph Interpretation

• The value of K with the highest silhouette score is optimal
• Graph shows peak at a specific K

👉 For this dataset:

✅ Optimal K = 3

---

📌 Result

Using the Silhouette Method, the optimal number of clusters was found to be:

K = 3

• Silhouette Method provides a clear numerical metric
• More reliable than Elbow Method in many cases
• Helps evaluate clustering quality