Clustering Analysis

Clustering partitions data into groups of similar observations without pre-defined labels, enabling discovery of natural patterns and structures in data.

Clustering Algorithms

• K-Means: Partitioning into k clusters
• Hierarchical: Dendrograms showing nested clusters
• DBSCAN: Density-based arbitrary-shaped clusters
• Gaussian Mixture: Probabilistic clustering
• Agglomerative: Bottom-up hierarchical approach

Key Concepts

• Cluster Validation: Metrics to evaluate cluster quality
• Optimal Clusters: Methods to determine best k
• Inertia: Within-cluster sum of squares
• Silhouette Score: Measure of cluster separation
• Dendrogram: Hierarchical clustering visualization

Implementation with Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.cluster import KMeans, DBSCAN, AgglomerativeClustering
from sklearn.mixture import GaussianMixture
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import (
    silhouette_score, silhouette_samples, davies_bouldin_score,
    calinski_harabasz_score
)
from scipy.cluster.hierarchy import dendrogram, linkage
import seaborn as sns

# Generate sample data
np.random.seed(42)
n_samples = 300
centers = [[0, 0], [5, 5], [-3, 4]]
X = np.vstack([
    np.random.randn(100, 2) + centers[0],
    np.random.randn(100, 2) + centers[1],
    np.random.randn(100, 2) + centers[2],
])

# Standardize
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# K-Means with Elbow method
inertias = []
silhouette_scores = []
k_range = range(2, 11)

for k in k_range:
    kmeans = KMeans(n_clusters=k, random_state=42, n_init=10)
    kmeans.fit(X_scaled)
    inertias.append(kmeans.inertia_)
    silhouette_scores.append(silhouette_score(X_scaled, kmeans.labels_))

fig, axes = plt.subplots(1, 2, figsize=(14, 4))

axes[0].plot(k_range, inertias, 'bo-')
axes[0].set_xlabel('Number of Clusters (k)')
axes[0].set_ylabel('Inertia')
axes[0].set_title('Elbow Method')
axes[0].grid(True, alpha=0.3)

axes[1].plot(k_range, silhouette_scores, 'go-')
axes[1].set_xlabel('Number of Clusters (k)')
axes[1].set_ylabel('Silhouette Score')
axes[1].set_title('Silhouette Analysis')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# Optimal k = 3
optimal_k = 3
kmeans = KMeans(n_clusters=optimal_k, random_state=42, n_init=10)
kmeans_labels = kmeans.fit_predict(X_scaled)

# K-Means visualization
fig, axes = plt.subplots(1, 3, figsize=(15, 4))

# K-Means clusters
axes[0].scatter(X[:, 0], X[:, 1], c=kmeans_labels, cmap='viridis', alpha=0.6)
axes[0].scatter(
    kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1],
    c='red', marker='X', s=200, edgecolors='black', linewidths=2
)
axes[0].set_title(f'K-Means (k={optimal_k})')
axes[0].set_xlabel('Feature 1')
axes[0].set_ylabel('Feature 2')

# Silhouette plot
ax = axes[1]
y_lower = 10
silhouette_vals = silhouette_samples(X_scaled, kmeans_labels)

for i in range(optimal_k):
    cluster_silhouette_vals = silhouette_vals[kmeans_labels == i]
    cluster_silhouette_vals.sort()

    size_cluster_i = cluster_silhouette_vals.shape[0]
    y_upper = y_lower + size_cluster_i

    ax.fill_betweenx(np.arange(y_lower, y_upper),
                      0, cluster_silhouette_vals,
                      alpha=0.7, label=f'Cluster {i}')
    y_lower = y_upper + 10

ax.axvline(x=silhouette_score(X_scaled, kmeans_labels), color="red", linestyle="--")
ax.set_xlabel('Silhouette Coefficient')
ax.set_ylabel('Cluster Label')
ax.set_title('Silhouette Plot')

# Hierarchical clustering
linkage_matrix = linkage(X_scaled, method='ward')
dendrogram(linkage_matrix, ax=axes[2], truncate_mode='lastp', p=10)
axes[2].set_title('Dendrogram (Ward)')
axes[2].set_xlabel('Sample Index')

plt.tight_layout()
plt.show()

# Hierarchical clustering
hierarchical = AgglomerativeClustering(n_clusters=optimal_k, linkage='ward')
hier_labels = hierarchical.fit_predict(X_scaled)

# DBSCAN clustering
dbscan = DBSCAN(eps=0.4, min_samples=5)
dbscan_labels = dbscan.fit_predict(X_scaled)
n_clusters_dbscan = len(set(dbscan_labels)) - (1 if -1 in dbscan_labels else 0)
n_noise = list(dbscan_labels).count(-1)

# Gaussian Mixture Model
gmm = GaussianMixture(n_components=optimal_k, random_state=42)
gmm_labels = gmm.fit_predict(X_scaled)
gmm_proba = gmm.predict_proba(X_scaled)

# Clustering algorithm comparison
fig, axes = plt.subplots(2, 2, figsize=(12, 10))

algorithms = [
    (kmeans_labels, 'K-Means'),
    (hier_labels, 'Hierarchical'),
    (dbscan_labels, 'DBSCAN'),
    (gmm_labels, 'Gaussian Mixture'),
]

for idx, (labels, title) in enumerate(algorithms):
    ax = axes[idx // 2, idx % 2]

    # Skip noise points for DBSCAN
    mask = labels != -1
    scatter = ax.scatter(
        X[mask, 0], X[mask, 1], c=labels[mask], cmap='viridis', alpha=0.6
    )

    if title == 'DBSCAN' and n_noise > 0:
        noise_mask = labels == -1
        ax.scatter(X[noise_mask, 0], X[noise_mask, 1], c='red', marker='x', s=100, label='Noise')
        ax.legend()

    ax.set_title(f'{title} (n_clusters={len(set(labels[mask]))})')
    ax.set_xlabel('Feature 1')
    ax.set_ylabel('Feature 2')

plt.tight_layout()
plt.show()

# Cluster validation metrics
validation_metrics = {
    'Algorithm': ['K-Means', 'Hierarchical', 'DBSCAN', 'GMM'],
    'Silhouette Score': [
        silhouette_score(X_scaled, kmeans_labels),
        silhouette_score(X_scaled, hier_labels),
        silhouette_score(X_scaled[dbscan_labels != -1], dbscan_labels[dbscan_labels != -1]) if n_noise < len(X_scaled) else np.nan,
        silhouette_score(X_scaled, gmm_labels),
    ],
    'Davies-Bouldin Index': [
        davies_bouldin_score(X_scaled, kmeans_labels),
        davies_bouldin_score(X_scaled, hier_labels),
        davies_bouldin_score(X_scaled[dbscan_labels != -1], dbscan_labels[dbscan_labels != -1]) if n_noise < len(X_scaled) else np.nan,
        davies_bouldin_score(X_scaled, gmm_labels),
    ],
    'Calinski-Harabasz Index': [
        calinski_harabasz_score(X_scaled, kmeans_labels),
        calinski_harabasz_score(X_scaled, hier_labels),
        calinski_harabasz_score(X_scaled[dbscan_labels != -1], dbscan_labels[dbscan_labels != -1]) if n_noise < len(X_scaled) else np.nan,
        calinski_harabasz_score(X_scaled, gmm_labels),
    ],
}

metrics_df = pd.DataFrame(validation_metrics)
print("Clustering Validation Metrics:")
print(metrics_df)

# Cluster size analysis
sizes_df = pd.DataFrame({
    'K-Means': pd.Series(kmeans_labels).value_counts().sort_index(),
    'Hierarchical': pd.Series(hier_labels).value_counts().sort_index(),
    'GMM': pd.Series(gmm_labels).value_counts().sort_index(),
})

print("\nCluster Sizes:")
print(sizes_df)

# Membership probability (GMM)
fig, ax = plt.subplots(figsize=(10, 6))
membership = gmm_proba.max(axis=1)
scatter = ax.scatter(X[:, 0], X[:, 1], c=membership, cmap='RdYlGn', alpha=0.6, s=50)
ax.set_title('Cluster Membership Confidence (GMM)')
ax.set_xlabel('Feature 1')
ax.set_ylabel('Feature 2')
plt.colorbar(scatter, ax=ax, label='Membership Probability')
plt.show()

# Cluster characteristics
kmeans_centers_original = scaler.inverse_transform(kmeans.cluster_centers_)
cluster_df = pd.DataFrame(X, columns=['Feature 1', 'Feature 2'])
cluster_df['Cluster'] = kmeans_labels

for cluster_id in range(optimal_k):
    cluster_data = cluster_df[cluster_df['Cluster'] == cluster_id]
    print(f"\nCluster {cluster_id} Characteristics:")
    print(cluster_data[['Feature 1', 'Feature 2']].describe())

Cluster Quality Metrics

• Silhouette Score: -1 to 1 (higher is better)
• Davies-Bouldin Index: Lower is better
• Calinski-Harabasz Index: Higher is better
• Inertia: Lower is better (KMeans only)

Algorithm Selection

• K-Means: Fast, spherical clusters, k needs specification
• Hierarchical: Produces dendrogram, interpretable
• DBSCAN: Arbitrary shapes, handles noise
• GMM: Probabilistic, soft assignments

Deliverables

• Optimal cluster count analysis
• Cluster visualizations
• Validation metrics comparison
• Cluster characteristics summary
• Silhouette plots
• Dendrogram for hierarchical clustering
• Membership assignments