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Machine learning / Classification and clustering

K-Means Clustering Algorithm Explained

14 min read, 16 sections

Contents
  1. K-Means Implementation from Scratch
  2. Mathematical Foundations of K-Means
  3. Elbow Method Implementation
  4. Silhouette Analysis Implementation
  5. K-Means++ Initialization
  6. Real-World Application - Customer Segmentation
  7. Handling High-Dimensional Data
  8. Mini-Batch K-Means Implementation
  9. Cluster Validation Metrics
  10. Online K-Means for Streaming Data
  11. Weighted K-Means Implementation
  12. Robust K-Means with Median Centers
  13. Real-World Application - Image Segmentation
  14. Parallel K-Means Implementation
  15. Additional Resources
  16. Results and Performance Metrics

K-Means Implementation from Scratch

The fundamental k-means clustering algorithm implements iterative refinement to partition n observations into k clusters. Each cluster is represented by the mean of its points, called the centroid. This implementation demonstrates the core algorithm without external libraries.

import numpy as np

class KMeans:
    def __init__(self, k=3, max_iters=100):
        self.k = k
        self.max_iters = max_iters
        
    def fit(self, X):
        # Randomly initialize centroids
        self.centroids = X[np.random.choice(X.shape[0], self.k, replace=False)]
        
        for _ in range(self.max_iters):
            old_centroids = self.centroids.copy()
            
            # Assign points to nearest centroid
            distances = np.sqrt(((X - self.centroids[:, np.newaxis])**2).sum(axis=2))
            self.labels = np.argmin(distances, axis=0)
            
            # Update centroids
            for i in range(self.k):
                self.centroids[i] = X[self.labels == i].mean(axis=0)
                
            # Check convergence
            if np.all(old_centroids == self.centroids):
                break
                
        return self.labels

# Example usage
X = np.random.randn(100, 2)  # Generate random 2D data
kmeans = KMeans(k=3)
labels = kmeans.fit(X)

Mathematical Foundations of K-Means

The k-means algorithm minimizes the within-cluster sum of squares (WCSS) through an objective function. This slide presents the mathematical formulation and demonstrates how to compute the objective function value.

def compute_wcss(X, labels, centroids):
    """
    Mathematical formulation of K-means objective:
    '''
    $$WCSS = \sum_{i=1}^{k} \sum_{x \in C_i} ||x - \mu_i||^2$$
    where:
    - k is number of clusters
    - Ci is the set of points in cluster i
    - μi is the centroid of cluster i
    """
    wcss = 0
    for i in range(len(np.unique(labels))):
        cluster_points = X[labels == i]
        centroid = centroids[i]
        wcss += np.sum((cluster_points - centroid) ** 2)
    return wcss

# Example usage
X = np.random.randn(100, 2)
kmeans = KMeans(k=3)
labels = kmeans.fit(X)
wcss = compute_wcss(X, labels, kmeans.centroids)
print(f"Within-cluster sum of squares: {wcss:.2f}")

Elbow Method Implementation

The elbow method helps determine the optimal number of clusters by plotting the WCSS against different k values. The "elbow" point represents diminishing returns in cluster compactness as k increases.

import matplotlib.pyplot as plt

def plot_elbow_curve(X, k_range):
    wcss_values = []
    for k in k_range:
        kmeans = KMeans(k=k)
        labels = kmeans.fit(X)
        wcss = compute_wcss(X, labels, kmeans.centroids)
        wcss_values.append(wcss)
    
    plt.figure(figsize=(10, 6))
    plt.plot(k_range, wcss_values, 'bo-')
    plt.xlabel('Number of Clusters (k)')
    plt.ylabel('Within-Cluster Sum of Squares')
    plt.title('Elbow Method for Optimal k')
    plt.grid(True)
    
    # Example usage
    X = np.random.randn(200, 2)
    k_range = range(1, 11)
    plot_elbow_curve(X, k_range)
    plt.show()

Silhouette Analysis Implementation

Silhouette analysis measures how similar an object is to its own cluster compared to other clusters. The silhouette score ranges from -1 to 1, where higher values indicate better-defined clusters and optimal cluster separation.

def silhouette_score(X, labels, centroids):
    def calculate_distance(point, cluster_points):
        return np.mean(np.sqrt(np.sum((cluster_points - point)**2, axis=1)))
    
    silhouette_scores = []
    for i, point in enumerate(X):
        # Find points in same cluster
        current_cluster = labels[i]
        cluster_points = X[labels == current_cluster]
        
        # Calculate a (average distance to points in same cluster)
        a = calculate_distance(point, cluster_points)
        
        # Calculate b (minimum average distance to points in different cluster)
        b = float('inf')
        for cluster in range(len(centroids)):
            if cluster != current_cluster:
                other_cluster_points = X[labels == cluster]
                avg_distance = calculate_distance(point, other_cluster_points)
                b = min(b, avg_distance)
        
        # Calculate silhouette score for this point
        silhouette = (b - a) / max(a, b)
        silhouette_scores.append(silhouette)
    
    return np.mean(silhouette_scores)

# Example usage
X = np.random.randn(100, 2)
kmeans = KMeans(k=3)
labels = kmeans.fit(X)
score = silhouette_score(X, labels, kmeans.centroids)
print(f"Silhouette Score: {score:.3f}")

K-Means++ Initialization

K-means++ initialization improves the standard k-means by selecting initial centroids that are far apart, leading to better convergence and final clustering results. This implementation demonstrates the probabilistic selection process.

def kmeans_plus_plus_init(X, k):
    n_samples = X.shape[0]
    centroids = [X[np.random.randint(n_samples)]]
    
    for _ in range(1, k):
        # Calculate distances from points to nearest centroid
        distances = np.array([min([np.sum((x-c)**2) for c in centroids]) 
                            for x in X])
        
        # Choose next centroid with probability proportional to distance squared
        probs = distances / distances.sum()
        cumulative_probs = np.cumsum(probs)
        r = np.random.random()
        
        for j, p in enumerate(cumulative_probs):
            if r < p:
                centroids.append(X[j])
                break
    
    return np.array(centroids)

class KMeansPlusPlus(KMeans):
    def fit(self, X):
        self.centroids = kmeans_plus_plus_init(X, self.k)
        return super().fit(X)

# Example usage
X = np.random.randn(100, 2)
kmeans_pp = KMeansPlusPlus(k=3)
labels = kmeans_pp.fit(X)

Real-World Application - Customer Segmentation

The following implementation demonstrates customer segmentation using k-means clustering on customer purchase data. The example includes data preprocessing, scaling, and visualization of customer segments.

import pandas as pd
from sklearn.preprocessing import StandardScaler

# Generate sample customer data
np.random.seed(42)
n_customers = 1000
customer_data = {
    'annual_income': np.random.normal(50000, 15000, n_customers),
    'spending_score': np.random.normal(50, 25, n_customers),
    'purchase_frequency': np.random.normal(10, 5, n_customers)
}
df = pd.DataFrame(customer_data)

# Preprocess and scale the data
scaler = StandardScaler()
X_scaled = scaler.fit_transform(df)

# Apply k-means clustering
kmeans = KMeansPlusPlus(k=4)
labels = kmeans.fit(X_scaled)

# Visualize results
plt.figure(figsize=(12, 8))
scatter = plt.scatter(df['annual_income'], df['spending_score'], 
                     c=labels, cmap='viridis')
plt.xlabel('Annual Income')
plt.ylabel('Spending Score')
plt.title('Customer Segments')
plt.colorbar(scatter, label='Cluster')
plt.show()

Handling High-Dimensional Data

When dealing with high-dimensional data, k-means requires additional considerations for distance calculations and visualization. This implementation includes dimensionality reduction using PCA before clustering and visualization tools.

import numpy as np
from sklearn.decomposition import PCA

class HighDimensionalKMeans:
    def __init__(self, k=3, n_components=2):
        self.k = k
        self.n_components = n_components
        self.pca = PCA(n_components=n_components)
        self.kmeans = KMeansPlusPlus(k=k)
        
    def fit_transform(self, X):
        # Reduce dimensionality
        X_reduced = self.pca.fit_transform(X)
        
        # Apply k-means
        self.labels = self.kmeans.fit(X_reduced)
        
        # Store explained variance
        self.explained_variance_ratio = self.pca.explained_variance_ratio_
        
        return X_reduced, self.labels

    def plot_clusters(self, X_reduced):
        plt.figure(figsize=(10, 6))
        scatter = plt.scatter(X_reduced[:, 0], X_reduced[:, 1], 
                            c=self.labels, cmap='viridis')
        plt.xlabel(f'PC1 ({self.explained_variance_ratio[0]:.2%} variance)')
        plt.ylabel(f'PC2 ({self.explained_variance_ratio[1]:.2%} variance)')
        plt.colorbar(scatter)
        plt.title('High-Dimensional Data Clusters')
        
# Example with 10-dimensional data
X = np.random.randn(500, 10)
hd_kmeans = HighDimensionalKMeans(k=4)
X_reduced, labels = hd_kmeans.fit_transform(X)
hd_kmeans.plot_clusters(X_reduced)

Mini-Batch K-Means Implementation

Mini-batch k-means processes subsets of the data in each iteration, making it more memory-efficient for large datasets. This implementation includes batch processing and incremental centroid updates.

class MiniBatchKMeans:
    def __init__(self, k=3, batch_size=100, max_iters=100):
        self.k = k
        self.batch_size = batch_size
        self.max_iters = max_iters
        
    def fit(self, X):
        n_samples = X.shape[0]
        # Initialize centroids using k-means++
        self.centroids = kmeans_plus_plus_init(X, self.k)
        
        for iteration in range(self.max_iters):
            # Sample mini-batch
            indices = np.random.choice(n_samples, self.batch_size)
            batch = X[indices]
            
            # Assign samples to nearest centroids
            distances = np.sqrt(((batch - self.centroids[:, np.newaxis])**2).sum(axis=2))
            batch_labels = np.argmin(distances, axis=0)
            
            # Update centroids using moving average
            for i in range(self.k):
                batch_cluster = batch[batch_labels == i]
                if len(batch_cluster) > 0:
                    learning_rate = 1.0 / (iteration + 1)
                    self.centroids[i] = (1 - learning_rate) * self.centroids[i] + \
                                      learning_rate * batch_cluster.mean(axis=0)
        
        # Final assignment for all points
        distances = np.sqrt(((X - self.centroids[:, np.newaxis])**2).sum(axis=2))
        self.labels = np.argmin(distances, axis=0)
        return self.labels

# Example usage with large dataset
X_large = np.random.randn(10000, 2)
mini_batch = MiniBatchKMeans(k=5, batch_size=500)
labels = mini_batch.fit(X_large)

Cluster Validation Metrics

A comprehensive set of metrics to validate cluster quality, including Calinski-Harabasz Index and Davies-Bouldin Index, helps in assessing clustering performance beyond the silhouette score.

def cluster_validation_metrics(X, labels, centroids):
    def calinski_harabasz_index(X, labels, centroids):
        n_samples = X.shape[0]
        n_clusters = len(centroids)
        
        # Between-cluster dispersion
        overall_mean = np.mean(X, axis=0)
        between_cluster_ss = sum(
            len(X[labels == i]) * np.sum((centroid - overall_mean) ** 2)
            for i, centroid in enumerate(centroids)
        )
        
        # Within-cluster dispersion
        within_cluster_ss = sum(
            np.sum((X[labels == i] - centroids[i]) ** 2)
            for i in range(n_clusters)
        )
        
        return (between_cluster_ss / (n_clusters - 1)) / \
               (within_cluster_ss / (n_samples - n_clusters))
    
    def davies_bouldin_index(X, labels, centroids):
        n_clusters = len(centroids)
        cluster_distances = np.zeros((n_clusters, n_clusters))
        cluster_dispersions = np.zeros(n_clusters)
        
        # Calculate cluster dispersions
        for i in range(n_clusters):
            cluster_points = X[labels == i]
            if len(cluster_points) > 0:
                cluster_dispersions[i] = np.mean(
                    np.linalg.norm(cluster_points - centroids[i], axis=1)
                )
        
        # Calculate davies-bouldin score
        score = 0
        for i in range(n_clusters):
            max_ratio = 0
            for j in range(n_clusters):
                if i != j:
                    ratio = (cluster_dispersions[i] + cluster_dispersions[j]) / \
                            np.linalg.norm(centroids[i] - centroids[j])
                    max_ratio = max(max_ratio, ratio)
            score += max_ratio
        
        return score / n_clusters

    ch_index = calinski_harabasz_index(X, labels, centroids)
    db_index = davies_bouldin_index(X, labels, centroids)
    
    return {
        'calinski_harabasz_index': ch_index,
        'davies_bouldin_index': db_index
    }

# Example usage
X = np.random.randn(500, 2)
kmeans = KMeans(k=4)
labels = kmeans.fit(X)
metrics = cluster_validation_metrics(X, labels, kmeans.centroids)
print(f"Validation Metrics:\n{metrics}")

Online K-Means for Streaming Data

This implementation handles streaming data by updating clusters incrementally as new data points arrive. The algorithm maintains running statistics and adapts centroids in real-time without storing all historical data.

class OnlineKMeans:
    def __init__(self, k=3):
        self.k = k
        self.n_samples = 0
        self.centroids = None
        self.counts = None
        
    def partial_fit(self, X):
        # Initialize centroids with first k points if not initialized
        if self.centroids is None:
            self.centroids = np.zeros((self.k, X.shape[1]))
            self.counts = np.zeros(self.k)
            for i in range(min(self.k, len(X))):
                self.centroids[i] = X[i]
                self.counts[i] = 1
                self.n_samples += 1
            return self
        
        # Process each new point
        for x in X:
            self.n_samples += 1
            # Find nearest centroid
            distances = np.sum((self.centroids - x) ** 2, axis=1)
            nearest_centroid = np.argmin(distances)
            
            # Update centroid
            self.counts[nearest_centroid] += 1
            lr = 1.0 / self.counts[nearest_centroid]
            self.centroids[nearest_centroid] += lr * (x - self.centroids[nearest_centroid])
        
        return self

    def predict(self, X):
        distances = np.sqrt(((X - self.centroids[:, np.newaxis])**2).sum(axis=2))
        return np.argmin(distances, axis=0)

# Example with streaming data
stream_kmeans = OnlineKMeans(k=3)

# Simulate data stream
for _ in range(5):
    batch = np.random.randn(100, 2)  # New batch of data
    stream_kmeans.partial_fit(batch)
    
    # Get predictions for current batch
    labels = stream_kmeans.predict(batch)
    
    # Plot current state
    plt.figure(figsize=(8, 6))
    plt.scatter(batch[:, 0], batch[:, 1], c=labels, cmap='viridis')
    plt.scatter(stream_kmeans.centroids[:, 0], stream_kmeans.centroids[:, 1], 
                c='red', marker='x', s=200, linewidths=3)
    plt.title(f'Online K-Means after {stream_kmeans.n_samples} samples')
    plt.show()

Weighted K-Means Implementation

Weighted k-means assigns different importance to data points during clustering, useful when certain observations are more significant or reliable than others.

class WeightedKMeans:
    def __init__(self, k=3, max_iters=100):
        self.k = k
        self.max_iters = max_iters
    
    def fit(self, X, weights):
        self.centroids = X[np.random.choice(X.shape[0], self.k, replace=False)]
        weights = np.array(weights).reshape(-1, 1)
        
        for _ in range(self.max_iters):
            old_centroids = self.centroids.copy()
            
            # Assign points to nearest centroid
            distances = np.sqrt(((X - self.centroids[:, np.newaxis])**2).sum(axis=2))
            self.labels = np.argmin(distances, axis=0)
            
            # Update centroids using weights
            for i in range(self.k):
                mask = (self.labels == i)
                if np.any(mask):
                    weighted_sum = np.sum(X[mask] * weights[mask], axis=0)
                    weight_sum = np.sum(weights[mask])
                    self.centroids[i] = weighted_sum / weight_sum
            
            # Check convergence
            if np.allclose(old_centroids, self.centroids):
                break
                
        return self.labels

# Example usage with weighted points
X = np.random.randn(200, 2)
# Generate weights based on distance from origin
weights = 1 / (1 + np.sqrt(np.sum(X**2, axis=1)))
weighted_kmeans = WeightedKMeans(k=3)
labels = weighted_kmeans.fit(X, weights)

# Visualize results with point sizes proportional to weights
plt.figure(figsize=(10, 6))
plt.scatter(X[:, 0], X[:, 1], c=labels, s=weights*1000, cmap='viridis', alpha=0.6)
plt.scatter(weighted_kmeans.centroids[:, 0], weighted_kmeans.centroids[:, 1], 
            c='red', marker='x', s=200, linewidths=3)
plt.title('Weighted K-Means Clustering')
plt.show()

Robust K-Means with Median Centers

This implementation uses medians instead of means for centroid calculation, making the algorithm more robust to outliers and non-spherical cluster shapes. The median-based approach provides better stability in the presence of noise.

class RobustKMeans:
    def __init__(self, k=3, max_iters=100):
        self.k = k
        self.max_iters = max_iters
        
    def fit(self, X):
        # Initialize centroids using k-means++
        self.centroids = kmeans_plus_plus_init(X, self.k)
        
        for _ in range(self.max_iters):
            old_centroids = self.centroids.copy()
            
            # Assign points to nearest centroid
            distances = np.sqrt(((X - self.centroids[:, np.newaxis])**2).sum(axis=2))
            self.labels = np.argmin(distances, axis=0)
            
            # Update centroids using median
            for i in range(self.k):
                cluster_points = X[self.labels == i]
                if len(cluster_points) > 0:
                    self.centroids[i] = np.median(cluster_points, axis=0)
            
            # Check convergence
            if np.allclose(old_centroids, self.centroids):
                break
                
        return self.labels

# Example with outliers
np.random.seed(42)
# Generate core clusters
X = np.vstack([
    np.random.randn(100, 2),
    np.random.randn(100, 2) + [4, 4],
    np.random.randn(100, 2) + [-4, 4]
])
# Add outliers
outliers = np.random.uniform(-10, 10, (20, 2))
X = np.vstack([X, outliers])

# Compare regular and robust k-means
regular_kmeans = KMeans(k=3)
robust_kmeans = RobustKMeans(k=3)

regular_labels = regular_kmeans.fit(X)
robust_labels = robust_kmeans.fit(X)

# Visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

ax1.scatter(X[:, 0], X[:, 1], c=regular_labels, cmap='viridis')
ax1.scatter(regular_kmeans.centroids[:, 0], regular_kmeans.centroids[:, 1], 
            c='red', marker='x', s=200, linewidths=3)
ax1.set_title('Regular K-Means')

ax2.scatter(X[:, 0], X[:, 1], c=robust_labels, cmap='viridis')
ax2.scatter(robust_kmeans.centroids[:, 0], robust_kmeans.centroids[:, 1], 
            c='red', marker='x', s=200, linewidths=3)
ax2.set_title('Robust K-Means')

plt.show()

Real-World Application - Image Segmentation

This implementation demonstrates k-means clustering for image segmentation, converting an image into a specified number of dominant colors. The example includes color space conversion and pixel clustering.

import numpy as np
from PIL import Image

class ImageSegmentation:
    def __init__(self, k=5):
        self.k = k
        self.kmeans = KMeansPlusPlus(k=k)
        
    def segment_image(self, image_array):
        # Reshape image to 2D array of pixels
        pixels = image_array.reshape(-1, 3)
        
        # Apply k-means clustering
        labels = self.kmeans.fit(pixels)
        
        # Replace each pixel with its centroid color
        segmented_pixels = self.kmeans.centroids[labels]
        
        # Reshape back to original image dimensions
        return segmented_pixels.reshape(image_array.shape)

    @staticmethod
    def load_and_process_image(image_path):
        # Load image and convert to numpy array
        img = Image.open(image_path)
        img_array = np.array(img) / 255.0  # Normalize to [0,1]
        return img_array
    
    @staticmethod
    def save_segmented_image(segmented_array, output_path):
        # Convert back to 0-255 range and save
        img_array = (segmented_array * 255).astype(np.uint8)
        img = Image.fromarray(img_array)
        img.save(output_path)

# Example usage
def segment_image_example(image_path, k=5):
    segmenter = ImageSegmentation(k=k)
    
    # Load and process image
    img_array = segmenter.load_and_process_image(image_path)
    
    # Perform segmentation
    segmented_array = segmenter.segment_image(img_array)
    
    # Display original and segmented images
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))
    ax1.imshow(img_array)
    ax1.set_title('Original Image')
    ax1.axis('off')
    
    ax2.imshow(segmented_array)
    ax2.set_title(f'Segmented Image (k={k})')
    ax2.axis('off')
    
    plt.show()

# Example call (assuming image path exists)
# segment_image_example('example_image.jpg', k=5)

Parallel K-Means Implementation

This implementation leverages multiprocessing to parallelize the computation of distances and cluster assignments, significantly improving performance for large datasets while maintaining clustering quality.

import multiprocessing as mp
from functools import partial

class ParallelKMeans:
    def __init__(self, k=3, max_iters=100, n_jobs=-1):
        self.k = k
        self.max_iters = max_iters
        self.n_jobs = mp.cpu_count() if n_jobs == -1 else n_jobs
        
    def _assign_clusters(self, X_chunk, centroids):
        distances = np.sqrt(((X_chunk - centroids[:, np.newaxis])**2).sum(axis=2))
        return np.argmin(distances, axis=0)
    
    def fit(self, X):
        # Initialize centroids using k-means++
        self.centroids = kmeans_plus_plus_init(X, self.k)
        
        # Split data into chunks for parallel processing
        chunk_size = len(X) // self.n_jobs
        chunks = [X[i:i + chunk_size] for i in range(0, len(X), chunk_size)]
        
        for _ in range(self.max_iters):
            old_centroids = self.centroids.copy()
            
            # Parallel assignment of clusters
            with mp.Pool(self.n_jobs) as pool:
                assign_func = partial(self._assign_clusters, centroids=self.centroids)
                labels_chunks = pool.map(assign_func, chunks)
            
            # Combine labels from all chunks
            self.labels = np.concatenate(labels_chunks)
            
            # Update centroids
            for i in range(self.k):
                cluster_points = X[self.labels == i]
                if len(cluster_points) > 0:
                    self.centroids[i] = cluster_points.mean(axis=0)
            
            # Check convergence
            if np.allclose(old_centroids, self.centroids):
                break
                
        return self.labels

# Performance comparison example
def compare_performance(X, k=3):
    # Standard K-means
    start_time = time.time()
    kmeans = KMeans(k=k)
    kmeans.fit(X)
    standard_time = time.time() - start_time
    
    # Parallel K-means
    start_time = time.time()
    parallel_kmeans = ParallelKMeans(k=k)
    parallel_kmeans.fit(X)
    parallel_time = time.time() - start_time
    
    print(f"Standard K-means time: {standard_time:.2f} seconds")
    print(f"Parallel K-means time: {parallel_time:.2f} seconds")
    print(f"Speedup: {standard_time/parallel_time:.2f}x")

# Example with large dataset
X_large = np.random.randn(100000, 10)
compare_performance(X_large, k=5)

Additional Resources

The following ArXiv papers provide comprehensive insights into k-means clustering algorithms, optimizations, and applications:

Results and Performance Metrics

def generate_performance_report(X, algorithms):
    results = {}
    for name, algo in algorithms.items():
        # Time execution
        start_time = time.time()
        labels = algo.fit(X)
        execution_time = time.time() - start_time
        
        # Calculate metrics
        metrics = {
            'execution_time': execution_time,
            'inertia': compute_wcss(X, labels, algo.centroids),
            'silhouette': silhouette_score(X, labels, algo.centroids)
        }
        
        # Add validation metrics
        metrics.update(cluster_validation_metrics(X, labels, algo.centroids))
        results[name] = metrics
    
    # Format and display results
    print("Performance Comparison:")
    print("-" * 50)
    for name, metrics in results.items():
        print(f"\n{name}:")
        for metric, value in metrics.items():
            print(f"  {metric}: {value:.4f}")
    return results

# Example usage
X = np.random.randn(1000, 2)
algorithms = {
    'Standard K-means': KMeans(k=3),
    'K-means++': KMeansPlusPlus(k=3),
    'Mini-batch K-means': MiniBatchKMeans(k=3),
    'Robust K-means': RobustKMeans(k=3)
}

performance_results = generate_performance_report(X, algorithms)

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