How the K-means Clustering Algorithm Works

When implementing the K-means clustering algorithm, the following steps are typically executed: Step 1: Determine the value of k, representing the number of clusters. This critical parameter can be optimized using techniques like the Elbow Method (plotting within-cluster sum of squares against k values) or Gap Statistic analysis. Step 2: Partition the data into k initial clusters. Training samples can be assigned randomly or systematically using methods like: - Taking the first k training samples as singleton clusters - Assigning remaining (N-k) samples to clusters with nearest centroids After each assignment, recalculate cluster centroids using the mean of all points in the cluster. In code, this involves computing centroid coordinates with: centroid = np.mean(cluster_points, axis=0) Step 3: Process each sample sequentially, calculating Euclidean distances to all cluster centroids. If a sample doesn't belong to the cluster with the nearest centroid: - Reassign the sample to the optimal cluster - Update centroids for both the gaining and losing clusters This distance calculation typically uses: distance = np.linalg.norm(sample - centroid) Step 4: Iterate Step 3 until convergence is achieved, indicated by no reassignments during a full pass through the dataset. The algorithm converges when cluster assignments stabilize. For enhanced performance, consider hybrid approaches incorporating spectral clustering (using graph Laplacians) or hierarchical clustering (building nested cluster trees) as preprocessing steps. The algorithm minimizes within-cluster variance through Lloyd's algorithm iterations, with time complexity O(n*k*i*d) where n=samples, k=clusters, i=iterations, d=dimensions.