SilhouetteScore

Silhouette score is a metric used to evaluate the quality of a clustering result. For a given labeling of data into clusters, it assigns each sample a silhouette coefficient between -1 and 1. The coefficient reflects how similar the sample is to its own cluster compared with points in the nearest neighboring cluster. A higher score indicates better clustering, while negative values suggest possible misassignment.

For a sample i in cluster A, a(i) is the average distance from i to all other points in A. For each other cluster B, compute the average distance from i to all points in B; b(i) is the minimum of these averages across all such clusters. The silhouette coefficient for i is s(i) = (b(i) - a(i)) / max(a(i), b(i)). The overall silhouette score is the mean of s(i) over all samples. Values close to 1 indicate well-separated clusters, around 0 indicate overlapping clusters or ambiguous assignments, and negative values indicate potential misclassifications.

Computation typically uses a distance metric, most commonly Euclidean distance, but any metric supported by the clustering method can be used. The silhouette score can be computed for a fixed number of clusters, and is often used to select a suitable k by comparing scores across different k values. Per-sample silhouette values can be visualized with a silhouette plot to assess cluster separation and cohesion.

Limitations include sensitivity to density variations and to clusters of different sizes, and it may be less informative for non-convex or irregularly shaped clusters. The calculation is generally O(n^2) in time complexity due to pairwise distances, making it expensive for large datasets. Implementations exist in many libraries, such as scikit-learn, as silhouette_score and silhouette_samples.