centersvary

Cent ersvary is a conceptual metric used in cluster analysis to quantify how much the centers of clusters vary across multiple partitions of a dataset. It is designed to capture the stability of the central positions that define clusters when a clustering algorithm is applied repeatedly under varying conditions, such as different bootstrap samples, subsamples, or initialization seeds.

Formally, for a dataset and a clustering run that yields k clusters with centers c1, c2, ..., ck, centersvary considers the variation of these centers across multiple runs. If runs r and s produce centers c_r,i and c_s,i for cluster i, a centersvary score can be defined as the average distance between corresponding centers across runs, often using Euclidean distance d(c_r,i, c_s,i). When cluster labels are not aligned across runs, a matching step (for example, via the Hungarian algorithm) may be needed to pair centers before computing distances. Variants may use the mean, median, or maximum of pairwise distances, or normalize by data scale.

Interpretation and use: A low centersvary value suggests stable, robust cluster centers and a reliably identified structure in the data; a high value points to instability, noise, or models that yield divergent centers under perturbations. Centersvary complements internal validity metrics such as silhouette or Davies-Bouldin and can inform model selection, such as choosing the number of clusters by balancing cohesion with center stability. It is particularly relevant in applications where the interpretability of cluster centers is important, such as in market segmentation or biological data analysis. Related concepts include cluster stability, consensus clustering, and bootstrap clustering.