Evaluation of Clusters Stability Based on Minkowski Ultrametrics

The main purpose of this paper is to suggest a family of Minkowski distances as a tool for measuring the stability of clusters yielded by a Cluster Analysis. Preliminarly, we recall the main properties of the Minkowski distances family and we introduce the Minkowski ultrametric distances. Latter a proof of the Theorem on the subdominant Minkowski ultrametric distances is given. Since the ultrametric distance between two units decreases whenever the Minkowski parameter increases, we suggest to evaluate the clusters stability by the identification of the Minkowski parameter that measures the convergence to stable clusters. The validity of the methodology is confirmed by an example of Cluster Analysis on a set of real data. This methodology can be applied to compare the cluster stability among various grouping criteria.