Finding the Number of Groups in Model-Based Clustering via Constrained Likelihoods

Abstract

Deciding the number of clusters k is one of the most difficult problems in Cluster Analysis. For this purpose, complexity-penalized likelihood approaches have been introduced in model-based clustering, such as the well known BIC and ICL criteria. However, the classification/mixture likelihoods considered in these approaches are unbounded without any constraint on the cluster scatter matrices. Constraints also prevent traditional EM and CEM algorithms from being trapped in (spurious) local maxima. Controlling the maximal ratio between the eigenvalues of the scatter matrices to be smaller than a fixed constant c ≥ 1 is a sensible idea for setting such constraints. A new penalized likelihood criterion which takes into account the higher model complexity that a higher value of c entails, is proposed. Based on this criterion, a novel and fully automatized procedure, leading to a small ranked list of optimal (k; c) couples is provided. Its performance is assessed both in empirical examples and through a simulation study as a function of cluster overlap.