Copula Functions in Model Based Clustering

Abstract

Model based clustering is common approach used in cluster analysis. Here each cluster is characterized by some kind of model, for example — multivariate distribution, regression, principal component etc. One of the most well known approaches in model based clustering is the one proposed by Banfield and Raftery (1993), where each class is described by multivariate normal distribution. Due to the eigenvalue decomposition, one gets flexibility in modeling size, shape and orientation of the clusters, still assuming general elliptical shape of the set of observations. In the paper we consider the other proposal based on the general stochastic approach in two versions:

• classification likelihood approach, where each observation comes from one of several populations;

• mixture approach, where observations are distributed as a mixture of several distributions.

We propose the use of the copula approach, by representing the multivariate distribution as the copula function of univariate marginal distributions. We give the theoretical bases for such an approach and the algorithms for practical use.

The discussed methods are illustrated by some simulation studies and real examples using financial data.