Mixture decomposition of distributions by copulas in the symbolic data analysis framework

This work investigates the situation in which each unit from a given set is described by some vector of p probability distributions. Our aim is to find simultaneously a “good” partition of these units and a probabilistic description of the clusters with a model using “copula functions” associated with each class of this partition. Different copula models are presented. The mixture decomposition problem is resolved in this general case. This result extends the standard mixture decomposition problem to the case where each unit is described by a vector of distributions instead of the traditional classical case where each unit is described by a vector of single (categorical or numerical) values. Several generalizations of some standard algorithms are proposed. All these results are first considered in the case of a single variable and then extended to the case of a vector of p variables by using a top-down binary tree approach.