Complex Dimensionality Reduction: Ultrametric Models for Mixed-Type Data

Abstract

The factorial latent structure of variables, if present, can be complex and generally identified by nested latent concepts ordered in a hierarchy, from the most specific to the most general one. This corresponds to a tree structure, where the leaves represent the observed variables and the internal nodes coincide with latent concepts defining the general one (i.e., the root of the tree). Although several methodologies have been proposed in the literature to study hierarchical relationships among quantitative variables, very little has been done for more general mixed-type data sets. Hence, it is of the utmost importance to extend these methods and make them suitable to the even more frequent availability of mixed-type data matrices, as complex real phenomena are often described by both qualitative and quantitative variables. In this work, we propose a new exploratory model to study the hierarchical statistical relationships among variables of mixed-type nature by fitting an ultrametric matrix to the general dependence matrix, where the former is one-to-one associated with a hierarchical structure.