Title: Ultrametric models for dimensionality reduction
Authors: Giorgia Zaccaria - University of Milano-Bicocca (Italy) [presenting]
Carlo Cavicchia - Erasmus University Rotterdam (Netherlands)
Maurizio Vichi - University La Sapienza, Rome (Italy)
Abstract: Many relevant multidimensional phenomena are characterized by nested latent concepts having different levels of abstraction, from the most specific to the most general. They can be represented by a tree-shape structure by supposing hierarchical relationships among observed variables. In literature, several methodologies have been proposed to both model the hierarchical relationships among observed variables that reflect unobserved ones, and assess the existence of latent variables of ``higher-order''. Nonetheless, these methodologies are usually developed with sequential procedures that do not optimize a unique objective function, and/or a confirmatory approach. We propose a new class of parsimonious, simultaneous, exploratory models which are based on the ultrametricity notion for matrices. The latter was introduced in mathematics and became widespread in statistics in relation to distances in hierarchical clustering. However, the definition of an ultrametric matrix differs from that of an ultrametric distance matrix and has interesting properties that make it useful for studying hierarchical relationships among variables, as to be one-to-one associated with a hierarchy of latent concepts. The proposal aims at identifying a parsimonious hierarchy by firstly partitioning the variables into groups, each one associated with a latent concept, and then inspecting their relationships via an ultrametric structure.