Discrete and Continuous Models for Two-way Data

Abstract

A continuous factorial model together with a discrete clustering one are fitted simultaneously to two-way data, with the aim to identify the best partition of the objects and the best partition of the variables, according to a least-squares loss function. In addition, the proposed methodology allows to detect simultaneously, factors describing classes of variables and centroids characterizing classes of objects.

The continuous and discrete models are fitted to two-way data by solving a least-squares modeling problem, mathematically restated as a quadratic constrained program with mixed variables.

An iterative alternating least-squares algorithm is proposed to give an efficient solution of the NP-hard minimization problem: starting from clusters centroids, in a number of reduced dimensions, a constrained orthogonal rotation allows to highlight classes of variables that better identify the classification of the objects; then new cluster centroids are computed and the partition of the objects is given by solving an assignment problem. At each step of the alternating least-squares algorithm the objective function is not increased, thus the algorithm converges to at least a local optimal solution of the problem.