Non-hierarchical clustering methods on factorial subspaces

Abstract

Cluster analysis (CA) aims at finding homogeneous group of individuals, where homogeneous is referred to individuals that present similar characteristics. Many CA techniques already exist, among the non-hierarchical ones the most known, thank to its simplicity and computational property, is k-means method. However, the method is unstable when the number of variables is large and when variables are correlated. This problem leads to the development of two-step methods, they perform a linear transformation of variable into a reduced number of uncorrelated factors and CA is applied on this factors. Two-steps methods minimize two different functions that can be in contrast between them and the first factorial step can in part obscure the clustering structure. Iterative factorial clustering methods overcome these issues; they perform a factorial step and a clustering step iteratively, optimizing a common criterion. In this thesis a new factorial clustering method is proposed: Factorial Probabilistic Distance Clustering (FPDC). It is based on Probabilistic Distance (PD) Clustering that is a non-parametric probabilistic method to find homogeneous groups, PD Clustering seeks for a set of K group centres maximising the empirical probabilities of belonging to a cluster of the n statistical units. As the number of variables tends to be large the solution tends to become unstable. FPDC consists of a two steps iterative procedure: linear transformation of the initial data using Tucker 3 decomposition and PD-clustering on the transformed data. This thesis also shows that Tucker3 decomposition is a consistent transformation to project original data in a subspace defined according to the PD-Clustering criterion. The integration of the PD Clustering and the Tucker3 factorial step makes the clustering more stable and permits to consider datasets with large number of variables and clusters having not elliptical form.