A Topological Discriminant Analysis

Abstract

In this paper, we propose a new discriminant approach, called Topological Discriminant Analysis, which use a proximity measure in a topological context. The results of any operation of clustering or classification of objects strongly depend on the proximity measure chosen. The user has to select one measure among many existing ones. Yet, from a discrimination point of view, according to the notion of topological equivalence chosen, some measures are more or less equivalent. The concept of topological equivalence uses the basic notion of local neighborhood. In a discrimination context, we first define the topological equivalence between the chosen proximity measure and the perfect discrimination measure adapted to the data considered, through the adjacency matrix induced by each measure, then propose a new topological method of discrimination using this selected proximity measure. To judge the quality of discrimination, in addition to the classical percentage of objects well classified, we define a criterion for topological equivalence of discrimination. The principle of the proposed approach is illustrated using a real data set with conventional proximity measures of literature for quantitative variables. The results of the proposed Topological Discriminant Analysis, associated to the “best” discriminating proximity measure, are compared with those of classical metric models of discrimination, Linear Discriminant Analysis and Multinomial Logistic Regression.