Comparison of proximity measures for a topological discrimination

Abstract

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, according to the notion of topological equivalence chosen, some measures are more or less equivalent. In this paper, we propose a new approach to compare and classify proximity measures in a topological structure and in a context of discrimination. The concept of topological equivalence uses the basic notion of local neighborhood. We define the topological equivalence between two proximity measures, in the context of discrimination, through the topological structure induced by each measure. We propose a criterion for choosing the "best" measure, adapted to the data considered, among some of the most used proximity measures for quantitative or qualitative data. The principle of the proposed approach is illustrated using two real datasets with conventional proximity measures of literature for quantitative and qualitative variables. Afterward, we conduct experiments to evaluate the performance of this discriminant topological approach and to test if the proximity measure selected as the "best" discriminant changes in terms of the size or the dimensions of the used data. The "best" discriminating proximity measure will be verified a posteriori using a supervised learning method of type Support Vector Machine, discriminant analysis or Logistic regression applied in a topological context.