Proximity measures in topological structure for discrimination

The choice of a proximity measure between objects has a direct impact on the results of any operation of classification, comparison, evaluation or structuring a set of objects. In many application fields, for a given problem, the user is prompted to choose one among the many existing proximity measures. However, according to the notion of topological equivalence chosen, some are more or less equivalent. In this paper, we propose a new comparison approach of proximity measures for the purpose of discrimination and in a new concept of topological equivalence. This approach exploits the concept of the local neighborhood. It defines discriminant equivalence between two proximity measures as having the same neighborhood structure on the objects of a set of explanatory continuous variables according to a target qualitative variable that we want to explain. According to the notion of topological equivalence based on the concept of neighborhood graphs, we use adjacency binary matrices, associated with proximity measure, Between and Within groups to classify. Some of the proximity measures are more or less equivalent, which means that they produce, more or less, the same discrimination results. We then propose to define the topological equivalence between two proximity measures through the topological structure induced by each measure. It believes that two proximity measures are topologically equivalent if they induce the same neighborhood structure on the objects in purpose of discrimination. The comparison adjacency matrix is a useful tool for measuring the degree of resemblance between two empirical proximity matrices in a discriminating context. To view these proximity measures, we propose an hierarchy of proximity measures which are grouped according to their degree of resemblance in a topological context of discrimination. We illustrate the principle of this approach on a simple real example of continuous explanatory data for about a dozen proximity measures of the literature.