Comparison of Proximity Measures: A Topological Approach

In many application domains, the choice of a proximity measure affect directly the result of classification, comparison or the structuring of a set of objects. For any given problem, the user is obliged to choose one proximity measure between many existing ones. However, this choice depend on many characteristics. Indeed, according to the notion of equivalence, like the one based on pre-ordering, some of the proximity measures are more or less equivalent. In this paper, we propose a new approach to compare the proximity measures. This approach is based on the topological equivalence which exploits the concept of local neighbors and defines an equivalence between two proximity measures by having the same neighborhood structure on the objects. We compare the two approaches, the pre-ordering and our approach, to thirty five proximity measures using the continuous and binary attributes of empirical data sets.