Topological comparisons of proximity measures

Abstract

In many fields of application, the choice of proximity measure directly affects the results of data mining methods, whatever the task might be: clustering, comparing or structuring of a set of objects. Generally, in such fields of application, the user is obliged to choose one proximity measure from many possible alternatives. According to the notion of equivalence, such as the one based on pre-ordering, certain proximity measures are more or less equivalent, which means that they should produce almost the same results. This information on equivalence might be helpful for choosing one such measure. However, the complexity O(n 4) of this approach makes it intractable when the size n of the sample exceeds a few hundred. To cope with this limitation, we propose a new approach with less complexity O(n 2). This is based on topological equivalence and it exploits the concept of local neighbors. It defines equivalence between two proximity measures as having the same neighborhood structure on the objects. We illustrate our approach by considering 13 proximity measures used on datasets with continuous attributes. © 2012 Springer-Verlag.