On the design of metric relations

Metric distances and the more general concept of dissimilarities are widely used tools in instance-based learning methods and very especially in the nearestneighbor classification technique. This paper contributes to the design of general dissimilarity measures to increase their utility. The ability to understand the main properties of a hand-crafted dissimilarity measure and to alter them if necessary greatly widens its applicability. Grounded upon a formal definition for a dissimilarity measure, together with a set of fundamental properties, a main body of results is presented concerning equivalence between dissimilarities. Moreover, the very important concepts of transitivity and aggregation are studied. Results in preserving transitivity under transformations and/or in aggregations are presented, with emphasis in the relationship between aggregation and transitivity. Further, the issue of dealing with special values (e.g. missing values) is studied. Although the focus is in dissimilarities, the particular case of metric distances (i.e. dissimilarities fulfilling the triangular inequality) is specifically covered, given its importance. Several examples of the use and potential utility of the results are worked out throughout the text