Understanding (dis)similarity measures

From a psychological point of view, a human being uses the notions of similarity and dissimilarity for problem solving, inductive reasoning, element categorization, or simply to search for information partially matching specific criteria. The ability to assess similarities between a newly given pattern and already known patterns is a distinctive feature of human thinking. It is therefore not strange that similarity and its dual concept dissimilarity are a fundamental part of many theories and applications in several fields, within or related to Artificial Intelligence, like Case Based Reasoning [1], Data Mining [2], Information Retrieval [3], Pattern Matching [4] or Neural Networks, as the Radial Basis Function network [5]. Many applications are characterized by the use of metrics for measuring differences between objects. Metric dissimilarities have been deeply studied but they are tied to a particular transitivity expression based on the triangle inequality. Very often metric (distance) functions are used due to our natural understanding of Euclidean spaces. However, not all metrics are Euclidean and many interesting dissimilarities are non-metric. 1 In a general sense, similarity and dissimilarity express a dual comparison between two elements. We argue that every property of a similarity should have a correspondence with one property of a dissimilarity and vice versa. This duality is commonly ignored, as well as some annoying properties (e.g.