A General Approach to Midpoint Theory and Aggregation of Quasimetrics

Abstract

Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the midpoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of "compromise" or "middle way" between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved. C 2013 Wiley Periodicals, Inc. MASSANET AND VALEROFrom the interest of information aggregation in the applied sciences, for instance Bioinformatics, Medical Diagnosis, and Decision Making, the notion of midpoint set for a metric to model the concept of "middle way" or "compromise" between two given positions emerges in Mathematics. In 1991, Herburt and Moszyńska provided a general description of the shape of midpoint sets for those metrics that are obtained via the aggregation, in the spirit of Borsík and Doboš, of a finite collection of another ones (see Ref. 6). Concretely, they showed that the midpoint set for a metric induced by aggregation can be expressed as the Cartesian product of the midpoint sets for each of the metrics that are merged. Later, in 2003, the notion of midpoint set was applied, through fuzzy sets, to Medicine by Nieto and Torres in Ref. 11. Concretely, they showed that in practical medicine given two descriptions of a patient, which are represented as fuzzy sets and provided by several expert raters, the new suitable average representation of the patient, as working description, does not reduce, in general, to the Euclidean midpoint. In fact, this working representation can be associated with a wide range of "middle ways" between the original provided patient descriptions (for a fuller treatment of data clustering in Medicine we refer the reader to Ref. 12). Inspired by the fact that the working representation does not concur with the Euclidean midpoint, and thus by the uselessness of Euclidean metric, Nieto and Torres studied, in the aforesaid reference, the shape of midpoint sets for the well-known Hamming metric defined between fuzzy sets. More recently, following the original work of Nieto and Torres, Casasnovas and Rosselló gave an explicit description of the midpoint sets, among others, for the weighted Hamming...