A study of similarity measures through the paradigm of measurement theory: the classic case

Similarity measures are used in various tasks dealing with the management of data or information, such as decision-making, case-based reasoning, cased-based information retrieval, recommendation systems and user profile analysis, to cite but a few. The paper aims at providing information on similarity measures that can help in choosing “a priori” one of them on the basis of the semantics behind this choice. To this end, we study similarity measures from the point of view of the ranking relation they induce on object pairs. Using a classic method of measurement theory, we establish necessary and sufficient conditions for the existence of a particular class of numerical similarity measures, representing a given binary relation among pairs of objects which express the idea of “no more similar than”. The above conditions are all (and only) the rules which are accepted when one decides to evaluate similarity through any element of a specific class of similarity measures. We exemplify the possible application of such conditions and the relevant results on a real-world problem and discuss them in the ambit of cognitive psychology. We consider here a classical context, while the fuzzy context will be studied in a companion paper.