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

  • Giulianella Coletti
  • Bernadette Bouchon-Meunier
Foundations
  • 35 Downloads

Abstract

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.

Keywords

Comparative similarities Boundary axioms Uniformity axioms Monotonicity axioms Independence axioms Representability by similarity measures 

Notes

Funding

Giulianella Coletti work was partially supported by Perugia University, funding of 2016 Research Projects, under grant: “Decisions under risk, uncertainty and imprecision”, by the Italian Ministry of Health under Grant J521I14001640001 (“Intelligent systems helping in decisions for the early alert and the dissuasion to the use of doping”).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly
  2. 2.Sorbonne Universités, UPMC Univ Paris 06, UMR 7606, LIP6ParisFrance
  3. 3.CNRS, UMR 7606, LIP6ParisFrance

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