A study of similarity measures through the paradigm of measurement theory: the classic case
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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.
KeywordsComparative similarities Boundary axioms Uniformity axioms Monotonicity axioms Independence axioms Representability by similarity measures
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”).
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Conflict of interest
The authors declare that they have no conflict of interest.
This article does not contain any studies with human participants or animals performed by any of the authors.
- Baioletti M, Coletti G, Petturiti D (2012) Advances in computational intelligence: 14th international conference on information processing and management of uncertainty in knowledge-based systems, IPMU 2012, Catania, Italy, July 9–13, 2012, Proceedings, Part III, Chapter. Weighted attribute combinations based similarity measures. Springer, Berlin, pp 211–220Google Scholar
- Boriah S, Chandola V, Kumar V (2008) Similarity measures for categorical data: a comparative evaluation. In: Proceedings of the 8th SIAM international conference on data mining, SIAM, pp 243–254Google Scholar
- Bouchon-Meunier B, Rifqi M, Lesot MJ (2008) Similarities in fuzzy data mining: from a cognitive view to real-world applications. In Zurada J, Yen G, Wang J (eds) Computational intelligence: research frontiers. WCCI 2008, vol 5050. Springer, LNCS, pp 349–367Google Scholar
- Bouchon-Meunier B, Coletti G, Lesot MJ, Rifqi M (2009) Towards a conscious choice of a similarity measure: a qualitative point of view. In: Sossai C, Ghemello G (eds) Symbolic and quantitative approaches to reasoning with uncertainty: Ecsqaru 2009 proceedings, vol 5590. Springer, LNAI, pp 542–553Google Scholar
- Bouchon-Meunier B, Coletti G, Lesot MJ, Rifqi M (2010) Towards a conscious choice of a fuzzy similarity measure: a qualitative point of view. In: Hllermeier E, Kruse R, Hoffmann F (eds) Computational intelligence for knowledge-based system design: IPMU 2010 proceedings, vol 6178. Springer, LNAI, pp 1–10Google Scholar
- Choi S-S, Cha S-H, Tappert CC (2010) A survey of binary similarity and distance measures. J Syst Cybern Inf 8(1):43–48Google Scholar
- Coletti G, Bouchon-Meunier B (2018) A study of similarity measures through the paradigm of measurement theory: the fuzzy case. SoftComputing (submitted) Google Scholar
- Coletti G, Petturiti D, Vantaggi B (2017) Fuzzy weighted attribute combinations based similarity measures. In: Proceedings of ECSQARU 2017 (Symbolic and quantitative approaches to reasoning with uncertainty), vol 10369. LNCS, pp 364–374Google Scholar
- Dvoraki J, Baume N, Botré Broséus J, Budgett R, Frey WO, Geyer H, Harcourt PR, Ho D, Howman D, Isola V, Lundby C, Marclay F, Peytavin A, Pipe A, Pitsiladis YP, Reichel C, Robinson N, Rodchenkov G, Saugy M, Sayegh S, Segura J, Thevis M, Vernec A, Viret M, Vouillamoz M, Zorzoli M (2014) Time for change: a roadmap to guide the implementation of the World Anti-Doping Code 2015. Br J Sports Med: BJSM 48:801–806CrossRefGoogle Scholar
- Hahn U, Ramscar M (eds) (2001) Similarity and categorization. Oxford University Press, OxfordGoogle Scholar
- Jaccard P (1908) Nouvelles recherches sur la distribution florale. Bull Soc Vaud Sci Nat 44:223–270Google Scholar
- Lesot MJ, Rifqi M (2010) Order-based equivalence degrees for similarity and distance measures. In: Hllermeier E, Kruse R, Hoffmann F (eds) Computational intelligence for knowledge-based systems design. IPMU 2010, vol 6178. Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, pp 19–28Google Scholar
- Penney GP, Weese J, Little JA, Desmedt P, Hill DLG, Hawkes DJ (1998) A comparison of similarity measures for use in 2-D-3-D medical image registration. In: Proceedings of MICCAI 1998: medical image computing and computer-assisted intervention MICCAI98, vol. 1496. LNCS, pp 1153–1161Google Scholar
- Sokal RR, Michener C (1958) A statistical method for evaluating systematic relationships. Univ Kansas Sci Bull 38:1409–1438Google Scholar
- Sokal RR, Sneath PHA (1963) Priciples of numerical taxonomy. W.H. Freeman, San FranciscoGoogle Scholar
- Sorensen T (1948) A method of establishing groups of equal amplitude in plant sociology based on similarity of species content and its application to analyses of the vegetation on Danish commons. K Dan Vidensk Selsk Biol Skr 5:1–34Google Scholar
- Toussaint GT (2004) A comparison of rhythmic similarity measures. In: Proceedings 5th international conference on music information retrievalGoogle Scholar
- Zhang Z, Huang K, Tan T (2006) Comparison of similarity measures for trajectory clustering in outdoor surveillance scenes. In: Proceedings of 18th international conference on pattern recognition (ICPR’06). IEEE. https://doi.org/10.1109/ICPR.2006.392