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

  • Giulianella ColettiEmail author
  • Bernadette Bouchon-Meunier


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.


Comparative 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”).

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.


  1. Anderberg MR (1973) Cluster analysis for applications. Academic Press, New YorkzbMATHGoogle Scholar
  2. 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
  3. Bertoluzza C, Di Bacco M, Doldi V (2004) An axiomatic characterization of the measures of similarity. Sankhya 66:474–486MathSciNetzbMATHGoogle Scholar
  4. Bhutani KR, Rosenfeld A (2003) Dissimilarity measures between fuzzy sets or fuzzy structures. Inf Sci 152:313–318MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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
  6. Bouchon-Meunier B, Rifqi M, Bothorel S (1996) Towards general measures of comparison of objects. Fuzzy Sets Syst 84:143–153MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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
  8. 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
  9. 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
  10. 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
  11. Coletti G, Bouchon-Meunier B (2018) A study of similarity measures through the paradigm of measurement theory: the fuzzy case. SoftComputing (submitted) Google Scholar
  12. Coletti G, Di Bacco M (1989) Qualitative characterization of a dissimilarity and concentration index. Metron XLVII:121–130MathSciNetzbMATHGoogle Scholar
  13. 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
  14. Couso I, Garrido L, Sànchez L (2013) Similarity and dissimilarity measures between fuzzy sets: a formal relational study. Inf Sci 229:122–141MathSciNetCrossRefzbMATHGoogle Scholar
  15. Cross VV, Sudkamp TA (2002) Similarity and compatibility in fuzzy set theory: assessment and applications. Studies in fuzziness and soft computing, vol 93. Springer, BerlinzbMATHGoogle Scholar
  16. Dice LR (1945) Measures of the amount of ecological association between species. Ecology 26:297–302CrossRefGoogle Scholar
  17. 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
  18. Filev P, Hadjiiski L, Sahiner B, Chan HP, Helvie MA (2005) Comparison of similarity measures for the task of template matching of masses on serial mammograms. Med Phys 32(2):515–529CrossRefGoogle Scholar
  19. Gilboa I, Schmeidler D (1995) Case-based decision theory. Q J Econ 110:605–639CrossRefzbMATHGoogle Scholar
  20. Gilboa I, Schmeidler D (1997) Act similarity in case-based decision theory. Econ Theory 9:47–61MathSciNetCrossRefzbMATHGoogle Scholar
  21. Gilboa I, Lieberman O, Schmeidler D (2006) A similarity-based approach to prediction. Rev Econ Stat 162(1):124–131MathSciNetzbMATHGoogle Scholar
  22. Ha V, Haddawy P (2003) Similarity of personal preferences: theoretical foundations and empirical analysis. Artif Intell 146:149–173MathSciNetCrossRefzbMATHGoogle Scholar
  23. Hahn U, Ramscar M (eds) (2001) Similarity and categorization. Oxford University Press, OxfordGoogle Scholar
  24. Hwang CM, Yang MS, Hung WL, Lee MG (2012) A similarity measure of intuitionistic fuzzy sets based on the Sugeno integral with its application to pattern recognition. Inf Sci 189:93–109MathSciNetCrossRefzbMATHGoogle Scholar
  25. Jaccard P (1908) Nouvelles recherches sur la distribution florale. Bull Soc Vaud Sci Nat 44:223–270Google Scholar
  26. Krantz D, Luce R, Suppes P, Tversky A (1971) Foundations of measurement, vol I. Academic Press, New YorkzbMATHGoogle Scholar
  27. 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
  28. Lesot MJ, Rifqi M, Benhadda H (2009) Similarity measures for binary and numerical data: a survey. Int J Knowl Eng Soft Data Paradig (KESDP) 1:63–84CrossRefGoogle Scholar
  29. Li Y, Qin K, He X (2014) Some new approaches to constructing similarity measures. Fuzzy Sets Syst 234:46–60MathSciNetCrossRefzbMATHGoogle Scholar
  30. Narens L (1974) Minimal conditions for additive conjoint measurement and qualitative probability. J Math Psychol 11:404–430MathSciNetCrossRefzbMATHGoogle Scholar
  31. Ochiai A (1957) Zoogeographic studies on the soleoid fishes found in Japan and its neighbouring regions. Bull Jpn Soc Sci Fish 22:526–30CrossRefGoogle Scholar
  32. Pelillo M (ed) (2013) Similarity-based pattern analysis and recognition. Advances in computer vision and pattern recognition. Springer, LondonzbMATHGoogle Scholar
  33. 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
  34. Rissland E (2006) AI and similarity. IEEE Intell Syst 21:33–49CrossRefGoogle Scholar
  35. Rogers DJ, Tanimoto TT (1960) A computer program for classifying plants. Science 132:1115–1118CrossRefGoogle Scholar
  36. Sokal RR, Michener C (1958) A statistical method for evaluating systematic relationships. Univ Kansas Sci Bull 38:1409–1438Google Scholar
  37. Sokal RR, Sneath PHA (1963) Priciples of numerical taxonomy. W.H. Freeman, San FranciscoGoogle Scholar
  38. 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
  39. Simmons S, Estes Z (2008) Individual differences in the perception of similarity and difference. Cognition 106(3):781–795CrossRefGoogle Scholar
  40. Suppes P, Krantz D, Luce R, Tversky A (1989) Foundations of measurement, vol II. Academic Press, New YorkzbMATHGoogle Scholar
  41. Toussaint GT (2004) A comparison of rhythmic similarity measures. In: Proceedings 5th international conference on music information retrievalGoogle Scholar
  42. Tversky A (1977) Features of similarity. Psychol Rev 84:327–352CrossRefGoogle Scholar
  43. 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.

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

Personalised recommendations