Advertisement

Logical Comparison Measures in Classification of Data—Nonmetric Measures

  • Kalle SaastamoinenEmail author
Conference paper
  • 1.1k Downloads
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

Abstract

In this chapter, we will create and use generalized combined comparison measures from t-norms (T) and t-conorms (S) for comparison of data. Norms are combined by the use of generalized mean, where t-norms give minimum and t-conorms give maximum compensation. From this intuitively thinking follows that when these norms are aggregated together, these new comparison measures should be able to find the best possible classification result in between minimum and maximum. We will use classification as our test bench for the suitability of these new comparison measures created. In these classification tasks, we have tested five different types of combined comparison measures (CCM), with t-norms and t-conorms. That were Dombi family, Frank family, Schweizer-Sklar family, Yager family, and Yu family. In classification, we used the following datasets: ionosphere, iris, and wine. We will compare the results achieved with CCM to the ones achieved with pseudo equivalences and show that these new measures tend to give better results.

Keywords

Ionos Iris Wine Similarity Comparison measure Logical Classification Data 

References

  1. 1.
    Tversky, A., Krantz, D.H.: The dimensional representation and the metric structure of similarity data. J. Math. Psychol. 7(3), 572–596 (1970)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Santini, S., Jain, R.: Similarity measures. IEEE Trans. Pattern Anal. Mach. Intell. 21(9), 871–883 (1999)CrossRefGoogle Scholar
  3. 3.
    France, R.K.: Weights and Measures: An Axiomatic Model for Similarity Computations. Internal Report, Virginia Tech (1994)Google Scholar
  4. 4.
    De Cock, M., Kerre, E.: Why fuzzy T-equivalence relations do not resolve the Poincar paradox, and related issues. Fuzzy Sets Syst. 133(2), 181–192 (2003)CrossRefGoogle Scholar
  5. 5.
    Tversky, A.: Features of similarity. Psychol. Rev. 84(4), 327–352 (1977)CrossRefGoogle Scholar
  6. 6.
    Zadeh, L.A., Fuzzy sets and their application to pattern classification and clustering analysis. In: Van Ryzin, J. (Ed.) Classification and Clustering. Academic Press, pp. 251–299 (1977)Google Scholar
  7. 7.
    Pal, S.K., Dutta-Majumder, D.K.: Fuzzy Mathematical Approach to Pattern Recognition. Wiley (Halsted), N. Y. (1986)Google Scholar
  8. 8.
    Saastamoinen, K.: Classification of data with similarity classifier. In: International Work-Conference on Time Series, Keynote Speech, ITISE2016 ProceedingsGoogle Scholar
  9. 9.
    Lowen, R.R.: Fuzzy Set Theory: Basic Concepts, Techniques, and Bibliography. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  10. 10.
    Bellman, R., Giertz, M.: On the analytic formalism of the theory of fuzzy sets. Inf. Sci. 5, 149–165 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kang, T., Chen, G.: Modifications of Bellman-Giertz’s theorem. Fuzzy Sets Syst. 94(3), 349–353 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Saastamoinen, K., Ketola, J.: Using generalized combination measure from Dombi and Yager type of T-norms and T-conorms in classification. In: Proceedings of the ECTI-CON 2005 Conference (2005)Google Scholar
  13. 13.
    Dyckhoff, H., Pedrycz, W.: Generalized means as model of compensative connectives. Fuzzy Sets Syst. 14, 143–154 (1984)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zimmermann, H.-J., Zysno, P.: Latent connectives in human decision making. Fuzzy Sets Syst. 4, 37–51 (1980)CrossRefGoogle Scholar
  15. 15.
    Bilgiç, T., Türkşen, I.B.: Measurement-theoretic justification of connectives in fuzzy set theory. Fuzzy Sets Syst. 76(3), 289–308 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Frank, M.J.: On the simultaneous associativity of \(F(x, y)\) and \(x+y-F(x, y)\). Aeqationes Math. 19, 194–226 (1979)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Schweizer, B., Sklar, A.: Associative functions and abstract semigroups. Publ. Math. Debr. 10, 69–81 (1963)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Yager, R.R.: On a general class of fuzzy connectives. Fuzzy Sets Syst. 4, 235–242 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yu, Y.: Triangular norms and TNF-sigma algebras. Fuzzy Sets Syst. 16, 251–264 (1985)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sklar, A.: Fonctions de r\(\acute{e}\)partition \(\acute{a}\) n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959)MathSciNetGoogle Scholar
  22. 22.
    Fisher, N.I.: Copulas. In: Kotz, S., Read, C.B., Banks, D.L. (eds.) Encyclopedia of Statistical Sciences, vol. 1, pp. 159–163. Wiley, New York (1997)Google Scholar
  23. 23.
    Hastie, T., Tibshirani, R.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics, Springer, New York (2001)CrossRefGoogle Scholar
  24. 24.
    Goldberg, D.E.: Real-coded genetic algorithms, virtual alphabets, and blocking. Technical Report 9001, University of Illinois at Urbana-Champain (1990)Google Scholar
  25. 25.
    Mantel, B., Periaux, J., Sefrioui, M.: Gradient and genetic optimizers for aerodynamic desing. In: ICIAM 95 Conference, Hamburgh (1995)Google Scholar
  26. 26.
    Michalewics, Z.: Genetic Algorithms + Data Structures = Evolution Programs Artificial Intelligence. Springer, New York (1992)CrossRefGoogle Scholar
  27. 27.
    Grefenstette, J.J.: Optimization of control parameters for genetic algorithms. IEEE Trans. Syst. Man Cybern. 16(1), 122–128 (1986)CrossRefGoogle Scholar
  28. 28.
    Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution—A Practical Approach to Global Optimization. Springer, Natural Computing Series (2005)zbMATHGoogle Scholar
  29. 29.
    UC Irvine Machine Learning Repository. http://archive.ics.uci.edu/ml/. Accessed 15 Jan 2017
  30. 30.
    Saastamoinen, K.: Many valued algebraic structures as measures of comparison, Acta Universitatis Lappeenrantaensis. Ph.D. Thesis (2008)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Military TechnologyNational Defence UniversityHelsinkiFinland

Personalised recommendations