Fitting Symmetric Fuzzy Measures for Discrete Sugeno Integration

  • Marek GągolewskiEmail author
  • Simon James
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 642)


The Sugeno integral has numerous successful applications, including but not limited to the areas of decision making, preference modeling, and bibliometrics. Despite this, the current state of the development of usable algorithms for numerically fitting the underlying discrete fuzzy measure based on a sample of prototypical values – even in the simplest possible case, i.e., assuming the symmetry of the capacity – is yet to reach a satisfactory level. Thus, the aim of this paper is to present some results and observations concerning this class of data approximation problems.


Sugeno integral Aggregation functions Machine learning Regression Approximation 



This study was supported by the National Science Center, Poland, research project 2014/13/D/HS4/01700. Data provided by


  1. 1.
    Anderson, D., Keller, J., Havens, T.: Learning fuzzy-valued fuzzy measures for the fuzzy-valued Sugeno fuzzy integral. Lecture Notes in Artificial Intelligence, vol. 6178, pp. 502–511 (2010)Google Scholar
  2. 2.
    Beliakov, G.: How to build aggregation operators from data. Int. J. Intell. Syst. 18, 903–923 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beliakov, G., Bustince, H., Calvo, T.: A Practical Guide to Averaging Functions. Springer (2016)Google Scholar
  4. 4.
    Beliakov, G., James, S.: Using linear programming for weights identification of generalized Bonferroni means in R. Lecture Notes in Computer Science, vol. 7647, pp. 35–44 (2012)Google Scholar
  5. 5.
    Bullen, P.: Handbook of Means and Their Inequalities. Springer Science+Business Media, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dukhovny, A.: Lattice polynomials of random variables. Stat. Probab. Lett. 77, 989–994 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gagolewski, M., Grzegorzewski, P.: S-statistics and their basic properties. In: Borgelt, C., et al. (eds.) Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Intelligent and Soft Computing, vol. 77, pp. 281–288. Springer (2010)Google Scholar
  8. 8.
    Gagolewski, M., Mesiar, R.: Monotone measures and universal integrals in a uniform framework for the scientific impact assessment problem. Inf. Sci. 263, 166–174 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hirsch, J.E.: An index to quantify individual’s scientific research output. Proc. Natl. Acad. Sci. 102(46), 16569–16572 (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Johnson, S.G.: The NLopt nonlinear-optimization package (2017),
  12. 12.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice Hall PTR, New Jersey (1995)zbMATHGoogle Scholar
  13. 13.
    Marichal, J.L.: Weighted lattice polynomials of independent random variables. Disc. Appl. Math. 156, 685–694 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mesiar, R., Gagolewski, M.: H-index and other Sugeno integrals: some defects and their compensation. IEEE Trans. Fuzzy Syst. 24(6), 1668–1672 (2016)CrossRefGoogle Scholar
  15. 15.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2006)zbMATHGoogle Scholar
  16. 16.
    Prade, H., Rico, A., Serrurier, M.: Elicitation of Sugeno integrals: a version space learning perspective. Lecture Notes in Computer Science, vol. 5722, pp. 392–401 (2009)Google Scholar
  17. 17.
    R Development Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2017),
  18. 18.
    Rios, L.M., Sahinidis, N.V.: Derivative-free optimization: a review of algorithms and comparison of software implementations. J. Global Optim. 56, 1247–1293 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology (1974)Google Scholar
  20. 20.
    Torra, V.: Learning weights for the quasi-weighted means. IEEE Trans. Fuzzy Syst. 10(5), 653–666 (2002)CrossRefGoogle Scholar
  21. 21.
    Torra, V., Narukawa, Y.: The \(h\)-index and the number of citations: two fuzzy integrals. IEEE Trans. Fuzzy Syst. 16(3), 795–797 (2008)CrossRefGoogle Scholar
  22. 22.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Syst. Man Cybern. 18(1), 183–190 (1988)CrossRefzbMATHGoogle Scholar
  23. 23.
    Yager, R.R., Kacprzyk, J. (eds.): The Ordered Weighted Averaging Operators. Theory and Applications. Kluwer Academic Publishers, Norwell (1997)Google Scholar
  24. 24.
    Yuan, B., Klir, G.J.: Constructing fuzzy measures: a new method and its application to cluster analysis. In: Proceedings of NAFIPS 1996, pp. 567–571 (1996)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Systems Research Institute, Polish Academy of SciencesWarsawPoland
  2. 2.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland
  3. 3.School of Information TechnologyDeakin UniversityBurwoodAustralia

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