Inclusion-Exclusion Integral and t-norm Based Data Analysis Model Construction

  • Aoi HondaEmail author
  • Yoshiaki Okazaki
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)


A data analysis model using the inclusion-exclusion integral and a new construction method of a model utilizing t-norms are proposed. This model is based on the integral with respect to the nonadditive measure and is constructed in three steps of specifications of monotone functions, t-norm and of monotone measures. The model has good description ability and can be applied flexibly to real problems. Applying this model to the data set of a multiple criteria decision making problem, the efficiency of the model is verified by comparing it with the classical linear regression model and with the Choquet integral model.


Monotone measure Inclusion-exclusion integral Möbius transform Interaction operator t-norm 



This work was supported by JSPS KAKENHI Grant Number 50271119.


  1. 1.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dombi, J.: A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 8, 149–163 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Draper, N.R., Smith, H.: Applied Regression Analysis. Wiley Series in Probability and Statistics, 2nd edn. Wiley, New York (1998)zbMATHGoogle Scholar
  4. 4.
    Dubois, S., Prade, H.: New results about properties, semantics of fuzzy set-theoretic operators. In: Wang, P.P., et al. (eds.) Fuzzy Sets, pp. 59–75. Plenum Press, New York (1980)CrossRefGoogle Scholar
  5. 5.
    Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst. 69(3), 279–298 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. A Q. J. Oper. Res. 6, 1–44 (2008). Springer, VerlagMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals: Theory and Applications. Physica, Heidelberg (2000)zbMATHGoogle Scholar
  8. 8.
    Hamacher, H.: Über Logische Aggregation Nicht-binär Explizierter Entscheidnungskriterien. Fischer, Frankfurt (1978)Google Scholar
  9. 9.
    Honda, A., Fukuda, R., Okamoto, J.: Rescaling for evaluations using inclusion-exclusion integral. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2014, Part I. CCIS, vol. 442, pp. 284–293. Springer, Heidelberg (2014)Google Scholar
  10. 10.
    Honda, A., Okamoto, J.: Inclusion-exclusion integral and its application to subjective video quality estimation. In: Kruse, R., Hoffmann, F., Hüllermeier, E. (eds.) IPMU 2010. CCIS, vol. 80, pp. 480–489. Springer, Heidelberg (2010)Google Scholar
  11. 11.
    Honda, A., Okazaki, Y.: Inclusion exclusion integral. In: Proceedings of the 12th Modeling Decisions for Artificial Intelligence (CD-ROM) (2015)Google Scholar
  12. 12.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. USA 28, 535–537 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mori, D., Honda, A., Uchida, M., Okazaki, Y.: Quality evaluations of network services using a non-additive set function. In: Proceedings of The 5th Modeling Decisions for Artificial Intelligence (CD-ROM) (2008)Google Scholar
  15. 15.
    Mori, T., Murofushi, T.: An analysis of evaluation model using fuzzy measure and the Choquet integral. In: Proceedings of the 5th Fuzzy System Symposium, pp. 207–212. Japan Society for Fuzzy Sets and Systems (1989)Google Scholar
  16. 16.
    Murofushi, T., Sugeno, M.: Fuzzy measures, fuzzy integrals. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals, pp. 3–41. Physica-Verlag, Heidelberg (2000)Google Scholar
  17. 17.
    Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific. J. Math. 10, 313–334 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sugeno, M.: Fuzzy measures and fuzzy integrals-a survey. In: Gupta, M.M., Saridis, G.N., Gaines, B.R. (eds.) Fuzzy Automata and Decision Processes, pp. 89–102. North Holland, Amsterdam (1977)Google Scholar
  19. 19.
    Tehrani, A.F., Cheng, W., Dembczyński, K., Hüllermeier, E.: Learning monotone nonlinear models using the Choquet integral. Mach. Learn 89, 183–211 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Torra, V.: Learning aggregation operators for preference modeling. In: Fürnkranz, J., Hüllermeier, E. (eds.) Preference Learning, pp. 317–333. Springer, Berlin (2011)Google Scholar
  21. 21.
    Yager, R.R.: On a general class of fuzzy connectives. Fuzzy Sets Syst. 4, 235–242 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Kyushu Institute of TechnologyIizukaJapan
  2. 2.Fuzzy Logic Systems InstituteIizukaJapan

Personalised recommendations