Fitting Aggregation Operators

  • Vojtěch HavlenaEmail author
  • Dana Hliněná
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9548)


This paper treats the problem of fitting aggregation operators to empirical data. Specifically, we are interested in modelling of the conjunction in human language. To our knowledge, the first attempt to see how humans “interpret” the conjunction for graded properties is due to the paper [1]. In that case, simply the minimum t-norm came out. Our results are different because our approach to the resolution is different. We have experimentally rated simple statements and their conjunctions. Then we have tried, on the basis of measured data, to find a suitable function, which corresponds to human conjunction. First, we discuss methods applicable to associative operators, t-norms. Next, we propose an algorithm for approximation of the t-norm’s generator based on the weighting method and Lawson-Hanson’s algorithm. Suitable modifications of the algorithm can generalize our solutions to aggregation operators. In this way we get new results for generated means which are well-known representatives of aggregation operators. Empirically measured data suggest that people do not understand conjunction necessarily as a commutative operation. Finally, we investigate the modelling of the conjunction via generated Choquet integral.


Aggregation Operators Choquet Integral Understand Conjunction Commutative Operation Restricted Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Hersh, H.M., Caramazza, A.: A fuzzy set approach to modifiers and vagueness in natural language. J. Exp. Psychol. Gen. 105, 254–276 (1976)CrossRefGoogle Scholar
  2. 2.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Boston (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Jenei, S., Fodor, J.C.: On continuous triangular norms. Fuzzy Sets Syst. 100(1–3), 273–282 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jenei, S.: On archimedean triangular norms. Fuzzy Sets Syst. 99(2), 179–186 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beliakov, G., Mesiar, R., Valaskova, L.: Fitting generated aggregation operators to empirical data. Int. J. Uncertainty, Fuzziness Knowl. Based Syst. 12(2), 219–236 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beliakov, G.: Fitting triangular norms to empirical data. In: Klement, E., Mesiar, R. (eds.) Analytic and Probabilistic Aspects of Triangular, pp. 261–272. Elsevier, Boston (2005)Google Scholar
  7. 7.
    De Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences. Springer, New York (2001)zbMATHGoogle Scholar
  8. 8.
    Björck, A.: Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia (1996)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, D., Plemmons, R.J.: Nonnegativity constraints in numerical analysis. In: The Birth of Numerical Analysis, pp. 109–139. World Scientific, Singapore (2010)Google Scholar
  10. 10.
    Havlena, V.: Konjunkce a disjunkce ve fuzzy logice. Bakalářská práce, FIT VUT v Brně (2015)Google Scholar
  11. 11.
    Knuth, D.E.: The Art of Computer Programming. Seminumerical Algorithms, vol. 2, 3rd edn. Addison-Wesley Longman Publishing Co., Inc., Boston (1997)zbMATHGoogle Scholar
  12. 12.
    Bunch, J.R., Hopcroft, J.E.: Triangular factorization and inversion by fast matrix multiplication. Math. Comput. 28(125), 231–236 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Žára, J., Beneš, B., Sochor, J.: Moderní počítačová grafika. Computer Press, a.s. (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Information TechnologyBrno University of TechnologyBrnoCzech Republic
  2. 2.Faculty of Electrical Engineering and CommunicationBrno University of TechnologyBrnoCzech Republic

Personalised recommendations