Fuzzy Optimization and Decision Making

, Volume 13, Issue 3, pp 273–286 | Cite as

Universal integrals based on copulas

  • Erich Peter KlementEmail author
  • Radko Mesiar
  • Fabio Spizzichino
  • Andrea Stupňanová


A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copula-based universal integrals as recently introduced in Klement et al. (IEEE Trans Fuzzy Syst 18:178–187, 2010). For the product copula, the family of integrals considered here contains the Choquet and the Shilkret integral, and it belongs to the class of decomposition integrals proposed in Even and Lehrer (Econ Theory, 2013) as well as to the class of superdecomposition integrals introduced in Mesiar et al. (Superdecomposition integral, 2013). For the upper Fréchet-Hoeffding bound, the corresponding hierarchical family contains only two elements: all but the greatest element coincide with the Sugeno integral.


Capacity Copula Universal integral Choquet integral  Sugeno integral Shilkret integral 



The work on this paper was supported by the “Technologie-Transfer-Förderung” Wi-2013-168051/WWin/Kai of the Upper Austrian Government and by the grants APVV-0073-10, GAČR P402/11/0378 and VEGA 1/0143/11.


  1. Bassan, B., & Spizzichino, F. (2005). Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. Journal of Multivariate Analysis, 93, 313–339.CrossRefzbMATHMathSciNetGoogle Scholar
  2. Beliakov, G., & James, S. (2011). Citation-based journal ranks: The use of fuzzy measures. Fuzzy Sets and Systems, 167, 101–119.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Benvenuti, P., Mesiar, R., & Vivona, D. (2002). Monotone set functions-based integrals. In E. Pap (Ed.), Handbook of measure theory, chapter 33 (pp. 1329–1379). Amsterdam: Elsevier Science.Google Scholar
  4. Choquet, G. (1954). Theory of capacities. Annales de l’Institut Fourier (Grenoble), 5, 131–295.CrossRefzbMATHMathSciNetGoogle Scholar
  5. Díaz, S., De Baets, B., & Montes, S. (2010). General results on the decomposition of transitive fuzzy relations. Fuzzy Optimization and Decision Making, 9, 1–29.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Durante, F., & Sempi, C. (2005). Semicopulæ. Kybernetika (Prague), 41, 315–328.zbMATHMathSciNetGoogle Scholar
  7. Even, Y., Lehrer, E. (2013). Decomposition-integral: Unifying Choquet and the concave integrals. Economic Theory doi: 10.1007/s00199-013-0780-0.
  8. Gagolewski, M., & Mesiar, R. (2014). Monotone measures and universal integrals in a uniform framework for the scientific impact assessment problem. Information Sciences. doi: 10.1016/j.ins.2013.12.004.
  9. Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.CrossRefzbMATHGoogle Scholar
  10. Klement, E. P., Kolesárová, A., Mesiar, R., & Stup\(\check{\rm n}\)anová, A. (2013). A generalization of universal integrals by means of level dependent capacities. Knowledge-Based Systems, 38, 14–18.Google Scholar
  11. Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular norms. Dordrecht: Kluwer.CrossRefzbMATHGoogle Scholar
  12. Klement, E. P., Mesiar, R., & Pap, E. (2004). Measure-based aggregation operators. Fuzzy Sets and Systems, 142, 3–14.CrossRefzbMATHMathSciNetGoogle Scholar
  13. Klement, E. P., Mesiar, R., & Pap, E. (2010). A universal integral as common frame for Choquet and Sugeno integral. IEEE Transactions on Fuzzy Systems, 18, 178–187.CrossRefGoogle Scholar
  14. Klement, E. P., Spizzichino, F., Mesiar, R., & Stup\(\check{\rm n}\)anová, A. (2013). Copula-based universal integrals. In Proceedings 2013 IEEE international conference on fuzzy systems (pp. 1065–1069). Hyderabad: IEEE.Google Scholar
  15. Mesiar, R., Li, J., & Pap, E. (2013). Superdecomposition integral (submitted for publication).Google Scholar
  16. Mesiar, R., & Stup\(\check{\rm n}\)anová, A. (2013). Decomposition integrals. Internat. J. Approx. Reason., 54, 1252–1259.Google Scholar
  17. Nelsen, R. B. (2006). An introduction to copulas (2nd ed., Vol. 139) Lecture Notes in Statistics. New York: Springer.Google Scholar
  18. Shilkret, N. (1971). Maxitive measure and integration. Indagationes Mathematicae, 33, 109–116.CrossRefMathSciNetGoogle Scholar
  19. Sklar, A. (1959). Fonctions de répartition à \(n\) dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229–231.MathSciNetGoogle Scholar
  20. Sugeno, M. (1974). Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology.Google Scholar
  21. Torra, V., & Narukawa, Y. (2008). The \(h\)-index and the number of citations: Two fuzzy integrals. IEEE Transactions on Fuzzy Systems, 16, 795–797.CrossRefMathSciNetGoogle Scholar
  22. Yager, R. R. (2013). Joint cumulative distribution functions for Dempster-Shafer belief structures using copulas. Fuzzy Optimization and Decision Making, 12, 393–414.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Erich Peter Klement
    • 1
    Email author
  • Radko Mesiar
    • 2
    • 3
  • Fabio Spizzichino
    • 4
  • Andrea Stupňanová
    • 5
  1. 1.Department of Knowledge-Based Mathematical SystemsJohannes Kepler UniversityLinzAustria
  2. 2.Department of Mathematics and Descriptive Geometry, Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovakia
  3. 3.ÚTIA AV ČRPragueCzech Republic
  4. 4.Department of MathematicsUniversity of Rome “La Sapienza”RomeItaly
  5. 5.Department of Mathematics and Descriptive Geometry, Faculty of Civil EngineeringSlovak University of TechnologyBratislavaSlovakia

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