Paper:

# Two-Dimensional Copulas as Important Binary Aggregation Operators

## Endre Pap^{*}, and Marta Takács^{**}

^{*}Department of Mathematics and Informatics, Faculty of Natural Sciences and Mathematics in Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia and Montenegro

^{**}Budapest Tech, John von Neumann Faculty of Informatics, H-1034 Budapest, Bécsi út 96.b, Hungary

We introduce 2-copulas (copulas, shortly) and recent related research results. We present invariant copulas and their application in the theory of aggregation operators. Copulas are transformed by increasing bijections at the unit interval and discuss copula attractors. We also present results on the approximation of associative copulas by strict and nilpotent triangular norms.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.10, No.4, pp. 522-526, 2006.

- [1] T. Calvo, G. Mayor, and R. Mesiar (Eds.), “Aggregation Operators. New Trends and Applications,” Physica-Verlag, Heidelberg, 2002.
- [2] P. Capéraà, A.-L. Fougères, and C. Genest, “Bivariate distributions with given extreme value attractor,” J. Multivariate Anal., 72, pp. 30-49, 2000.
- [3] G. Choquet, “Theory of capacities,” Ann. Inst. Fourier (Grenoble), 5 (1953-1954), pp. 131-292.
- [4] I. Cuculescu, and R. Theodorescu, “Extreme value attractors for star unimodal copulas,” C. R. Math. Acad. Sci. Paris, 334, pp. 689-692, 2002.
- [5] W. F. Darsow, B. Nguyen, and E. T. Olsen, “Copulas and Markov processes,” Illinois J. Math., 36: pp. 600-642, 1992.
- [6] J. C. Fodor, and M. Roubens, “Fuzzy Preference Modelling and Multicriteria Decision Support,” Kluwer Academic Publishers, Dordrecht, 1994.
- [7] M. J. Frank, “On the simultaneous associativity of F(x, y) and x+y-F(x, y),” Aequationes Math., 19, pp. 194-226, 1979.
- [8] J. Galambos, “The Asymptotic Theory of Extreme Order Statistics,” Robert E. Krieger Publishing, Melbourne (FL), 1987.
- [9] C. Genest, and L.-P. Rivest, “A characterization of Gumbel’s family of extreme value distributions,” Statist. Probab. Lett., 8, pp. 207-211, 1989.
- [10] O. Hadžić, and E. Pap, “Fixed Point Theory in Probabilistic Metric Spaces,” Kluwer Academic Publishers, Dordrecht, 2001.
- [11] H. Imaoka, “On a subjective evaluation model by a generalized fuzzy integral,” Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 5: pp. 517-529, 1997.
- [12] E. P. Klement, R. Mesiar, and E. Pap, “Triangular Norms,” Kluwer Academic Publishers, Dordrecht, 2000.
- [13] E. P. Klement, R. Mesiar, and E. Pap, “Uniform approximation of associative copulas by strict and non-strict copulas,” Illinois J. Math., 45, pp. 1393-1400, 2001.
- [14] E. P. Klement, R. Mesiar, and E. Pap, “Invariant copulas,” Kybernetika (Prague), 38, pp. 275-285, 2002.
- [15] E. P. Klement, R. Mesiar, and E. Pap, “Measure-based aggregation operators,” Fuzzy Sets and Systems, 142, pp. 3-14, 2004.
- [16] E. P. Klement, R. Mesiar, and E. Pap, “Archimax copulas and invariance under transformations,” C. R. Math. Acad. Sci. Paris, 340, pp. 755-758, 2005.
- [17] E. P. Klement, R. Mesiar, and E. Pap, “Transformations of copulas,” Kybernetika (Prague), 41, pp. 425-434, 2005.
- [18] G. J. Klir, and T. A. Folger, “Fuzzy Sets, Uncertainty, and Information,” Prentice Hall, Englewood Cliff, 1988.
- [19] A. Kolesárova, “1-Lipschitz aggregation operators and quasi copulas,” Kybernetika (Prague), 39, pp. 615-629, 2003.
- [20] P. Mikusiński, and M. D. Taylor, “A remark on associative copulas,” Comment. Math. Univ. Carolin., 40, pp. 789-793, 1999.
- [21] R. B. Nelsen, “An Introduction to Copulas,” Lecture Notes in Statistics 139, Springer, New York, 1999.
- [22] E. Pap, “Null-Additive Set Functions,” Kluwer Academic Publishers, Dordrecht, 1995.
- [23] J. Pickands, “Multivariate extreme value distributions,” Bull. Inst. Internat. Statist., 49, pp. 859-878 (with a discussion, pp. 894-902), 1981.
- [24] T. Rückschlossová, “Aggregation Operators and Invariantness,” Ph.D. thesis, Slovak University of Technology, Bratislava, 2003.
- [25] B. Schweizer, and A. Sklar, “Probabilistic Metric Spaces,” North-Holland, New York, 1983.
- [26] A. Sklar, “Fonctions de répartition à
*n*dimensions et leurs marges,” Publ. Inst. Statist. Univ. Paris, 8, pp. 229-231, 1959. - [27] J. A. Tawn, “Bivariate extreme value theory: models and estimation,” Biometrika, 75, pp. 397-415, 1988.

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