Characterization of Bivariate Quadratic Transformations of Quasi-copulas

Document Type : Research Paper


Department of Mathematics, Faculty of Science, Chiang Mai University


In this study, we focus on bivariate transformations of trivariate quasi-copulas. We characterize necessary and sufficient conditions of transformations that can be written in the form of compositions between two quasi-copulas and a quadratic polynomial function. The conditions only depend on the coefficients of the quadratic polynomial. The set of these coefficients is convex with linear-section boundary and it lies on the seven-dimensional Euclidean space. All extreme points of this set have been characterized via CAS and can be used to construct quasi-copulas. Construction examples are also given.


Main Subjects

[1] C. Alsina, R. B. Nelsen, B. Schweizer, On the characterization of a class of binary operations on distribution functions, Statistics and Probability Letters, 17(2) (1993), 85-89.
[2] R. C. Archibald, Mathematics before the Greeks, Science, 71(1831) (1930), 109-121.
[3] J. J. Arias-Garc´ıa, R. Mesiar, B. De Baets, A hitchhiker's guide to quasi-copulas, Fuzzy Sets and Systems, 393 (2020), 1-28.
[4] G. Beliakov, A. Pradera, T. Calvo, Aggregation functions: A guide for practitioners, Springer Publishing Company, Incorporated, 2008.
[5] P. Boonmee, P. Chanthorn, Quadratic transformations of multivariate semi-copulas, Thai Journal of Mathematics, 18(4) (2020), 1917-1931.
[6] P. Boonmee, S. Tasena, Quadratic transformation of multivariate aggregation functions, Dependence Modeling, 8(1)(2020), 254-261.
[7] V. Boonyasri, S. Tasena, Aggregation function constructed from copula, Carpathian Journal of Mathematics, 39(2)(2023), 383-401.
[8] O. Csisz´ar, J. Fodor, Threshold constructions of aggregation functions, Proceeding of IEEE 9th International Conference on Computational Cybernetics (ICCC), Tihany, Hungary, July 2013.
[9] I. Cuculescu, R. Theodorescu, Copulas: Diagonals, tracks, Revue Roumaine de Math´ematiques Pures et Appliqu´ees, 46(6) (2001), 731-742.
[10] M. Deck´y, R. Mesiar, A. Stup˘nanov´a, Deviation-based aggregation functions, Fuzzy Sets and Systems, 332 (2018), 29-36.
[11] Y. Dodge, The concise encyclopedia of statistics, Springer Science and Business Media, 2008.
[12] M. Gagolewski, Data fusion: Theory, methods, and applications, Institute of Computer Science Polish Academy of Sciences, 2015, 290 pages. DOI:10.5281/zenodo.6960306.
[13] C. Genest, J. J. Quesada Molina, J. A. Rodr´ıguez Lallena, C. Sempi, A characterization of quasi-copulas, Journal of Multivariate Analysis, 69(2) (1999), 193-205.
[14] M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap, Aggregation functions, Cambridge University Press, 2009.
[15] M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap, Aggregation functions: Construction methods, conjunctive, dis-junctive and mixed classes, Information Sciences, 181(1) (2011), 23-43.
[16] A. Koles´arov´a, G. Mayor, R. Mesiar, Quadratic constructions of copulas, Information Sciences, 310 (2015), 69-76.
[17] A. Koles´arov´a, R. Mesiar, On linear and quadratic constructions of aggregation functions, Fuzzy Sets and Systems, 268 (2015), 1-14.
[18] A. Koles´arov´a, R. Mesiar, J. Kalick´a, On a new construction of 1-Lipschitz aggregation functions, quasi-copulas and copulas, Fuzzy Sets and Systems, 226 (2013), 19-31.
[19] A. Mesiarov´a-Zem´ankov´a, R. Mesiar, Y. Su, Ordinal sum constructions for aggregation functions on the real unit interval, Iranian Journal of Fuzzy Systems, 19(1) (2022), 83-96.
[20] S. Tasena, Characterization of quadratic aggregation functions, IEEE Transactions on Fuzzy Systems, 27(4) (2019), 824-829.
[21] S. Tasena, Polynomial copula transformations, International Journal of Approximate Reasoning, 107 (2019), 65-78.
[22] V. Torra, Y. Narukawa, Modeling decisions: Information fusion and aggregation operators, Springer Science and Business Media, 2007. 
[23] S. Wisadwongsa, S. Tasena, Bivariate quadratic copula constructions, International Journal of Approximate Reasoning, 92 (2018), 1-19.