Copula-like operations on finite settings

Abstract

This paper deals with discrete copulas considered as a class of binary aggregation operators on a finite chain. A representation theorem by means of permutation matrices is given. From this characterization, we study the structure of associative discrete copulas and a theorem of decomposition of any discrete copula in terms of associative discrete copulas is obtained. Finally, some aspects concerning their extension to copulas are dealt with.

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

“…The copula density function c(u, v) can be approximated by the discrete density function [93][94][95], which motivates another development of the copula with the entropy theory [18,[30][31][32]. Suppose the probability P(i, j) = pij, 0 ≤ i, j ≤ n, is the discrete copula probability partitioned within the interval [0 1] × [0 1] on the point (xi, yi).…”

Coimplications in the set of discrete fuzzy numbers

“…For this reason, many authors have studied in last years operations defined on a finite chain L n , usually called discrete operations. For instance, tnorms and t-conorms were characterized in [20], uninorms and nullnorms in [14], idempotent uninorms in [9], a non-commutative version of nullnorms in [10], weighted means in [13], smooth aggregation functions in [17], copulas in [19] and also implications functions in [15] and [16]. It is proved in [20] that only the number of elements of the finite chain L n is relevant when we deal with monotonic operations on L n , and so the finite chain used in many of the mentioned works is the most simple one L n = {0, 1, .…”

Sklar's Theorem in Finite Settings

“…T HIS paper should be viewed as a second part of the one entitled "Copula-like operations on finite setting" [6] where the authors dealt with a class of binary operations defined on a finite chain . A discrete copula on is a 2-increasing function with 0 and as annihilator and neutral elements, respectively.…”

“…On the other hand, note that copulas can be viewed as special aggregation operators and several papers have appeared in this sense (see [2], [4]- [6], and [8]). Paper [6] from the same authors is mainly devoted to discrete copulas in a finite setting from this point of view, proving for instance that there is a close relation between discrete copulas and discrete triangular norms.…”

“…Paper [6] from the same authors is mainly devoted to discrete copulas in a finite setting from this point of view, proving for instance that there is a close relation between discrete copulas and discrete triangular norms. One of the results proved in [6] is that the class of associative discrete copulas coincides with the class of divisible (or smooth) discrete t-norms (see [3] and [7] for the characterization of this class of operators). Now, we are interested in studying copulas from the statistical point of view and, in particular, in the statement and proof of Sklar's theorem [11] in a finite setting.…”