A characterization of quasi-copulas
Journal of Multivariate Analysis
Quasi-copulas and copulas on a discrete scale
Soft Computing - A Fusion of Foundations, Methodologies and Applications
On a family of copulas constructed from the diagonal section
Soft Computing - A Fusion of Foundations, Methodologies and Applications
An Introduction to Copulas (Springer Series in Statistics)
An Introduction to Copulas (Springer Series in Statistics)
Matrix representation of discrete quasi-copulas
Fuzzy Sets and Systems
Representation of discrete quasi-copulas through non-square matrices
Proceedings of the 2008 conference on Artificial Intelligence Research and Development: Proceedings of the 11th International Conference of the Catalan Association for Artificial Intelligence
Solution of an open problem for associative copulas
Fuzzy Sets and Systems
Copula-Like Operations on Finite Settings
IEEE Transactions on Fuzzy Systems
IEEE Transactions on Fuzzy Systems
Sklar's Theorem in Finite Settings
IEEE Transactions on Fuzzy Systems
Matrix representation of meet-irreducible discrete copulas
Fuzzy Sets and Systems
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In this paper a matrix representation for discrete copulas and quasi-copulas defined on a non-square grid I"nxI"m of [0,1]^2 is given. Two special cases are studied, irreducible discrete copulas and quasi-copulas (those that cannot be expressed as non-trivial convex combination of other ones) and those of minimal range. This study is divided into two cases depending on whether n divides m or not. In the first case, it is proved that copulas and quasi-copulas with minimal range admit a representation through a special kind of matrices, that they are always irreducible and that this implication becomes an equivalence in the case of copulas. Moreover, an algorithm to express any discrete copula as a convex combination of irreducible ones is given. However, the case when n does not divide m is quite more complex and it is proved that the previous results do not longer hold in this case. Similar results are given only for a subclass of copulas and quasi-copulas, those whose associated matrices are given by blocks.