Representation of discrete quasi-copulas through non-square matrices

Authors:
Isabel Aguiló;Jaume Suòer;Joan Torrens
Affiliations:
Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca. Spain;Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca. Spain;Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca. Spain
Venue:
Proceedings of the 2008 conference on Artificial Intelligence Research and Development: Proceedings of the 11th International Conference of the Catalan Association for Artificial Intelligence
Year:
2008

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Abstract

In this paper a matrix representation for discrete quasi-copulas defined on a non-square grid In× Im of [0,1]2 with m=kn, k≥1 is given. The special case of irreducible discrete quasi-copulas (those that cannot be expressed as nontrivial convex combination of other ones) is also studied. It is proved that quasi-copulas of minimal range admit a matrix representation through a special class of matrices. In the case of copulas, it is proved that irreducible copulas coincide with those of minimal range. Moreover, an algorithm to express any discrete copula as a convex combination of irreducible ones is given.