Geometry of Discrete Copulas

TL;DR

This paper explores the geometric structure of discrete copulas, revealing their representation as polytopes and connecting them to well-known polytopes like the Birkhoff polytope, enhancing understanding for applications in multivariate modeling.

Contribution

It introduces a geometric representation of ultramodular and convex discrete copulas as polytopes, generalizing key results and linking to established polytopes in discrete geometry.

Findings

Discrete copulas can be represented as convex polytopes.

Connections established between copula polytopes and Birkhoff and alternating sign matrix polytopes.

Generalizations of known polytope results to the context of discrete copulas.

Abstract

Multivariate distributions are fundamental to modeling. Discrete copulas can be used to construct diverse multivariate joint distributions over random variables from estimated univariate marginals. The space of discrete copulas admits a representation as a convex polytope which can be exploited in entropy-copula methods relevant to hydrology and climatology. To allow for an extensive use of such methods in a wide range of applied fields, it is important to have a geometric representation of discrete copulas with desirable stochastic properties. In this paper, we show that the families of ultramodular discrete copulas and their generalization to convex discrete quasi-copulas admit representations as polytopes. We draw connections to the prominent Birkhoff polytope, alternating sign matrix polytope, and their most extensive generalizations in the discrete geometry literature. In doing so, we generalize some well-known results on these polytopes from both the statistics literature and the discrete geometry literature.