Geometry 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.

Extreme Points of Polytopes of Discrete Copulas

“…We now present the definition of discrete copulas on non-uniform grid domains, originally introduced in [18], and the associated convex polytopes.…”

“…A similar correspondence holds between the as the alternating transportation polytope [11,26]. The definition of discrete (quasi-) copulas is further generalized in [18] to the discrete (quasi-) copulas on arbitrary grid domains. There, the authors also show a connection between discrete (quasi-) copulas and other classes of (alternating) transportation polytopes.…”

“…Another work shows an application to copula selection in portfolio optimization [13]. In [18], the authors use tools from discrete geometry to define the polytopal representations of subfamilies of discrete copulas with desirable stochastic properties. Such representations can be helpful to design a hypothesis test for a negative dependence property [7].…”

“…On the mathematical side, discrete copulas (and discrete quasi-copulas) have been extensively studied in terms of their matrix representations [1,2,9,12,14,15,16,21]. In [18], the authors highlight that the space of discrete copulas with fixed marginal distributions of finite support of size p and q, respectively, corresponds to a polytope known as the generalized Birkhoff polytope [6]. A similar correspondence holds between the as the alternating transportation polytope [11,26].…”

Expected value of the one-dimensional earth mover’s distance

“…For results concerning the geometry of J µν see [3] and [14] where they are referred to as transportation polytopes and discrete copulas respectively. The earth mover's distance is defined as ‫,‪(µ‬ބލޅ‬ ν) = inf…”