Title: Geometry of discrete copulas
Authors: Elisa Perrone - University of Massachusetts Lowell (United States) [presenting]
Liam Solus - KTH Royal Institute of Technology (Sweden)
Caroline Uhler - Massachusetts Institute of Technology (United States)
Abstract: Discrete copulas serve as a useful tool for modeling dependence among random variables. The space of discrete copulas admits a representation as a convex polytope which has been exploited in entropy-copula methods relevant to environmental sciences. We further analyze geometric features of discrete copulas with prescribed stochastic properties. In particular, we show that bivariate discrete copulas with a property known as ultramodularity have polytopal representations, thereby opening the door to applying linear optimization techniques in the identification of ultramodular copulas. We first draw connections to the Birkhoff polytope, alternating sign matrix polytope, and their more extensive generalizations in discrete geometry. Then, we identify the minimal collection of bounding affine inequalities of the polytope of ultramodular discrete copulas, and present techniques to construct subsets of its vertices. Finally, we discuss how to possibly exploit the introduced polytopal representations to develop new theory in applied fields, such as meteorology and climatology.