Elisa Perrone (TU/e)

 Geometric structure in dependence models and applications.

Abstract: The growing availability of data makes it challenging yet crucial to model complex dependence traits. For example, hydrological and financial data typically display tail dependences, non-exchangeability, or stochastic monotonicity. Copula functions serve as mathematical tools for capturing these complex traits and constructing accurate dependence models that resemble the underlying distributions of data. This talk explores the geometric properties of copulas to address dependence modeling challenges in applications. In particular, we study the class of discrete copulas, i.e., restrictions of copulas on uniform grid domains, which admits representations as convex polytopes. We give a geometric characterization of discrete copulas with desirable stochastic constraints in terms of the properties of their associated convex polytopes. In doing so, we draw connections to the Birkhoff polytopes, thereby unifying and extending results from both the statistics and the discrete geometry literature. Then, we further consolidate the statistics/discrete geometry bridge by showing the significance of our geometric findings to construct entropy-copula models useful in hydrology. The talk is based on joint work with Caroline Uhler (MIT, USA) and Liam Solus (KTH, Sweden).