Geometry of Discrete Copulas for Weather Forecasting

Copula functions are largely employed in applied statistics as a flexible tool to describethe behavior of the dependence between random variables. In fact, the joint distribution function of any d-dimensional random vector can be described via a d-dimensional copula by fixing the univariate margins of the distribution. While in the continuous setting this copula is uniquely determined on the entire unit hypercube, in the discrete setting the uniqueness only holds on a proper subset. This limitation has led to the introduction of the notion of discrete copulas, which are restrictions of copula functions to uniform grids of their domain. Despite their fascinating mathematical properties and their importance for empirical modeling in the applied sciences, the potential of Multivariate Discrete Copulas (MDC) is yet to be thoroughly explored. The aim of this project is to explore and define the geometry of MDC in order to allow their application to weather forecasting problems, specifically in the context of ensemble postprocessing of weather forecasts. More precisely, our proposed research will study the geometry of MDC to develop new methodology for weather forecasting that overcomes the following two problems, namely to develop methodology that (1) can perform forecasting for ensemble systems with non- exchangeable members, and (2) can deal with repeated values, i.e., ties, in the numerical weather prediction ensemble forecasts. Our research will be based on a geometric description and a statistical understanding of the facets and vertices of the polytope of MDC. Our project will answer fundamental questions related to mathematical and statistical features of MDC, and allow applications of these functions to new weather forecasting scenarios. Addressing these questions requires innovative research among the areas of discrete geometry, statistics, and weather forecasting. This interdisciplinary approach will advance the state-of-the-art of these fields, establishing new bridges between them. Additionally, the project will directly contribute to weather forecasting in Austria by applying the developed methodology to forecasts from the Austrian regional ensemble systems. In addition, this project will directly contribute to weather forecasting in Austria by applying the developed methodology to weather forecasts from Austrian regional ensemble systems. Research on the theoretical aspects of the project will be conducted at the Massachusetts Institute of Technology (MIT, USA), under the supervision of Dr. Caroline Uhler. Research on the weather forecasting applications will be carried out at the Zentralanstalt für Meteorologie und Geodynamik (ZAMG, Austria) under the supervision of Dr. Yong Wang. Moreover, this project will benefit from international collaborators, establishing new cooperations between Austrian research institutions and various institutions abroad.

No additional funding sources recorded.