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.

Statistical models describe, in a mathematical way, the uncertainty of a phenomenon influenced by different factors. My research focus is on analyzing a particular class of statistical models, which describe how various factors interact with each other. Such models are based on a mathematical tool called copulas. The goal of my Erwin-Schrödinger project was to investigate the properties of fascinating mathematical objects called discrete copulas to make them useful for applications. The project answered fundamental questions related to mathematical and statistical features of discrete copulas, thereby allowing for addressing new modeling challenges in applied fields. The project contributed to building a novel bridge between apparently unconnected research areas, namely, discrete geometry, statistics, and environmental sciences. First, we introduced and studied previously unknown geometric objects that relate to a class of discrete copulas with desirable properties useful in applications. Second, we exploited the geometric properties of the newly introduced objects to develop novel tools for statistical modeling and solve real-world problems in environmental sciences, such as modeling rainfall totals in hydrology. Third, we applied discrete copula methods to improve weather forecasting in Austria. The interdisciplinary research conducted within my FWF project is the result of collaborative work with leading researchers in discrete geometry, statistics, and weather forecasting from all around the world. Besides collaborating with my host at the Massachusetts Institute of Technology (MIT, USA), Prof. Caroline Uhler, the project allowed me to work jointly with researchers at the Zentralanstalt für Meteorologie und Geodynamik (ZAMG, Austria) to address research challenges on weather forecasting in Austria. I believe that the connections established in the frame of my FWF project will flourish in the upcoming years, resulting in numerous joint papers and project proposals aimed at exploring new promising research paths.