Modeling anisotropy and non‐stationarity through physics‐informed spatial regression

Many spatially dependent phenomena that are of interest in environmental problems are characterized by strong anisotropy and non-stationarity. Moreover, the data are often observed over regions with complex conformations, such as water bodies with complicated shorelines or regions with complex orography. Furthermore, the distribution of the data locations may be strongly inhomogeneous over space. These issues may challenge popular approaches to spatial data analysis. In this work, we show how we can accurately address these issues by spatial regression with differential regularization. We model the spatial variation by a Partial Differential Equation (PDE), defined upon the considered spatial domain. This PDE may depend upon some unknown parameters that we estimate from the data through an appropriate profiling estimation approach. The PDE may encode some available problem-specific information on the considered phenomenon, and permit a rich modeling of anisotropy and non-stationarity. The performances of the proposed approach are compared to competing methods through simulation studies and real data applications. In particular, we analyze rainfall data over Switzerland, characterized by strong anisotropy, and oceanographic data in the Gulf of Mexico, characterized by non-stationarity due to the Gulf Stream.