Generalized Spatio-Temporal Regression with PDE Penalization

We develop a novel generalised linear model for the analysis of data distributed over space and time. The model involves a nonparametric term 5, a smooth function over space and time. The estimation is carried out by the minimization of an appropriate penalized negative log-likelihood functional, with a roughness penalty on 5 that involves space and time differential operators, in a separable fashion, or an evolution partial differential equation. The model can include covariate information in a semi-parametric setting. The functional is discretized by means of finite elements in space, and B-splines or finite differences in time. Thanks to the use of finite elements, the proposed method is able to efficiently model data sampled over irregularly shaped spatial domains, with complicated boundaries. To illustrate the proposed model we present an application to study the criminality in the city of Portland, from 2015 to 2020.