Space-Time Geostatistical Models with both Linear and Seasonal Structures in the Temporal Components

Abstract: We provide a novel approach to model space-time random fields where the temporal argument is decomposed into two parts. The former captures the linear argument, which is related, for instance, to the annual evolution of the field. The latter is instead a circular variable describing, for instance, monthly observations. The basic intuition behind this construction is to consider a random field defined over space (a compact set of the $d$-dimensional Euclidean space) across time, which is considered as the product space $\mathbb{R} \times \mathbb{S}^1$, with $\mathbb{S}^1$ being the unit circle. Under such framework, we derive new parametric families of covariance functions. In particular, we focus on two classes of parametric families. The former being parenthetical to the Gneiting class of covariance functions. The latter is instead obtained by proposing a new Lagrangian framework for the space-time domain considered in the manuscript. Our findings are illustrated through a real dataset of surface air temperatures. We show that the incorporation of both temporal variables can produce significant improvements in the predictive performances of the model. We also discuss the extension of this approach for fields defined spatially on a sphere, which allows to model space-time phenomena over large portions of planet Earth.