ARMA approximation of a Non-separable Spatio-Temporal Model with Fractional Smoothnesses in Space and Time

2026-04-29

AI Summary

Existing spatiotemporal modeling approaches struggle to flexibly capture arbitrary fractional smoothness under nonseparable covariance structures and lack computationally efficient frameworks. This work addresses these limitations by constructing a nonseparable model based on the diffusion-based Matérn covariance through a spatiotemporal fractional stochastic partial differential equation. By introducing a rational approximation for temporal discretization, the method yields a low-order vector autoregressive moving average (VARMA) process. It provides, for the first time, an efficient approximation of arbitrary fractional spatiotemporal smoothness, with rigorous pointwise convergence of the covariance function and explicit convergence rates. Numerical experiments demonstrate that even a low-order VARMA accurately approximates the true process, enables effective parameter recovery, and significantly improves predictive performance when temporal smoothness is correctly specified, as successfully illustrated in modeling daily mean temperatures over mainland France across three months.

Abstract

The Matérn covariance model is ubiquitous in spatial modelling, but there is no default choice for spatio-temporal modelling. In this paper, we consider the recently proposed ``diffusion-based'' extension of the spatial Matérn covariance model to a spatio-temporal non-separable covariance model that allows fractional smoothnesses in space and in time. The model is described in terms of a space-time fractional stochastic partial differential equation, but currently proposed computational approaches have strong restrictions on the possible smoothnesses in time. We propose a discretization method based on rational approximations in time to handle arbitrary smoothnesses, which leads to a vector autoregressive moving average process (VARMA). We prove that the covariance function of the approximation converges pointwise, determine explicit convergence rates as a function of spatial and temporal resolutions and the accuracy of the rational approximation, and conduct numerical verification to demonstrate small pointwise error for low orders of the VARMA process. Through a simulation study, we demonstrate that the parameters can be estimated back and that correctly specifying the temporal smoothness is especially important for forecasting. The approach is illustrated for three months of daily mean temperatures in mainland France.

Problem

• spatio-temporal modeling
• fractional smoothness
• non-separable covariance
• Matérn covariance
• stochastic partial differential equation

Innovation

• non-separable spatio-temporal model
• fractional smoothness
• rational approximation
• VARMA process
• convergence rate