Non-stationary Spatial Modeling Using Fractional SPDEs

TL;DR

This paper introduces a flexible non-stationary Gaussian random field model using fractional SPDEs, allowing spatially varying anisotropy and smoothness, with efficient estimation and demonstrated applications in ocean salinity and precipitation prediction.

Contribution

It develops a novel SPDE-based framework for modeling non-stationary GRFs with spatially varying parameters, including priors and efficient gradient-based estimation methods.

Findings

Reliable estimation requires at least 500 observations.

Penalization effectively prevents overfitting.

Application-dependent benefits of non-stationarity and fractional smoothness.

Abstract

We construct a Gaussian random field (GRF) that combines fractional smoothness with spatially varying anisotropy. The GRF is defined through a stochastic partial differential equation (SPDE), where the range, marginal variance, and anisotropy vary spatially according to a spectral parametrization of the SPDE coefficients. Priors are constructed to reduce overfitting in this flexible covariance model, and parameter estimation is done with an efficient gradient-based optimization approach that combines automatic differentiation with sparse matrix operations. In a simulation study, we investigate how many observations are required to reliably estimate fractional smoothness and non-stationarity, and find that one realization containing 500 observations or more is needed in the scenario considered. We also find that the proposed penalization prevents overfitting across varying numbers of observation locations. Two case studies demonstrate that the relative importance of fractional smoothness and non-stationarity is application dependent. Non-stationarity improves predictions in an application to ocean salinity, whereas fractional smoothness improves predictions in an application to precipitation. Predictive ability is assessed using mean squared error and the continuous ranked probability score. In addition to prediction, the proposed approach can be used as a tool to explore the presence of fractional smoothness and non-stationarity.