Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy

TL;DR

This paper introduces a novel class of non-stationary spatial Gaussian random fields with local anisotropy, constructed via SPDEs with spatially varying coefficients, enabling more flexible and interpretable spatial modeling.

Contribution

The paper develops a new method to create non-stationary Gaussian random fields with local anisotropy using SPDEs with variable coefficients, enhancing model flexibility and interpretability.

Findings

SPDE-based non-stationary GMRFs are promising for physical interpretability.

Parameters controlling anisotropy can be estimated in a Bayesian framework.

Challenges remain before practical application of these models.

Abstract

Gaussian random fields (GRFs) constitute an important part of spatial modelling, but can be computationally infeasible for general covariance structures. An efficient approach is to specify GRFs via stochastic partial differential equations (SPDEs) and derive Gaussian Markov random field (GMRF) approximations of the solutions. We consider the construction of a class of non-stationary GRFs with varying local anisotropy, where the local anisotropy is introduced by allowing the coefficients in the SPDE to vary with position. This is done by using a form of diffusion equation driven by Gaussian white noise with a spatially varying diffusion matrix. This allows for the introduction of parameters that control the GRF by parametrizing the diffusion matrix. These parameters and the GRF may be considered to be part of a hierarchical model and the parameters estimated in a Bayesian framework. The results show that the use of an SPDE with non-constant coefficients is a promising way of creating non-stationary spatial GMRFs that allow for physical interpretability of the parameters, although there are several remaining challenges that would need to be solved before these models can be put to general practical use.