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

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. Specifically, using a form of diffusion equation driven by Gaussian white noise with a diffusion matrix that varies with position. 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 useful way of creating non-stationary spatial GMRFs that allows for physical interpretability of the parameters.