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Gaussian random fields (GRFs) constitute an important part of spatialmodelling, but can be computationally infeasible for general covariancestructures. An efficient approach is to specify GRFs via stochastic partialdifferential equations (SPDEs) and derive Gaussian Markov random field (GMRF)approximations of the solutions. We consider the construction of a class ofnon-stationary GRFs with varying local anisotropy, where the local anisotropyis 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 whitenoise with a spatially varying diffusion matrix. This allows for theintroduction of parameters that control the GRF by parametrizing the diffusionmatrix. These parameters and the GRF may be considered to be part of ahierarchical model and the parameters estimated in a Bayesian framework. Theresults show that the use of an SPDE with non-constant coefficients is apromising way of creating non-stationary spatial GMRFs that allow for physicalinterpretability of the parameters, although there are several remainingchallenges that would need to be solved before these models can be put togeneral practical use.

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