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

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Gaussian random elds (GRFs) play an important part in spatial mod- elling, but can be computationally infeasible for general covariance structures. An ecient approach is to specify GRFs via stochastic partial dierential equations (SPDEs) and derive Gaussian Markov random eld (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 coecients in the SPDE to vary with position. This is done by using a form of diusion equation driven by Gaussian white noise with a spatially varying diu- sion matrix. This allows for the introduction of parameters that control the GRF by parametrizing the diusion 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 coecients 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.