Spatial Modelling and Inference with SPDE-based GMRFs

In recent years, stochastic partial differential equations (SPDEs) have been shown to provide a usefulway of specifying some classes of Gaussian random fields. The use of an SPDEallows for the construction of a Gaussian Markov random field (GMRF) approximation, which has verygood computational properties, of the solution.In this thesis this kind of construction is considered for a specificspatial SPDE with non-constant coefficients, a form of diffusion equation driven by Gaussian white noise. The GMRF approximation is derived from the SPDE by a finite volume method. The diffusion matrixin the SPDE provides a way of controlling the covariancestructure of the resulting GMRF.By using different diffusion matrices, itis possible to construct simple homogeneous isotropic and anisotropic fields and more interesting inhomogeneous fields. Moreover, it is possible to introduce random parametersin the coefficients of the SPDE and consider the parametersto be part of a hierarchical model. In this way onecan devise a Bayesian inference scheme for theestimation of the parameters. In this thesis twodifferent parametrizations of the diffusion matrixand corresponding parameter estimations are considered.The results show that the use of an SPDE with non-constant coefficients provides a useful way of creating inhomogeneousspatial GMRFs