Models and Methods for Random Fields in Spatial Statistics with Computational Efficiency from Markov Properties

The focus of this work is on the development of new random field models and methods suitable for the analysis of large environmental data sets. A large part is devoted to a number of extensions to the newly proposed Stochastic Partial Differential Equation (SPDE) approach for representing Gaussian fields using Gaussian Markov Random Fields (GMRFs). The method is based on that Gaussian Matérn field can be viewed as solutions to a certain SPDE, and is useful for large spatial problems where traditional methods are too computationally intensive to use. A variation of the method using wavelet basis functions is proposed and using a simulation-based study, the wavelet approximations are compared with two of the most popular methods for efficient approximations of Gaussian fields. A new class of spatial models, including the Gaussian Matérn fields and a wide family of fields with oscillating covariance functions, is also constructed using nested SPDEs. The SPDE method is extended to this model class and it is shown that all desirable properties are preserved, such as computational efficiency, applicability to data on general smooth manifolds, and simple non-stationary extensions. Finally, the SPDE method is extended to a larger class of non-Gaussian random fields with Matérn covariance functions, including certain Laplace Moving Average (LMA) models. In particular it is shown how the SPDE formulation can be used to obtain an efficient simulation method and an accurate parameter estimation technique for a LMA model. A method for estimating spatially dependent temporal trends is also developed. The method is based on using a space-varying regression model, accounting for spatial dependency in the data, and it is used to analyze temporal trends in vegetation data from the African Sahel in order to find regions that have experienced significant changes in the vegetation cover over the studied time period. The problem of estimating such regions is investigated further in the final part of the thesis where a method for estimating excursion sets, and the related problem of finding uncertainty regions for contour curves, for latent Gaussian fields is proposed. The method is based on using a parametric family for the excursion sets in combination with Integrated Nested Laplace Approximations (INLA) and an importance sampling-based algorithm for estimating joint probabilities.