Design utility methods for preferentially-sampled spatial data

Spatial preferential sampling occurs when the choice of sampling locations at which a spatial process of interest is measured is stochastically dependent on the values of this process. Ignoring such a sampling scheme when mapping the spatial process leads to inaccurate predictions, particularly at locations further from the sampling sites. This may be dealt with by jointly modelling the sampling process with the spatial process. Existing methods for this require that the sampling locations be independent of one another. In this thesis we dispense with such an unrealistic requirement and model the sampling process as a whole. This is achieved by defining a whole-design utility function over the space of possible sampling designs to assign a `usefulness' to each design based on a set of possible experimenter preferences and some (unknown) strength of preference parameters. We may then assume that the probability that any design is selected is proportional to this utility. We shall give particular attention to utility functions which balance a preference for high values with a preference for even coverage of the region of interest. This whole-design approach presents a variety of challenges which we will address, such as the definition of suitable utility functions and the selection of suitable algorithms for the fitting of such models, in particular, because the the design distribution may introduce intractable normalising constants into the likelihood function of the spatial process. We shall describe methods for replacing these with suitable approximations, explore methods for efficiently drawing samples of designs required to form these approximations, and define a class of utility functions, the properties of which allow for the application of a combinatorially-based method to improve accuracy. We shall demonstrate the effectiveness of our model via a simulation study and application to various spatial data sets. Finally, we shall consider methods of combining multiple data sets in order to get better estimates of strength of preference.