Multivariate spatial modelling in a Bayesian setting

The statistical modelling of spatial data plays an important role in the geological and environmental sciences. Multivariate spatial modelling techniques have recently surfaced as important inferential tools of spatial data analysis. This dissertation studies multivariate spatial processes as sources of spatial data and suggests an unifying approach to tackle several questions of spatial data analysis such as prediction at arbitrary sites, regression of spatial variables and modelling of spatial interaction data. ^ The dissertation first looks at the continuity properties of a spatial process. For inferential analysis of spatial data, a spatial stochastic process is often adopted. The specification of the process has implications for the smoothness of process realizations. The ideas of almost sure continuity, mean square continuity and mean square differentiability are clarified and extended. These notions are helpful to the spatial modeller seeking greater insight into these smoothness issues. ^ The thesis next focuses upon the modelling of point referenced data that are used for capturing spatial association and for spatial prediction, typically in the presence of explanatory variables that are possibly misaligned with the response. Three inference questions are formalized. The first question, which we call interpolation, seeks to infer about a missing response at an observed explanatory location. The second, referred to as prediction, infers about a response at a location with the explanatory variable unobserved. The last, regression, investigates the functional relationship between the response and the explanatory variable. Departures from the Gaussian framework are considered. ^ The dissertation also focuses on multivariate spatial processes under a regression structure. In this context a spatial regression model is fit to investigate the relationship between the selling price of real estates and the time they remain in the market before they sell. This helps to clarify an intriguing and highly misunderstood relationship. Multivariate spatial processes also help in analysing spatial interaction data. Here the responses are related to pairs of sites. These responses need not follow a Gaussian model. A hierarchical model is used to capture spatial interaction at the second stage using a bivariate Gaussian process, giving rise to very flexible classes of models.