Surfaces, or spatial fields, may be known imperfectly or incompletely. In the presence of such uncertainty, a hierarchical spatial statistical approach to making inference on these surfaces, or important summaries of them, offers a coherent approach. It separates the uncertainties into components, namely those represented in a data model, those arising from a scientifically motivated process model, and those due to a lack of perfect knowledge of the parameters of the models. Inference is Bayesian (either fully Bayesian or empirical Bayesian), and hence it is based on the posterior distribution of the unknowns given the data. A core methodology in this project is the use of spatial random effects (SRE) models to represent the spatial dependence in the process model. These may simply provide a flexible class of random surfaces or they may include effects from physical laws of the phenomenon under study. The methodological research lies at the interface of two important properties: statistical optimality (obtained from summaries of the posterior distribution)and computational speed.
The problem addressed in this project, namely spatial prediction (or mapping) of surfaces, is fundamental to large areas of science and engineering. It is recognized that Geographic Information Systems (GISs) are powerful tools that have superior database and visualization tools for mapping. An important function of a GIS is to convert georeferenced data into spatially coherent images that convey a lot of information visually. The proposed research adds uncertainty (in data, models, parameters) into spatial modeling and shows how maps can be made optimally. Variability measures associated with the optimal maps allow two maps to be compared and statistically meaningful changes to be detected.
