| In recent years, the advent of global positioning systems, satellite imagery, etc., has eased the ability to obtain geo-referenced data. Consequently many scientists are eager to develop more realistic spatial models for a variety of applications. The research presented in this dissertation opens by illustrating the importance of incorporating spatial dependence into the model selection process by making comparisons to traditional model selection techniques. The dissertation begins with a derivation of the Akaike Information Criteria (AIC) statistic for geostatistical models. The AIC is a commonly used order selection procedure in statistics. In this dissertation, the form and use of the AIC is considered in the spatial context. The derivation of the spatial AIC statistic relies critically on asymptotic normality of parameter estimates. This motivates the remainder of the dissertation which sets out to show that the maximum likelihood estimators (MLE) of the correlation parameters are asymptotically normally distributed regardless of sampling design. We begin by considering the Ornstein-Uhlenbeck process, the continuous analogue of the discrete first order autoregressive Gaussian process [AR(1)] in one-dimension. This process corresponds to exponentially correlated data in the spatial lexicon and is characterized by a range parameter which describes the strength of the correlation between two locations as a function of the distance between them. We develop the asymptotic distribution for the MLE of the range parameter for randomly spaced observations along a one-dimensional transect. We also provide simulation results that corroborate the theoretical results and we identify the optimal sampling design. Next we examine the behavior of the MLEs for the correlation parameters for the Matern class of correlation functions in both one- and two-dimensions through simulation. The Matern function is characterized by two parameters: a range parameter that dictates the "sphere of correlation" and a smoothness parameter which describes the behavior of the function at close proximities. We adopt the mean integrated squared error (MISE) as one measure to compare the performance of the estimates of several sampling designs. |
