Jenise L. Swall
Associate Professor, Department of Statistical Science and Operations ResearchVirginia Commonwealth University
Nonstationary Spatial Modeling using a Process Convolution Approach
This dissertation develops a process convolution approach to modeling the covariance structure of non-stationary Gaussian processes. This methodology is useful in the analysis of many real-life processes in which the usual assumptions of stationarity and/or isotropy are not appropriate. Implementation and related computational issues are also discussed in detail. Various traditional modeling strategies are available for modeling processes that are stationary, an assumption that is often justifiable and leads to a reasonable analysis. However, some examples seem to require alternative models that can account for a more hetergeneous spatial structure. With a review of some of these procedures, we note that our method has the advantages of providing insight into the extent and nature of the non-stationarity that exists, and presenting this information graphically and clearly. We develop an hierarchical specification for covariance structure that can be used with a single realization from the spatial process and that can provide estimates about uncertainty in the spatial process. Our approach defines a spatial process z(s) as a convolution of a Gaussian white noise process and a series of convolution kernels k(s). A one-to-one mapping exists between each one of these kernels and its one standard deviation ellipse, even as the kernel is stretched and rotated. We choose the bivariate normal distribution as the form of each kernel, but other forms could be selected instead. These convolution kernels are then specified at each location by a pair of parameters denoting the location of one of the focus points of the ellipse and by another paramter that serves to shrink or expand the kernel. We use MCMC to explore the posterior distribution of the model's parameters. This provides a simple way to estimate the variability of the parameters. In conjunction with this, we provide explanation of the many computational issues that are encountered when employing such a hierarchical covariance structure. This includes some complicated parameter updating mechanisms to help speed the convergence of the parameters to their posterior distributions using MCMC, as well as the development of modified Cholesky algorithm that handles semi-positive definite matrices in a sensible way. Finally, we summarize the results, comparing the results using the process convolution approach with other, more traditional methods. Possible extensions and directions for future research serve to conclude the discussion.