Abstract
In this thesis, we develop two approaches for approximating nonstationary spatial covariance function and predicting missing data. The first approach is proposed for handling spatial data with multiple replications and the second is for spatial data with single replication. The first method represents a spatial process as a linear combination of some local basis functions with uncorrelated random coefficients plus some independent stationary processes. The covariance function estimation problem is formulated as a regression problem and a constrained least squared method proposed by Tibshirani (1996) is applied for selecting appropriate basis functions and stationary processes, and estimating parameters. The best linear unbiased prediction is then used for prediction. The second approach is constructed on the wavelet domain. It can be applied to lattice data with single measurement. In wavelet domain, the log variances of the transformed data are assumed to follow the conditional autoregressive model (Besag, 1974). Bayesian inference is implemented for estimation and prediction. Since missing data are allowed in this approach, it can be adopted to irregularly spaced data by mapping irregular data on a finer grid. Both approaches are computationally efficient for handling large data sets since the covariance matrix of either method is diagonal or nearly diagonal under a certain transformation. Simulation experiments show that the proposed methods approximate both stationary and nonstationary dependence structures very well. Both methods also perform well in prediction for real applications.