Space-time models have become increasingly popular in scientific studies. Gaussian random fields (GRFs) are an important tool of this field. Developing covariance structures for GRFs, parameter estimation, and prediction in big spatial datasets are central challenges in spatio-temporal modeling. My thesis consists of three main chapters dealing with these main challenges in this area. In chapter 2, we investigate the problem of equivalence of GRFs with stationary increments, and obtain sufficient conditions for equivalence in terms of the behavior of the spectral measures at infinity. Further, the main results are applied to a rich family of nonstationary space-time models with possible anisotropy behavior. In chapter 3, we propose a nonstationary parametric model, in which the underlying Gaussian random field may have different regularities in different directions, thus can be applied to model anisotropy. Using thetheory of equivalence of Gaussian measures under nonstationary assumption, strongconsistency of the tapered likelihood based estimation of the variance component under fixed domain asymptotics are derived by putting mild conditions on the spectral behavior ofthe tapering covariance function. In chapter 4, we merge two ideas of factor modeling and low-rank spatial processes approximation to develop and propose a class of hierarchical low-rank spatial factor models which offer a rich and flexible modeling option for dealing with large vector of outcomes observed at large number of locations. A Markov chain Monte Carlo algorithm is developed for estimation, and further, the full posterior distributions are recovered in a Bayesian predictive framework.
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