Spatial-temporal autologistic models are useful models for binary data that are measured repeatedly over time on a spatial lattice. They can account for effects of potential covariates and spatial-temporal statistical dependence among the data. However, the traditional parametrization of spatial-temporal autologistic model presents difficulties in interpreting model parameters across varying levels of statistical dependence, where its non-negative autocovariates could bias the realizations toward 1. In order to achieve interpretable parameters, a centered spatial-temporal autologistic regression model has been developed. Two efficient statistical inference approaches, expectation-maximization pseudo-likelihood approach (EMPL) and Monte Carlo expectation-maximization likelihood approach (MCEML), have been proposed. Also, Bayesian inference is considered and studied. Moreover, the performance and efficiency of these three inference approaches across various sizes of sampling lattices and numbers of sampling time points through both simulation study and a real data example have been studied. In addition, We consider the imputation of missing values is for spatial-temporal autologistic regression models. Most existing imputation methods are not admissible to impute spatial-temporal missing values, because they can disrupt the inherent structure of the data and lead to a serious bias during the inference or computing efficient issue. Two imputation methods, iteration-KNN imputation and maximum entropy imputation, are proposed, both of them are relatively simple and can yield reasonable results. In summary, the main contributions of this dissertation are the development of a spatial-temporal autologistic regression model with centered parameterization, and proposal of EMPL, MCEML, and Bayesian inference to obtain the estimations of model parameters. Also, iteration-KNN and maximum entropy imputation methods have been presented for spatial-temporal missing data, which generate reliable imputed values with the reasonable efficient imputation time.

In many areas of the agriculture, biological, physical and social sciences, spatial lattice data are becoming increasingly common. In addition, a large amount of lattice data shows not only visible spatial pattern but also temporal pattern (see, Zhu et al. 2005). An interesting problem is to develop a model to systematically model the relationship between the response variable and possible explanatory variable, while accounting for space and time effect simultaneously. Spatial-temporal linear model and the corresponding likelihood-based statistical inference are important tools for the analysis of spatial-temporal lattice data. We propose a general asymptotic framework for spatial-temporal linear models and investigate the property of maximum likelihood estimates under such framework. Mild regularity conditions on the spatial-temporal weight matrices will be put in order to derive the asymptotic properties (consistency and asymptotic normality) of maximum likelihood estimates. A simulation study is conducted to examine the finite-sample properties of the maximum likelihood estimates. For spatial data, aside from traditional likelihood-based method, a variety of literature has discussed Bayesian approach to estimate the correlation (auto-covariance function) among spatial data, especially Zheng et al. (2010) proposed a nonparametric Bayesian approach to estimate a spectral density. We will also discuss nonparametric Bayesian approach in analyzing spatial data. We will propose a general procedure for constructing a multivariate Feller prior and establish its theoretical property as a nonparametric prior. A blocked Gibbs sampling algorithm is also proposed for computation since the posterior distribution is analytically manageable.