Hierarchical spatio-temporal models are used for modeling space-time responses and temporally and spatially correlations of the data is considered via Gaussian latent random field with Matérn covariance function. The most important interest in these models is estimation of the model parameters and the latent variables, and is predict of the response variables at new locations and times. In this paper, to analyze these models, the Bayesian approach is presented. Because of the complexity of the posterior distributions and the full conditional distributions of these models and the use of Monte Carlo samples in a Bayesian analysis, the computation time is too long. For solving this problem, Gaussian latent random field with Matern covariance function are represented as a Gaussian Markov Random Field (GMRF) through the Stochastic Partial Differential Equations (SPDE) approach. Approximatin Baysian method and Integrated Nested Laplace Approximation (INLA) are used to obtain an approximation of the posterior distributions and to inference about the model. Finally, the presented methods are applied to a case study on rainfall data observed in the weather stations of Semnan in 2013.

The prevalence of Covid-19 is greatly affected by the location of the patients. From the beginning of the pandemic, many models have been used to analyze the survival time of  Covid-19 patients. These models often use the Gaussian random field to include this effect in the survival model. But the assumption of Gaussian random effects is not realistic. In this paper, by considering a spatial skew Gaussian random field for random effects and a new spatial survival model is introduced. Then, in a simulation study, the performance of the proposed model is evaluated.  Finally, the application of the model to analyze the survival time data of Covid-19 patients in Tehran is presented.

Identifying the best prediction of unobserved observation is one of the most critical issues in spatial statistics. In this line, various methods have been proposed, that each one has advantages and limitations in application. Although the best linear predictor is obtained according to the Kriging method, this model is applied for the Gaussian random field. The uncertainty in the distribution of random fields makes researchers use a method that makes the nongaussian prediction possible. In this paper, using the Projection theorem, a non-parametric method is presented to predict a random field. Then some models are proposed for predicting the nongaussian random field using the nearest neighbors. Then, the accuracy and precision of the predictor will be examined using a simulation study. Finally, the application of the introduced models is examined in the prediction of rainfall data in Khuzestan province.

The Gaussian random field is commonly used to analyze spatial data. One of the important features of this random field is having essential properties of the normal distribution family, such as closure under linear transformations, marginalization and conditioning, which makes the marginal consistency condition of the Kolmogorov extension theorem. Similarly, the skew-Gaussian random field is used to model skewed spatial data. Although the skew-normal distribution has many of the properties of the normal distribution, in some definitions of the skew-Gaussian random field, the marginal consistency property is not satisfied. This paper introduces a stationery skew-Gaussian random field, and its marginal consistency property is investigated. Then, the spatial correlation model of this skew random field is analyzed using an empirical variogram. Also, the likelihood analysis of the introduced random field parameters is expressed with a simulation study, and at the end, a discussion and conclusion are presented.