In many applications we have to encountered with bounded dependent variables. Beta regression model can be used to deal with these kinds of response variables. In this paper we aim to study Spatially correlated responses in the unit interval. Initially we introduce Spatial beta generalized linear mixed model in which the Spatial correlation is captured through a random effect. Then the performances of the proposed model is evaluated via a simulation study, implementing Bayesian approach for parameter estimation.Finally the application of this model on two real data sets about migration rate and divorce rate in Iran are presented.

A common scientific purpose in Spatial data analysis is prediction of a random field in unmeasured sites based on measured data in some sample sites. If the random field is Gaussian with parametric mean and covariance functions, optimal predictor and its mean square error can be determined. But in some applications, the data give evidence of non-Gausian features. In this case, if a nonlinear transformation of the random field is Gaussian, the Spatial prediction is carried out. When the transformation is unknown, we assumed that it is belong to a certain parametric family of transformations. If the maximum likelihood estimators of the model parameters is determined and plugged in optimal predictor, optimality of the obtained predictor is doubt and often, we can't determine its MSE. Instead, in this paper, using the Bayesian approach, we determine the optimal predictor and its MSE. In a numerical example our method is used to deriving the Bayesian Spatial prediction of rainfall at a given site.

In this article a Spatial model is presented for extreme values with marginal generalized extreme value (GEV) distribution. The Spatial model would be able to capture the multi-scale Spatial dependencies. The small scale dependencies in this model is modeled by means of copula function and then in a hierarchical manner a random field is related to location parameters of marginal GEV distributions in order to account for large scale dependencies. Bayesian inference of presented model is accomplished by offered Markov chain Monte Carlo (MCMC) design, which consisted of Gibbs sampler, random walk Metropolis-Hastings and adaptive independence sampler algorithms. In proposed MCMC design the vector of location parameters is updated simultaneously based on devised multivariate proposal distribution. Also, we attain Bayesian Spatial prediction by approximation of the predictive distribution. Finally, the estimation of model parameters and possibilities for capturing and separation of multi-scale Spatial dependencies are investigated in a simulation example and analysis of wind speed extremes.

Gaussian random field is usually used to model Gaussian Spatial data. In practice, we may encounter non-Gaussian data that are skewed. One solution to model skew Spatial data is to use a skew random field. Recently, many skew random fields have been proposed to model this type of data, some of which have problems such as complexity, non-identifiability, and non-stationarity. In this article, a flexible class of closed skew-normal distribution is introduced to construct valid stationary random fields, and some important properties of this class such as identifiability and closedness under marginalization and conditioning are examined. The reasons for developing valid Spatial models based on these skew random fields are also explained. Additionally, the identifiability of the Spatial correlation model based on empirical variogram is investigated in a simulation study with the stationary skew random field as a competing model. Furthermore, Spatial predictions using a likelihood approach are presented on these skew random fields and a simulation study is performed to evaluate the likelihood estimation of their parameters.

Spatial prediction of a Gaussian random field in unmeasured sites based on precise observations is easily carried out. But, in practice, because of measurement errors, data contain noise. We assume that noises are independent random variable with distribution (0,t2)and they are also independent of the interest random field. If parameters of the mean, Govariance function and t2 are known, the optimal predictor and its MSE could be determined by usual methods. But, these methods are not desirable when some of the model parameters are random variables. We use the Bayesian approach to determine the optimal predictor and its MSE.

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 neighbours. 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.