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Abstract: In many environmental studies, the collected data are usually spatially dependent. Determination of the spatial correlation structure of the data and prediction are two important problem in statistical analysis of spatial data. To do so, often, a parametric variogram model is fitted to the empirical variogram of the data by estimating the unknown parameters of the mentioned variogram. Since there are no closed formulas for the variogram parameters estimator, they are usually computed numerically. Therefore, the precision measures of the variogram parameters estimator and spatial prediction can be calculated using bootstrap methods. Lahiri (2003) proposed the moving block bootstrap method for spatial data, in which observations are divided into several moving blocks and resampling is done from them. Since, in this method, the presence of boundary observations in the resampling blocks have less selection chance than the other observations, therefore, the estimator of the precision measures would be biased. In this paper, revising the moving block bootstrap method, the separate block bootstrap method was presented for estimating the precision measures of the variogram parameters estimator and spatial prediction. Then its usage was illustrated in an applied example.

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‎Modeling and efficient estimation of the trend function is of great importance in the estimation of variogram and prediction of spatial data. In this article, the support vector regression method is used to model the trend function. Then the data is de-trended and the estimation of variogram and prediction is done. On a real data set, the prediction results obtained from the proposed method have been compared with Spline and kriging prediction methods through cross-validation.  The criterion for choosing the appropriate method for prediction is to minimize the root mean square of the error. The prediction results for several positions with known values were left out of the data set (for some reason) and were obtained for new positions. The results show the high accuracy of prediction (for all positions and elimination positions) with the proposed method compared to kriging and spline.

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Spatial regression models are used to analyze quantitative spatial responses based on linear and non-linear relationships with explanatory variables. Usually, the spatial correlation of responses is modeled with a Gaussian random field based on a multivariate normal distribution. However, in practice, we encounter skewed responses, which are analyzed using skew-normal distributions. Closed skew-normal distribution is one of the extended families of skew-normal distributions, which has similar properties to normal distributions. This article presents a hierarchical Bayesian analysis based on a flexible subclass of closed skew-normal distributions. Given the time-consuming nature of Monte Carlo methods in hierarchical Bayes analysis, we have opted to use the variational Bayes approach to approximate the posterior distribution. This decision was made to expedite the analysis process without compromising the accuracy of our results. Then, the proposed model is implemented and analyzed based on the real earthquake data of Iran.