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Skew spatial data often are modeled by using skew Gaussian random field. The main problem is that simulations from this random field are very time consuming for some parameter values and large dimensions. Also it is impossible in some cases and requires using of an approximation methods. One a spatial statistics branch often used to determine the natural resources such as oil and gas, is analysis of seismic data by inverse model. Bayesian Gaussian inversion model commonly is used in seismic inversion that the analytical and computational can easily be done for large dimensions. But in practice, we are encountered with the variables that are asymmetric and skewed. They are modeled using skew distributions. In Bayesian Analysis of closed skew Gaussian inversion model, there is an important problem to generate samples from closed skew normal distributions. In this paper, an efficient algorithm for the realization of the Closed Skew Normal Distribution is provided with higher dimensions. Also the Closed Skew T Distribution is offered that include heavy tails in the density function and the simulation algorithm for generating samples from the Closed Skew T Distribution is provided. Finally, the discussion and conclusions are presented.

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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.

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Prediction of spatial variability is one of the most important issues in the analysis of spatial data. So predictions are usually made by assuming that the data follow a spatial model. In General, the spatial models are the spatial autoregressive (SAR), the conditional autoregressive and the moving average models. In this paper, we estimated parameter of SAR(2,1) model by using maximum likelihood and obtained formulas for predicting in SAR models, including the prediction within the data (interpolation) and outside the data (extrapolation). Finally, we evaluate the prediction methods by using image processing data.

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In this paper parameters of spatial covariance functions have been estimated using block composite likelihood method. In this method, the block composite likelihood is constructed from the joint densities of paired spatial blocks. For this purpose, after differencing data, large data sets are splited into many smaller data sets. Then each separated blocks evaluated separately and finally combined through a simple summation. The advantage of this method is that there is no need to inverse and to find determination of high dimensional matrices. The simulation shows that the block composite likelihood estimates as well as the pair composite likelihood. Finally a real data is analysed.

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Uncertainty is an inherent characteristic of biological and geospatial data which is almost made by measurement error in the observed values of the quantity of interest. Ignoring measurement error can lead to biased estimates and inflated variances and so an inappropriate inference. In this paper, the Gaussian spatial model is fitted based on covariate measurement error. For this purpose, we adopt the Bayesian approach and utilize the Markov chain Monte Carlo algorithms and data augmentations to carry out calculations. The methodology is illustrated using simulated data.

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Geostatistical spatial count data in finite populations can be seen in many applications, such as urban management and medicine. The traditional model for analyzing these data is the spatial logit-binomial model. In the most applied situations, these data have overdispersion alongside the spatial variability. The binomial model is not the appropriate candidate to account for the overdispersion. The proper alternative is a beta-binomial model that has sufficient flexibility to account for the extra variability due to the possible overdispersion of counts. In this paper, we describe a Bayesian spatial beta-binomial for geostatistical count data by using a combination of the integrated nested Laplace approximation and the stochastic partial differential equations methods. We apply the methodology for analyzing the number of people injured/killed in car crashes in Mashhad, Iran. We further evaluate the performance of the model using a simulation study.

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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.

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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.

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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.

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An important issue in many cities is related to crime events, and spatio–temporal Bayesian approach leads to identifying crime patterns and hotspots. In Bayesian analysis of spatio–temporal crime data, there is no closed form for posterior distribution because of its non-Gaussian distribution and existence of latent variables. In this case, we face different challenges such as high dimensional parameters, extensive simulation and time-consuming computation in applying MCMC methods. In this paper, we use INLA to analyze crime data in Colombia. The advantages of this method can be the estimation of criminal events at a specific time and location and exploring unusual patterns in places.

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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 processes are widely used in data analysis, specifically image processing. In image processing, examining periodic images is one of the most critical challenges. To investigate this issue, we can use periodically correlated spatial processes. To this end, it is necessary to determine whether the images are periodic or not, and if they are, what type of period it is. In the current study, we first introduce and express the properties of periodically correlated spatial processes. Then, we present a spatial periodogram to determine the period of periodically correlated spatial processes. Finally, we will elaborate on its usage to recognize the periodicity of the images.

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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. 

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In some instances, the occurrence of an event can be influenced by its spatial location, giving rise to spatial survival data. The accurate and precise estimation of parameters in a spatial survival model poses a challenge due to the complexity of the likelihood function, highlighting the significance of employing a Bayesian approach in survival analysis. In a Bayesian spatial survival model, the spatial correlation between event times is elucidated using a geostatistical model. This article presents a simulation study to estimate the parameters of classical and spatial survival models, evaluating the performance of each model in fitting simulated survival data. Ultimately, it is demonstrated that the spatial survival model exhibits superior efficacy in analyzing blood cancer data compared to conventional models.

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The spatial generalized linear mixed models are often used, where the latent variables representing spatial correlations are modeled through a Gaussian random field to model the categorical spatial data. The violation of the Gaussian assumption affects the accuracy of predictions and parameter estimates in these models. In this paper, the spatial generalized linear mixed models are fitted and analyzed by utilizing a stationary skew Gaussian random field and employing an approximate Bayesian approach. The performance of the model and the approximate Bayesian approach is examined through a simulation example, and implementation on an actual data set is presented.

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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.

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In this article, autoregressive spatial regression and second-order moving average will be presented to model the outputs of a heavy-tailed skewed spatial random field resulting from the developed multivariate generalized Skew-Laplace distribution. The model parameters are estimated by the maximum likelihood method using the Kolbeck-Leibler divergence criterion. Also, the best spatial predictor will be provided. Then, a simulation study is conducted to validate and evaluate the performance of the proposed model. The method is applied to analyze a real data.

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Financial and economic indicators, such as housing prices, often show spatial correlation and heterogeneity. While spatial econometric models effectively address spatial dependency, they face challenges in capturing heterogeneity. Geographically weighted regression is naturally used to model this heterogeneity, but it can become too complex when data show homogeneity across subregions. In this paper, spatially homogeneous subareas are identified through spatial clustering, and Bayesian spatial econometric models are then fitted to each subregion. The integrated nested Laplace approximation method is applied to overcome the computational complexity of posterior inference and the difficulties of MCMC algorithms. The proposed methodology is assessed through a simulation study and applied to analyze housing prices in Mashhad City.