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.

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.

Image segmentation is an important task in image processing and computer vision which attract many researchers attention. There are a couple of information sets pixels in an image: statistical and structural information which refer to the feature value of pixel data and local correlation of pixel data, respectively. Markov Random field (MRF) is a tool for modeling statistical and structural information at the same time. Fuzzy Markov Random field (FMRF) is a MRF in fuzzy space which handles fuzziness and Randomness of data simultaneously. This paper propose a new method called FMRFC which is model clustering using FMRF and applying it in application of image segmentation. Due to the similarity of FMRF model structure and image neighbourhood structure, exploiting FMRF in image segmentation makes results in acceptable levels. One of the important tools is Cellular learning automata (CLA) for suitable initial labelling of FMRF. The reason for choosing this tool is the similarity of CLA to FMRF and image structure. We compared the proposed method with several approaches such as Kmeans, FCM, and MRF and results demonstratably show the good performance of our method in terms of tanimoto, mean square error and energy minimization metrics.

Although strong ground motion networks are expanding, near-source strong motion recordings are still sparse .In this article it is planned to characterize the level and variability of strong ground motion in near field of large earthquakes due to source effects. We have developed a stochastic rupture model that characterizes the variability and spatial complexity of slip as observed in past earthquakes. We model slip heterogeneity on the fault plane as a spatial Random field for 21 near source earthquakes. The data follows a von Karman autocorrelation function (ACF), for which the correlation lengths (a) increase with the source dimensions .These stochastic slip distributions are used to develop the temporal behavior of slip using physically consistent with stochastic-dynamic earthquake source models .It means that we can use this model to simulate realistic strong ground motion in order to characterize the variability of source effects in the near-field of large earthquakes. For earthquakes with large fault aspect ratios, we observe substantial differences of the correlation length in the along-strike (ax) and downdip (az) directions. Increasing correlation length with increasing magnitude can be understood using concepts of dynamic rupture propagation. The power spectrum of the slip distribution can also be well described with a  fractal distribution in which the fractal dimension D remains scale invariant, accounting for larger ‘‘asperities ’’ for large-magnitude events.Our stochastic slip model can be used to generate scenario earthquakes for near-source ground motion simulations.

In recent years, some statisticians have studied the signal detection problem by using the Random field theory. In this paper we have considered point estimation of the Gaussian scale space Random field parameters in the Bayesian approach. Since the posterior distribution for the parameters of interest does not have a closed form, we introduce the Markov Chain Monte Carlo (rv'ICrv'IC) algorithm to approximate the Bayesian estimations. vVc have also applied the proposed procedure to real flv'IRI data, collected by the Montreal Neurological Institute.

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.

2Gaussian 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.

Untrained shear strength is the main parameter in most problems concerned with short-term stability or total stress analysis states (TSA). Mechanism of soil deposit formation leads to inherent variability in soil strength and stiffness parameters.Inherent variability as the primary source of uncertainty in geotechnical problems consists of deterministic and stochastic components. In this paper, a generic deterministic trend is proposed by utilizing a good amount of well-documented in-situ test data. The new concept of transformation depth was introduced as the depth where it changes from a decreasing trend to an increasing one. Random field theory and local average subdivision (LAS) technique was employed in order to produce realization of untrained shear strength. Untrained shear strength was assumed to inherit a deterministic trend in vertical direction while preserving its stochastic behavior in horizontal direction.