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%{\large\bf Abstract}\\
%In this thesis, we introduce ranking functions as  tools for comparing two fuzzy numbers.Then we discuss about some common ranking function w%
%hich are used for solving fuzzy linear programmin problems. Finally, introduce a general class of fuzzy linear programming called fully fuzzified linaer programming is introduced
%and solve it using ranking function.\\                                                                                                                                                                                                                            
                                                                                                                                                                                                        
                                                                                                                                                                                                                 
 
                                                                                                                                                                                                                                                                                                                                                                                                                                                        
%{\large\bf Keywords:} 
   %IBN Rings، Type Of A Ring، Leavitt Algebras، Leavitt Path Algebras، K_{\circ} , \mathbb{Z}-Grading  
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{\bf Abstract:}\\
Often in practice the data on the mortality of a living unit with a correlation of location of the observations in space are studied. One of the most important issues in this type of analysis of survival data with spatial dependence is estimation of the parameters and prediction of the unknown values in known sites based on observations vector.
In this thesis to analyze this type of survival  data, Cox regression model with piecewise exponential function is used as a hazard and spatial dependence as a Gaussian random field and as a latent variable is added to the model. Because there is no closed form for the posterior distribution and the full conditional distributions, also long computing for Markov Chain Monte Carlo of algorithms, to analyze the model is used the approximate Bayesian approach. In practical an example of how to implement an approximate Bayesian approach is presented.\\
%{\large\bf Abstract}\\
%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 thesis, spatial-temporal Gaussian model is introduced in an area of dense spatial, and 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  through the Stochastic Partial Differential Equations  approach. then, Approximatin Baysian method and Integrated Nested Laplace Approximation  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. \\
\textbf{Keywords:} 
 \textit{Approximate Bayesian Analysis, Parametric Model, Cox Regression, Latent Gaussian Random Field.}  
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\Large\bf\lr{Semnan University}
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\Large
\Large\bf\lr{Faculty of Mathematics, Statistics and Computer Science}
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\Large\bf\lr{A Thesis Submitted in Partial Fulfillment of the
Requirements for the Degree of Master of Science (M.Sc.) in Statistics}
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{\Large\bf\lr{Bayesian analysis of survival data with spatial dependence }}\\
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%\Large\bf  \lr{~~~~ Compact Multipliers on some of Banach Algebras ~~~~~~~~}}\\
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{\Large\sf By:} \par\vskip 1cm
{\LARGE \lr{Ahdieh Azizi}} \par 
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{\Large\sf Supervisor:}\par\vskip 1cm
{\LARGE\bf\lr{Dr. Fatemeh Hosseini}}\par
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{\Large\sf Advisor:} \par\vskip 1cm
{\LARGE \bf\lr{ Dr. Omid Karimi}} \par 
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{\Large 2016}
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