In recent years, fuzzy regression has emerged as a powerful tool for modeling relationships between independent and dependent variables under uncertainty. In classical linear regression, significant variations in data can reduce the accuracy and reliability of results. Therefore, optimal design for fuzzy regression models is of great importance. In this regard, this study examines and develops the D-optimal method for fuzzy regression models to improve the accuracy, efficiency, and parameter estimation. The fuzzy regression scheme is chosen from optimal design points. The proposed method is examined in the context of models with three classical techniques, and the results show that the suggested approach can expand the applications of fuzzy regression models under complex conditions with data containing significant uncertainty.

  Clustering longitudinal data is a complex task that requires taking into account the similarity of individual trajectories despite scattered and irregular observed times. Clustering is a widely used statistical technique in various fields, such as unsupervised machine learning, data mining, pattern recognition, image analysis, and bioinformatics. This thesis reviews and studies several multivariate longitudinal data clustering algorithms as well as introduces a new clustering algorithm called ClusterMLD. This new method shows promise in identifying meaningful patterns in high-dimensional longitudinal data. The algorithms have been compared using simulation studies and real data to evaluate their performance.

Abstract In many challenges related to Bayesian inference, we face with some models that have certain complexities and it is necessary to calculate the likelihood function, which is difficult or impossible to calculate. This complexity makes it impossible to get the posterior distribution which is the basis of Bayesian inferences; so, as a solution, simulation methods can be used to estimate the model. One of the methods used in this field is the Approximate Bayesian Computation via >Key­words: Approximate­BayesianComputation, Kullback- Leibler (K-L) divergence, Bayesian inference,likelihood function, summary statistics,      

One of the topics of interest in statistics is the time of occurrence of a particular event. Therefore, a sub-field called survival analysis has been created in statistics. In general, survival analysis is a set of statistical methods for analyzing data in which the outcome variable is the time until the occurrence of a specific event. In survival analysis, time variable usually is called survival time. Because this variable determines how long a person "survived" during the follow-up period. Also, because usually in this type of analysis, the desired events are death, illness or other individual experiences, desired event usually called failure. However, failure does not necessarily have a negative meaning, for example, it can be the time until the birth of the first child after marriage (as the moment of starting the study). Many survival analyses are faced with a fundamental problem called censoring. Censoring occurs when we have partial survival time information but do not know the exact survival time.\\\\ With the expansion of science and the progress of various data analysis methods, survival data analysis methods are also progressing, and the application of this science in medical data and other fields is increasing.\\\\ One of the common statistical methods for analyzing survival data is Cox proportional hazard regression, this model does not have the optimal performance when we are faced with the problem of high dimensions, an alternative method that is introduced in this thesis is the use of support vector machine, which is one of the techniques in machine learning and it can work well with high-dimensional problems, and it also does not need to hold the usual regression assumptions that we have in classical statistics.\\\\ The original version of the support vector machine does not have the ability to deal with survival data due to the presence of censors. A naive idea is to exclude the censored samples from the study, as a large amount of information will be lost. In this thesis, by making changes on the constraints of the support vector machine optimization problem, we arrive at a version of this method that is suitable for the analysis of survival data and uses the information of censors.This version is called survival support vector machine. Finally, for a case study, we will use the survival support vector machine method to analyze it and compare it with classical methods in statistics such as Cox proportional hazards regression.  

In linear models, when the number of independent variables is large, it is common to use methods such as step-by-step, forward, backward, etc. to find optimal models among possible models. But these methods of finding the optimal model do not find the best model in the absolute sense and produce different results case by case. A model found with these methods may be the best in terms of the mean squared error or it may be the best in terms of the coefficient of determination. However, using these methods requires removing a number of independent variables from the analysis, which can be misleading or at least limiting in some applications. When the researcher's goal in determining the models is to predict new observations, the use of models obtained from these variable elimination methods can have a greater effect on losing information and reducing accuracy. Our goal in this thesis is to obtain a model based on the combination of simple models in such a way that this combination of models is the most reliable (in various meanings such as minimum MSE or maximum information) instead of using independent variable elimination approaches. and ...) to produce forecasts.  

   In this paper, the aim is to investigate the dependent competing risk model under middle censoring using some bivariate distributions such as Marshall-olkin bivariable Weibull distribution and Marshall-olkin bivariable Gompertz distribution and to compare the maximum likelihood estimation approach and Bayesian approach. To achieve this, subset of real data as well as simulations using R software are used, and the maximum likelihood estimates and Bayesian estimation are compared with each other.

  Regression in Statistics means returning to an average or a verage value, Statisticians have always examined the relation ship between Variables, One of the most common models that fit data, Are regression models. Regression analysis is a Statistical method for analyzing and modeling multivariate data. Aspecial type of regression model is the high dimensional regression model in Which the Volume of Variables independent of the sample size is greater, that is, when is p > n, in these models, because the matrix X is not, complete column rank, therefore estimating the least squares ?ˆOLS is not obtained uniquely and estimating the parameters will not be a good predictor. for this reason, in recent years, methods called penalty regression or contraction methods have been used. such as ridge, lasso, group lasso and elastic net, that in this thesis, the lasso convex function is used. lasso is defind as the L1 norm of the parameters, that ? is the vector of regression coefficients and ? is the penalizing parameter. larger value of ? exerts higher penalty on regression coefficients, resulting in the inclusion of fewer variables in the model. and conversely commonly, a sequence of ? value are generated, and then variables are detected for each value of the series. Thereafter, a value of ? is chosen by k-fold cross-validation, and corresponding set of predictors are included in the model. also for simulation results in this Thesis, InfTh and BIC have been used, and we discuss all these issues in R software. Keywords Weibull regression, Penalty methods, Shrinkage methods, Lasso, Criteria of information theory, Bayesian information criteria

In many applications, the data subject to inverstigation are in the form of location or geographical positions of some events in a specific region. In the present thesis, we are facing data related to location of corneal endothelium cells of 153 individual. Here, for proper analysis of this data, we have linked spatial point patterns to these images so that we can classify a group of very similar images into identical clusters based on clustering algorithms related to space point patterns. To investigate the cases of dissimilarity, the nearest neighbor distance function, empty space function, K-reply function, etc, have been used.

In analyzing big data, time of computations is increased, so in data mining algorithms cannot use all the data. Therefore, using sampling methods in big data set is a good solution.\\\\In statistical studies of multivariate populations, obtaining information over all variation range of variables is very important. Since it is difficult or impossible to select all data, the required information can be obtained by survey a subpopulation as a sample. In such cases, the appropriate sample can be selected by LPM2-kdtree method.\\\\Also, in big data analysis, selection bias is very important. In this thesis, in order to decrease the bias by using importance sampling a method is explained. Finally, in a numerical study on two real populations, the spatially balance of LPM2-kdtree and decreasing selection bias of the sampling design that uses importance sampling are evaluated.\\\\ \\textbf{Keywords:} {Big Data}, {Clustering}, {Data Mining}, {Inverse sampling}, {Knowledge Discovery}, {Non-probability sampling}, {Selection bias} . \\end{latin}

   One of the fundamental issues in statistics is the modeling of random phenomena. Generally, statistical models are used to represent random structures, predicting future behavior of variables, deduction and extraction of information from data. In the meantime, the copula function as a model for multivariate and dependent observations has attracted the attention of many users in recent studies. In fact, an alternative way to model the dependency structure between multivariate data without imposing any assumptions on the marginal distributions based on the structure of the copula functions has been proposed which considers defects such as linear correlation coefficient, asymmetry and sequence dependence. Copula functions are functions to create a multivariate common distribution. Or, in other words, it establishes relationship between multivariate distribution functions and one-dimensional marginal distribution functions. These functions of the marginal distributions have a continuous uniform distribution over interval (0,1). In addition, copula functions are useful in obtaining dependency on both sides of the distribution by using the tail dependence. In this thesis, the selection of copula functions is performed using maximum likelihood method and prediction method. And the appropriate copula function on the Automotive and Metal Industry Stock Exchange data for years 88-96 based on the maximum likelihood estimation method and the Akaike information criterion as well as on the prediction method has been determined and used to provide optimal forecasts on the data.

Our research is applied in terms of purpose. Because the proposed model lays solutions to improve the premium determination and generally improve the performance of insurance companies.We offer model forecasting methods to determine the premium rate,That detects data exploration and modeling. Among these methods, the accelerated gradient is a method in composite Poisson model. Since the main variables and interaction effects used in the models are, therefore, a tree accelerated gradient algorithm with the name TDboost offer visited. Also for data with a large zero accumulation The methods will be provided to make the premium forecast possible . First, we will discuss the definitions and concepts required in insurance science . So we introduce and examine the accelerated gradient tree model .In the third chapter, we implement a model for the survey of the database composite Poisson with insurance studies data.In the fourth chapter, we will analyze and compare non-parametric models using data sets, And finally, we will conclude our suggestions and conclusions.  

  ‏در آمار يكي از ابزار مهم براي تحليل داده ها در مدل هاي آماري برآورد پارامترها و انتخاب متغير مناسب است و روش هاي مختلفي براي آن وجود دارد. دو مورد از معروف ترين آن ها در حالت كلاسيك روش كمترين مربعات معمولي و روش بر‎‏آورد درستنمايي ماكسيمم است. اما در رگرسيون با بعد بالا‏، به علت وقوع مشكل بيش برآورد نمي توان از اين روش ها استفاده كرد‏، پس محقق سعي مي كند با كمك روش هاي انقباضي مانند رگرسيون جريمه دار اين مشكل را حل كند. در حالي كه آمار بيزي براي برآورد پارامترها از برآورد حالت پسين استفاده مي كند. زماني كه با رگرسيون با بعد بالا مواجه مي شود‏، تلاش مي كند كه اين مشكل را با استفاده از روش هاي   تنظيم بيزي (انقباضي بيزي) حل كند. اين روش ها تعداد متغيرهاي پيش بيني كننده و پيچيدگي مدل را كاهش مي دهند و برآورد پارامترها و انتخاب متغير ها را ساده تر مي كنند. بنابراين در اين پايان نامه اين روش ها را مورد بحث و بررسي قرار مي دهيم

گاهي اوقات ممكن است نمونه اي كه مشاهده مي كنيم نمونه اي اريب از جامعه باشد. به اين معنا كه تمام اعضا از شانس برابري براي انتخاب شدن در نمونه برخوردار نيستند. براي حل اين مشكل از نسخه اريب-طول كه نسخه ي وزني شده از متغير تصادفي اصلي جامعه است، استفاده مي شود. در نمونه گيري اريب-طول شانس حذف شدن هر واحد از نمونه نااريب، متناسب با طول عمر آن واحد مي باشد. حال آن كه در برخي حالات ممكن است شانس حذف شدن متناسب با طول عمر واحد تحت مطالعه نباشد، از اين رو براي حل اين مشكل مي توان از توزيع هاي وزني استفاده كرد. در اين پايان نامه توزيع اصلي جامعه را گسسته در نظر گرفتيم و سپس با استفاده از تابع ميانگين مانده عمر، توزيع پواسون بريده شده وزني شده و توزيع پواسون آماسيده در صفر بريده شده وزني شده را معرفي كرديم و ويژگي هاي آن ها را مورد بررسي قرار داديم، همچنين نشان داديم مشاهداتي كه ميانگين مانده عمر بزرگ تري دارند شانس بيشتري براي انتخاب شدن در نمونه را دارند. توزيع هاي وزني شده داراي كاربردهاي فراواني در مبحث تحليل بقا و قابليت اعتماد مي باشند، به همين دليل علاوه بر روش شبيه سازي، برخي موارد كاربرد آن با استفاده از داده هاي واقعي نيز تشريح شده است. 

   Variable selection is one of the most important topics in statistical modeling which is widely used in statistical applications. In this thesis, penalized regression models are used for selection important variables and in order to accelerate the estimation procedure of regression coefficients from partial likelihood of recurrent event data, the coordinate descent algorithm is applied. Using real longitudinal data from 230 patients with schizophrenia admitted to Farabi Hospital in Kermanshah from 01/01/1395 to 12/12/1397, each experienced more than one recurrence was able to select important variables. We have differentiated from the large number of covariate variables included in this data and finally fitted the models. Keywords: Longitudinal Data, penalty Regression, Partial Likelihood, Recurrent Event Data, Coordinate   Descent Algorithm

  در آمار يكي از مهم ترين ابزارها براي تجزيه و تحليل داده ها ، به دست آوردن برآورد مناسب يك تابع است كه روش هاي مختلفي براي آن به وجود آمده است. يكي از معروف ترين و ساده ترين روش هاي برآورد، روش كمترين مربعات معمولي است كه در شرايط مطلوب مزيت هاي فراواني دارد. اين روش در رگرسيون بعد بالا كاربردي ندارد و دليل اين مساله هم اين است كه در رگرسيون بعد بالا به علت زياد بودن متغير هاي پيشگو، سبب دشوار شدن تفسير مدل و كاهش دقت در برآورد مي شود. در چنين شرايطي محقق مي تواند با كاهش متغير هاي پيشگو و درواقع حذف متغير هاي كم اثر با استفاده از روش هاي خاصي كه بيان مي گردند، سبب بهتر شدن تفسير اين مدل ها شود. دراين پايان نامه ابتدا ضمن معرفي روش هايي كه به روش هاي انقباضي معروف هستند، روش هايي همانند لاسو ، لاسو گروهي، ريج، بريج و الاستيك نت مورد بررسي قرار مي گيرند. از طرف ديگر   مدل هاي با ضرايب متغير نيز از جمله مهم ترين ابزارها براي كشف الگوهاي حركتي در بسياري از علوم از جمله: سرمايه گذاري مالي، اپيدميولوژي، علوم سياسي،علوم پزشكي، اكولوژي و غيره هستند. اين مدل ها بسط طبيعي مدل هاي كلاسيك پارامتري هستند كه با تفسير پذيري خوب، محبوبيت زيادي در تجزيه و تحليل داده ها به دست آورده اند. انعطاف پذيري و تفسير پذيري بالاي اين مدل ها در دهه اخيرسبب شده كه تحولات شگرف و جالبي در روش شناسي، نظري و كاربردي در اين زمينه پديد آيد. در ادامه ضمن معرفي مدل هاي با ضرايب متغير به بررسي مهم ترين روش برآورد پارامتر در اين مدل ها كه روش استفاده از تابع هسته است پرداخته مي شود. همچنين مدل هاي با ضرايب متغير تعميم يافته را معرفي خواهيم كرد و با استفاده از روش رگرسيون بريج در مدل هاي بعد بالا، برآورد پارامتر ها در اين حالت را نيز مورد بحث و بررسي قرار خواهيم داد. در نهايت انتخاب متغير براي مدل هاي با ضريب متغير تعميم يافته بعد بالا را مورد بررسي قرار مي دهيم كه از روش انقباضي لاسو گروهي در اين بحث استفاده مي كنيم ، همچنين براي انتخاب بهترين زير مجموعه از متغير هاي موجود از روش اطلاع بيزي تعميم يافته استفاده مي كنيم و تمام اين مسائل را در نرم افزار R مورد بحث و بررسي قرار خواهيم داد.

In many environmental applications involving spacial and spatio-temporal data, covariance functions are the fundamental tools for modeling dependent data which observed over space and (or) time. Constructing nonseparable spatio-temporal variogram and covariance functions is one of the issues related to spatio-temporal geostatistics for space-time data. Limitations on the number of locations and high values of parameters motivate the need for practical and efficient method to estimate the parameters of covariance models for spatial interpolation, or kriging. In this paper, we discuss different types of spatio-temporal data, also we combine these methods with Bees algorithm (BA) to explore the spatiotemporal correlation structure of monthly ground-water data in Ilam city. Finally, we predict the level of some spacial stations using spatio-temporal kriging which the efficiency of model was assessed by Mean of Kriging Standard Deviations (MKSD). Keywords: Kriging, Spatio-temporal data, Bee Algorithm

In this thesis, we give a new distribution of a new >- F(HCF) distribution. Since the Burr distribution of the most special case of this family is theHyperbolic Cosine New Burr distribution. We also discuss simulation method , also maximumlikelihood minimum spaccing estimators of the parameters of the distribution . The new distributioncan be use effectively in the analysis of survival data .Show that the distribution of (HCB)Compared to distributions New Burr (NB), Weibull distribution (W), Log normal(LN) is flexibieto fit the data.  

  In this thesis, using variational methods and Critical Point Theory, we will investigate the existence and the multiplicity of solutions to   boundary value problems, including Neumannand Dirichlet problems, impulsive problems, fractional differential equations, vaiableexponent equations and etc. Moreover, in some of the results, the positivity and non-triviality of the solutiond will be discussed. Regarding the problems under study, we consider the nonlinear parts as continuous or Carathéodory functions. Then we build the related functionals and discuss the multiplicity of their critical points, obtaining multiple solutions which are the already obtained critical points. Finally, we illuminate the obtained results by presenting various examples.

  Regression is used to predict and express variations of a variable based on other variables. In fact, statistical regression analysis and statistical techniques are used to examine and model the relationship between variables. Linear regression is one of the most widely used statistical tools. Modern regression applications for large data sets have created new challenges. when the number of variables approaches or exceeds the number of samples, We will consider a new method in this thesis that applies to many genomic databases. This method is called the Ising Bayesian Approximation (BIA). This method is used to quickly calculate the latent probabilities for the fit of the characteristic in the linear regression L2 compensated. From a practical point of view, BIA provides an algorithm for computing effective Bayesian paths for regression compensated L2. Using this method, it is possible to calcugate the latent likelihood of fit for a collection of dataset data, as is commonly found in genomic studies. The importance of this study is to consider the relationship between the features when evaluating meaningful statistics in the large data set. When the number of properties is high, even low correlations can lead to a decrease in the characteristics of the later probabilities. In this thesis, We also show that choosing a probability threshold for examining the importance of high-demention issues is usually not logical. Instead, BIA can be used as part of a two-step process in which BIA is used to quickly eliminate inappropriate variables, that is, variables that have a lower rating in the later probability before a rigorous validation process, from the computational point for the deduction of coefficients regression to be used the computational influence of BIA and the existence of a natural threshold for the penalized parameter are used, the two-step process is appropriate

 for analyises emax model can that baysian D-optimal desition for E-max model

In statistics, an important tool for data analysis is the proper estimation of a function that there are several ways to do this. One of the most well-known methods for estimating functions is the ordinar‏y least squares method, which has many advantages in desirable conditions.However, in ‎h‎igh dimension‎‏s regression, ‎due to the presence of a large number of predictor variables in the model, the interpretation of the model is difficult, and the usual least squares can not be used in the usual way. In this case, the researcher tries to reduce the number of predictor variables. One of the methods that is effective in this regard is the use of shrinkage methods whose effect on the size of the parameters and their tendency to zero. In this thesis, we discuss several types of shrinkage methods.Keywords:‎Ordinar‏y Least Squares, Shrinkage Methods, Lasso, Ridge, Elastic net, Cross-Validation, Model Selectio  

  A special category of nonparametric inference includes types of hypothesis tests in the commu-nity under study. The basis of the methods of testing the classical nonparametric hypothesisis that the data, the hypotheses and the method of obtaining the test are crisp and Withoutimprecise . But in practice and in the real world, there are many situations that the exact andconsistency of the above are unrealistic. In these cases, the methods of testing the statisticalhypothesis in the classical state of efficiency and credibility are not necessary. The theory offuzzy sets is the perfect way to formulate and analyze issues in these fuzzy states. In generalnon-parametric tests can be carried out in the following three phases

  ‎In statistical study of spatial two-dimensional societies‎, ‎one of the purposes of which is preparing different plans‎, ‎collecting data from all area is of great importance‎. Since enumerating‎ all points a ‎wide‎ area is difficult and in some cases impossible‎, ‎then the required data should be collected only for a part of that area as the sample‎. ‎In a conventional and non-spatial population in which the location of the sample unit is not considered‎, ‎the main assessment criterion for sampling is efficiency of the estimator‎. ‎In sampling two-dimensional areas‎, ‎in addition to efficiency and estimators precision‎, ‎well-ballance of all the area is considered as well with due consideration to sampling methods in two-dimensional ‎population‎‎, ‎in this thesis‎, the relate ‎sampling methods are studied‎, also their well-spread‎, ‎and efficiencies are measured.

  In this thesis, we give a new distribution of a new class of distributions called the Hyperbolic Cosine – F (HCF) distribution.Since the Weibull distribution of the most popular and most widely used distributions in reliability and lifetime data analysis. A special case of this family is the Hyperbolic Cosine Weibull distribution . We also discuss simulation method, also maximum likelihood minimum spaccings estimators of the parameters of the distribution.And continue   we have discussed   with the second type of censorship increasingly on the distribution, to estimate the parameters. The new distribution can be used effectively in the analysis of survival data.

 In this thesis‎, ‎we the first study point process and their models and summary functions‎. ‎Then‎, ‎we discussion global envelope test for spatial processes‎. ‎Finally‎, ‎we compare power of stated test with other test for simulated and real data‎.  In this thesis‎, ‎we the first study point process and their models and summary functions‎. ‎Then‎, ‎we discussion global envelope test for spatial processes‎. ‎Finally‎, ‎we compare power of stated test with other test for simulated and real data‎.

In many from life ‎testing and reliability ‎studies, the experimenter may not always obtain complete information on failure times for all experimental units. for Example, individuals in a clinical trial may drop out of the study, may have to be terminated for lack of funds or in an industrial experiment, units may break accidentally. ‎Therefore, one has to remove some units prior to failure for saving time and cost associated with testing. Data obtained from such experiments are called censored. The most common censoring schemes are type I and type II. The Type I and Type II censoring schemes have major deficiency in that they only do not allow removal of units at points other than the terminal point of the experiment. Due to experimenter use a versatile scheme of censoring called progressive.‎This thesis has been focused on the scenario of progressive Type II censoring. A problem associated with this scheme is that the total testing duration might be unacceptably long. To address this issue, a hybrid variant of the ‎progressive ‎‎censoring scheme was proposed in which imposing a time limit T on the test. Although this hybrid ‎progressive ‎‎censoring scheme controls the total testing duration not larger than T, it is possible that the effective sample size is very small or even zero in which usual efficient statistical inference may not be feasible. To strike a balance between the ‎total ‎testing ‎time ‎and ‎the ‎efficiency ‎in ‎statistical ‎inference, ‎Ng ‎et.al ‎(2009)‎ proposed an adaptive Type II progressive censoring scheme.In this thesis influential methods for progressive Type II censoring and adaptive Type II progressive censoring under some statistical distributions are studied. We ‏‎obtain‎ both maximum likelihood estimators, approximate maximum likelihood estimators and observed information matrix for the unknown parameters. We simulated values of the estimates parameters, a comparison of the values variance and covariance of the estimators with those obtained from the corresponding observed information matrix and coverage probabilities for pivotal quantities based estimators. Various interval estimation methods for the unknown parameters such as asymptotic confidence intervals with both observed information matrix and fisher information matrix, percentile bootstrap and bootstrap-t confidence intervals are obtained. Then these methods are compared in terms of their expected lengths and coverage probabilities using simulation.

 Comparing risks play an important role in insurance statistics. One of the ways to compare risks is by using the measures of risk. In actuarial literature, considering tail variabilities of risks which are   low frequency and   high severity losses is vital. While in many cases, comparing based on different risk measures will be followed various results as well as using a risk measure in different cases. Furthermore, we cannot also propose an explicit expressions for risk measure under a special statistical distribution.   Because of these limitations, actuaries should use stochastic orders for ordering risks. Therefore, comparison of random risks with using the   functions of probability   distributions as Tail ,   top Loss,   excess-Mean functions and etc is more helpful than comparing based on some   umerical criteria associated distributions. The comparison of the Random risks with using mentioned functions which usually produce partial orders among probability distributions is called “Stochastic Orders.”In this thesis, firstly, The risk concept, some measures of risk and type of stochastic orders are introduced which allow us to compare variabilites between random variables. Then, the relationships between the excess wealth order and other familiar stochastic variability orders (Dispersive Order, Stop Loss, convex, star and mean-excess) has been studied. In the reminder of this thesis, some characterizations of variability stochastic orders are showed by using the usual and distortion risk measures.

  The allocation of redundant component(s) in a system so as to optimize the lifetime of the system is quite ‌considerable inreliability theory. In engineering systems, the k-out-of-n system is a very important structure, whichworks if and only if at least k components are operating. In this thesis, we consider the problems of optimal allocation of R redundancies to n components of   k-out-of-n systemsand series systems, as particular case of k-out-of-n systems when k=n,with respect to some stochastic orderings.Two commonly used methods to allocate redundancy are active redundancy and standby redundancy. We discuss the allocation of active redundancies to series and k-out-of-n systems in the three cases with i.i.d components and redundancies, i.i.d components and i.i.d redundancies and stochastically ordered components. We also compare the lifetimes of series systems arising out of one standby redundancy. Optimal allocation of redundancies in -out-of-   ystems

  The epidemic type aftershock sequence (ETAS) model of serves as the baseline model for describing the behavior of earthquake clusters. The ETAS model is a space-time point process that is based on the empirical laws driven from the descriptive study of earthquakes statistics. The model assumes that earthquakes are produced by two sources: background seismic activities and clusters of aftershocks that are produced with occurrence of large earthquakes. Parameters of the common ETAS model are fixed and clusters are assumed to be isotropic. Since the earthquake that occurred in past are heterogeneous and clusters of aftershocks are not isotropic, so in the present thesis we study an improved version of the ETAS model which allows the parameters to vary spatial.

  Queuing theory has many applications in various fields such as airports, hospitals, manufactories, computer and programming, telecommunications and so on. In operating, find servers that serves to customers with equal speeds is very difficult, so discussion of using from heterogeneous servers is arised. The purpose of this thesis is presentation subjects to correct management of queues with heterogeneous servers.In this thesis, how to calculate the maximum likelihood estimates for the parameters of M/M/2 queuing model with heterogeneous servers, is described. Working vacation problem in M/M/2 queuing model with heterogeneous servers, is checked. Waiting time asymptotics in M/G/2   queuing model with heterogeneous servers is earned. In following several versions of the job assignment problem for M/M/m queue with heterogeneous servers is considered and it is shown that the optimal policy that minimizes the mean waiting time has a threshold structure. It is shown that how to compute the optimal thresholds and the impact of heterogeneity in server speeds on mean waiting times is studied.Keywords:Multiserver queue, Heterogeneous server, Individual optimality, Social optimality, Working vacation, Slowly varying function, Regularly varying distribution.