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Markov chains are memoryless stochastic processes that undergoes transitions from one state to another state on a state space having the property that, given the present,the future is conditionally independent of the past. Under general conditions, the markov chain has a stationary distribution and the probability distribution of the markov chain, independent of the staring state, converges to it’s stationary distribution.
We use this fact to construct markov chain monte carlo, which are a class of algorithms for sampling from probability distributions based on constructing a markov chain that has the desired distribution as its stationary distribution. The state of a chain after a large... 

The chapter 1 begins the discussion of a theory of large-scale simultaneous hypothesis testing now under development in the statistics literature. Furthermore,this chapter introduces the False Discovery Rate (FDR) and Empirical Bayes approach. In chapter 2, the frequentist viewpoints to the simultaneous hypothesis testing is mentioned. apter 3 describes the break through paper of the Benjamini and Hochberg published in 1995. Chapter 4 provides new criteria for error and represents an outstanding method of controlling FDR by J.D. Storey. The first part of chapter 5 discusses a paper related to control of FDR for variable selection in linear model setting by E.Candes and R. Barber. In the rest... 

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this dissertation we study the spherical ensemble and its local repelling property by... 

Nowadays, machine learning is used to solve various problems, and different machine learning models have increasingly become a part of our daily lives. This raises the importance of a more detailed examination of the models' outputs. In sensitive areas such as medical uses, this issue becomes even more critical, and using these models without having a framework to quantify the uncertainty of the outputs can lead to significant challenges and hinder their broader application. Conformal prediction is one of the frameworks that can help quantify the uncertainty of outputs. This framework is model-agnostic, meaning that we can use it for any model, including black-box models. The outputs of this... 

Application of optimal control in cancer modeling is studied through both linear and nonlinear modeling of the dynamics in ordinary differential equations. At the outset, a fairly straight-forward analysis of a linear model in presented. Through comparably simple machinery, this seminal work published at early 2000s covers some of most important techniques previously developed. The model here is infinite- dimensional, taking different number of gene amplifications into account. Thereafter by surveying recently published papers, the literature is reviewed and different lines of progress is followed, culminating in detailed study of a specific approach which is theoretically of interest.... 

The problem of learning using high volume of data as stream, has attracted much attention recently. In this thesis, the problem is modeled and analized using Online Convex Optimization tools [1], [2]. General performance bounds are stated and clarified in this framework [8]. Using the practical experience in Online Decision Making (e.g., predicting price in Stock Market), the need for a more flexible model, which adapts to changes in problem, is presented. In this thesis, after reviewing the literature and online convex optimization framework, we will define ”Concept Drift”, which describes changes in the dynamics of the problem and the statistical tools to detect it [13], [5]. And finally,... 

Strongly Rayleigh measures are an important class of negatively dependent (repulsive) probability measures. These measures are defined via a geometric condition, called “real stability”, on their generating polynomials, and have interesting probabilistic properties. On one important property of negatively dependent measures is their rigid dependence structure. In other words, the it is impossible for these measures to have strong overall dependencies. In this thesis, we study two manifestations if this phenomenon: (1) paving property and (2) tail triviality. Informally, the paving property states that it is possible to partition the set of the components of every strongly Rayleigh random... 

Nowadays, because of high volume and growth of data in industrial organizations and productive factories, registration and storing of data have forgotten manual and tradition styles for which using automation and mechanized machinery and systems has been a necessary task. In order to reach to this revolution, need to some tools, facilities and methods which can fulfill this requirement is felt strongly. Therefore, high volume of data is considered as an advantage because based on precise analysis it is possible to make logical management decisions with less risk. During last years, statistical and numerical methods and simulation were used to discover knowledge and information when one of... 

Random Polytopes, the first occurrence of which dates back to the famous Sylvester’s four points problem in the 1860s, is a branch of geometric probability, typically concerning the convex hull of some random points chosen from a convex subset of Rd. In this thesis we have studied some special kind of random polytopes; the one that is the convex hull of some independent random points chosen from a convex body (a convex, compact set with interior point) according to the uniform distribution. It was a new approach from A. Rényi and R. Sulanke in 1963 to consider this type when the number of random points tends to infinity.This thesis consists of three main parts: The first part is devoted to... 

“How can we generate a random point with uniform distribution over a convex body ?” According to it’s applications, it’s important for a solution to this problem to be applicable in high dimensions. Here, we are interested in algorithms with polynomial order with respect to the dimension. All existing methods for dealing with this problem are based on the Markov chain Monte Carlo method, i.e. a random walk is constructed in such that its stationary distribution is the uniform distribution over. Then, after simulating “enough” steps of this random walk, the distribution of the resulting point is “approximately” uniform. The real problem in Monte Carlo method is analyzing its “mixing time”,... 

The purpose of this thesis is to introduce and review a recent methods in simultaneous hypothesis testing. False discovery rates, Benjamini and Hochberg’s FDR Control Algorithm, is the great success story of the new methodology. Much of what follows is an attempt to explain that success in empirical Bayes terms.The later chapters are at pains to show the limitations of current largescale statistical practice: Which cases should be combined in a single analysis? How do we account for notions of relevance between cases? What is the correct null hypothesis? How do we handle correlations? Some helpful theory is provided in answer, but much of the argumentation is by example, with graphs and... 

Determinantal processes are a special family of stochastic processes that arise in physics (fermions), random matrices (eigenvalues), and in combinatorics (random spanning trees and non-intersecting paths). These processes have repelling property (points close to each other are chosen with low probability). Because of this repelling property, determinantal processes are approporiat for modeling some physical quantities (e.g. the position of electrons). Their probabilistic structure is described by operators on complex vector spaces and their eigenvalues. Determinantal processes have interesting properties, e.g. number of points in a region is a sum of independent Bernoulli random variables.... 

With Cramer’s model we have a probability measure on the power set of N. This probability measure is concentrated on the set that its elements are that subsets of N which number of their elements up to a certain natural number is asymptotically equal with the number of primes up to the same number. Let Pc be a sample obtained from this probability measure and consider 8n 2 N, an counts the number of ways that ncan be represented as a multiplication of some elements of Pc, such that changing the arrangement of factors in a representation does not introduce a new one. In this thesis, we prove that limn!1 a1++an n almost surely exists and is positive  

Semi-analytical solutions for the nonlinear, one dimensional wave equation have been investigated. The aim of this procedure is to deliver fast and yet accurate approaches for solving the abovementioned equasions. These solutions make a good alternative for full-numerical methods, which are usually time consuming and combersome. Therefore the proposed methods may find complete priority, considering design goals. Lack of a fast numerical method for solving the nonlinear, steady state cases, make the proposed approaches relevant, dealing theses problems. By employing the presented methods, It is possible to effieciently simulate the behavior of the space-wave packet, incident on the nonlinear... 

In this thesis, while a comprehensive study of different methods for the characterization of the optical fibers is done, a unique and effective method is being introduced for the characterization of the dispersion coefficient of Highly Nonlinear Fibers (HNLFs). The proposed method is based on the Brillouin Optical Time Domain Analysis (BOTDA) of a wave generated by the Four Wave Mixing (FWM) interaction. The current method, which includes an experimental scheme and an algorithm for solving the inverse problem, offers high sensitivity and experimental accuracy at the longitudinal resolution of 1 meter. The noise level has been considerably reduced by understanding different sources of the... 

Birkhoff's ergodic theorem in dynamical systems and ergodic theory, and the strong law of large numbers in probability theory are among the fundamental theorems of the two fields, which are closely related. Thus Birkhoff's ergodic theorem directly yields the strong law of large numbers. Attempts were then made to express some limit theorems in probability theory in the form of dynamic systems, such as the central limit theorem, which was expressed in the form of dynamic systems, and even generalizations of It was also obtained. In this paper, we will investigate the above and similar connections between probability limit theorems and well-known theorems in ergodic theory  

Travel time estimation is a central issue in the urban transportation industry and is the basis of many analyses and services in businesses related to this area. In the past few years, various statistical approaches have been devised to solve this problem. The purpose of this dissertation is to review existing methods by focusing on segment-based approaches for urban travel time estimation. A big challenge is the small amount of data in hand compared to the size of the urban network. Exploring historical data and extracting correlation between urban network segments leads to modeling the urban traffic condition and travel time estimation in one specific time interval of the day