The main theme of this dissertation is the study of polytopes arising from binary multi-way contingency tables and characteristic imsets for Bayesian networks. Firstly, we study on three-way tables whose entries are independent Bernoulli ran- dom variables with canonical parameters under no three-way interaction generalized linear models. Here, we use the sequential importance sampling (SIS) method with the conditional Poisson (CP) distribution to sample binary three-way tables with the sufficient statistics, i.e., all two-way marginal sums, fixed. Compared with Monte Carlo Markov Chain (MCMC) approach with a Markov basis (MB), SIS procedure has the advantage that it does not require expensive or prohibitive pre-computations. Note that this problem can also be considered as estimating the number of lattice points inside the polytope defined by the zero-one and two-way marginal constraints. The theorems in Chapter 2 give the parameters for the CP distribution on each column when it is sampled. In this chapter, we also present the algorithms, the simulation results, and the results for Samson’s monks data. Bayesian networks, a part of the family of probabilistic graphical models, are widely applied in many areas and much work has been done in model selections for Bayesian networks. The second part of this dissertation investigates the problem of finding the optimal graph by using characteristic imsets, where characteristic imsets are defined as 0-1 vector representations of Bayesian networks which are unique up to Markov equivalence. Characteristic imset polytopes are defined as the convex hull of all characteristic imsets we consider. It was proven that the problem of finding optimal Bayesian network for a specific dataset can be converted to a linear programming problem over the characteristic imset polytope [51]. In Chapter 3, we first consider characteristic imset polytopes for all diagnosis models and show that these polytopes are direct product of simplices. Then we give the combinatorial description of all edges and all facets of these polytopes. At the end of this chapter, we generalize these results to the characteristic imset polytopes for all Bayesian networks with a fixed underlying ordering of nodes. Chapter 4 includes discussion and future work on these two topics.

The majority of aircraft structures are designed to be damage-tolerant such that safe operation can continue in the presence of minor damage. It is necessary to schedule inspections so that minor damage can be found and repaired. It is generally not possible to perform structural inspections prior to every flight. The scheduling is traditionally accomplished through a deterministic set of methods referred to as Damage Tolerance Analysis (DTA). DTA has proven to produce safe aircraft but does not provide estimates of the probability of failure of future flights or the probability of repair of future inspections. Without these estimates maintenance costs cannot be accurately predicted. Also, estimation of failure probabilities is now a regulatory requirement for some aircraft. The set of methods concerned with the probabilistic formulation of this problem are collectively referred to as Probabilistic Damage Tolerance Analysis (PDTA). The goal of PDTA is to control the failure probability while holding maintenance costs to a reasonable level. This work focuses specifically on PDTA for fatigue cracking of metallic aircraft structures. The growth of a crack (or cracks) must be modeled using all available data and engineering knowledge. The length of a crack can be assessed only indirectly through evidence such as non-destructive inspection results, failures or lack of failures, and the observed severity of usage of the structure. The current set of industry PDTA tools are lacking in several ways: they may in some cases yield poor estimates of failure probabilities, they cannot realistically represent the variety of possible failure and maintenance scenarios, and they do not allow for model updates which incorporate observed evidence. A PDTA modeling methodology must be flexible enough to estimate accurately the failure and repair probabilities under a variety of maintenance scenarios, and be capable of incorporating observed evidence as it becomes available. This dissertation describes and develops new PDTA methodologies that directly address the deficiencies of the currently used tools. The new methods are implemented as a free, publicly licensed and open source R software package that can be downloaded from the Comprehensive R Archive Network. The tools consist of two main components. First, an explicit (and expensive) Monte Carlo approach is presented which simulates the life of an aircraft structural component flight-by-flight. This straightforward MC routine can be used to provide defensible estimates of the failure probabilities for future flights and repair probabilities for future inspections under a variety of failure and maintenance scenarios. This routine is intended to provide baseline estimates against which to compare the results of other, more efficient approaches. Second, an original approach is described which models the fatigue process and future scheduled inspections as a hidden Markov model. This model is solved using a particle-based approximation and the sequential importance sampling algorithm, which provides an efficient solution to the PDTA problem. Sequential importance sampling is an extension of importance sampling to a Markov process, allowing for efficient Bayesian updating of model parameters. This model updating capability, the benefit of which is demonstrated, is lacking in other PDTA approaches. The results of this approach are shown to agree with the results of the explicit Monte Carlo routine for a number of PDTA problems. Extensions to the typical PDTA problem, which cannot be solved using currently available tools, are presented and solved in this work. These extensions include incorporating observed evidence (such as non-destructive inspection results), more realistic treatment of possible future repairs, and the modeling of failure involving more than one crack (the so-called continuing damage problem). The described hidden Markov model / sequential importance sampling approach to PDTA has the potential to improve aerospace structural safety and reduce maintenance costs by providing a more accurate assessment of the risk of failure and the likelihood of repairs throughout the life of an aircraft. / Statistics

Mark-recapture methods use repeated captures of individually identifiable animals to provide estimates of properties of populations. Different models allow estimates to be obtained for population size and rates of processes governing population dynamics. State-space models consist of two linked processes evolving simultaneously over time. The state process models the evolution of the true, but unknown, states of the population. The observation process relates observations on the population to these true states. Mark-recapture models specified within a state-space framework allow population dynamics models to be embedded in inference ensuring that estimated changes in the population are consistent with assumptions regarding the biology of the modelled population. This overcomes a limitation of current mark-recapture methods. Two alternative approaches are considered. The "conditional" approach conditions on known numbers of animals possessing capture history patterns including capture in the current time period. An animal's capture history determines its state; consequently, capture parameters appear in the state process rather than the observation process. There is no observation error in the model. Uncertainty occurs only through the numbers of animals not captured in the current time period. An "unconditional" approach is considered in which the capture histories are regarded as observations. Consequently, capture histories do not influence an animal's state and capture probability parameters appear in the observation process. Capture histories are considered a random realization of the stochastic observation process. This is more consistent with traditional mark-recapture methods. Development and implementation of particle filtering techniques for fitting these models under each approach are discussed. Simulation studies show reasonable performance for the unconditional approach and highlight problems with the conditional approach. Strengths and limitations of each approach are outlined, with reference to Soay sheep data analysis, and suggestions are presented for future analyses.

Neste trabalho definimos o processo de exclusão simples simétrico em tempo discreto sobre grafos por meio de permutações com restrições sobre os índices dos vértices dos grafos. O processo é uma generalização das permutações dos índices do grafo completo. Apresentamos algoritmos de Monte Carlo e de amostragem sequencial por importância para amostrar permutações com restrições inspirados pelo problema análogo de calcular permanentes. Como aplicação, utilizamos esses algoritmos para estimar os tempos de relaxação do processo de exclusão simples simétrico em tempo discreto sobre grafos aleatórios densos de Erdös-Rényi com laços / In this work we define the symmetric simple exclusion process in discrete time over graphs by means of suitably restricted permutations over the labels of the vertices of the graphs. The process is a generalization of the shuffling of labels on the complete graph. Straightforward Monte Carlo and sequential importance sampling algorithms to sample restricted permutations inspired by the related problem of computing permanents are discussed. We illustrate the formalism by estimating the relaxation times of the symmetric simple exclusion process in discrete time over dense loop-augmented Erdös-Rényi random graphs

Neste trabalho definimos o processo de exclusão simples simétrico em tempo discreto sobre grafos por meio de permutações com restrições sobre os índices dos vértices dos grafos. O processo é uma generalização das permutações dos índices do grafo completo. Apresentamos algoritmos de Monte Carlo e de amostragem sequencial por importância para amostrar permutações com restrições inspirados pelo problema análogo de calcular permanentes. Como aplicação, utilizamos esses algoritmos para estimar os tempos de relaxação do processo de exclusão simples simétrico em tempo discreto sobre grafos aleatórios densos de Erdös-Rényi com laços / In this work we define the symmetric simple exclusion process in discrete time over graphs by means of suitably restricted permutations over the labels of the vertices of the graphs. The process is a generalization of the shuffling of labels on the complete graph. Straightforward Monte Carlo and sequential importance sampling algorithms to sample restricted permutations inspired by the related problem of computing permanents are discussed. We illustrate the formalism by estimating the relaxation times of the symmetric simple exclusion process in discrete time over dense loop-augmented Erdös-Rényi random graphs

This thesis is divided into two parts. In the first part we describe a new Monte Carlo algorithm for the consistent and unbiased estimation of multidimensional integrals and the efficient sampling from multidimensional densities. The algorithm is inspired by the classical splitting method and can be applied to general static simulation models. We provide examples from rare-event probability estimation, counting, optimization, and sampling, demonstrating that the proposed method can outperform existing Markov chain sampling methods in terms of convergence speed and accuracy. In the second part we present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.

Development of dynamic state estimation techniques and their applications in problems of identification in structural engineering have been taken up. The thrust of the study has been the identification of structural systems that exhibit nonlinear behavior, mainly in the form of constitutive and geometric nonlinearities. Methods encompassing both linearization based strategies and those involving nonlinear filtering have been explored. The applications of derivative-free locally transversal linearization (LTL) and multi-step transversal linearization (MTrL) schemes for developing newer forms of the extended Kalman filter (EKF) algorithm have been explored. Apart from the inherent advantages of these methods in avoiding gradient calculations, the study also demonstrates their superior numerical accuracy and considerably less sensitivity to the choice of step sizes. The range of numerical illustrations covers SDOF as well as MDOF oscillators with time-invariant parameters and those with discontinuous temporal variations. A new form of the sequential importance sampling (SIS) filter is developed which explores the scope of the existing SIS filters to cover nonlinear measurement equations and more general forms of noise involving multiplicative and (or) Gaussian/ non-Gaussian noises. The formulation of this method involves Ito-Taylor’s expansions of the nonlinear functions in the measurement equation and the development of the ideal ispdf while accounting for the non-Gaussian terms appearing in the governing equation. Numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear Euler–Bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms, i.e. the unscented particle filter. The study demonstrates the applicability of diverse range of mathematical tools including Magnus’ functional expansions, theory of SDE-s, Ito-Taylor’s expansions and simulation and characterization of the non-Gaussian random variables to the problem of nonlinear structural system identification.