Mapping of mortality rates has been a valuable public health tool. We describe novel Bayesian methods for constructing maps which do not depend on a post stratification of the estimated rates. We also construct posterior modal maps rather than posterior mean maps. Our methods are illustrated using mortality data from chronic obstructive pulmonary diseases (COPD) in the continental United States. Poisson regression models have attracted much attention in the scientific community for their superiority in modeling rare events (including mortality counts from COPD). Christiansen and Morris (JASA 1997) described a hierarchical Bayesian model for heterogeneous Poisson counts under the exchangeability assumption. We extend this model to include latent classes (groups of similar Poisson rates unknown to an investigator). Also, it is standard practice to construct maps using quantiles (e.g., quintiles) of the estimated mortality rates. For example, based on quintiles, the mortality rates are cut into 5 equal size groups, each containing $20\%$ of the data, and a different color is applied to each of them on the map. A potential problem is that, this method assumes an equal number of data in each group, but this is often not the case. The latent class model produces a method to construct maps without using quantiles, providing a more natural representation of the colors. Typically, for rare events, the posterior densities of the rates are skewed, making the posterior mean map inappropriate and inaccurate. Thus, although it is standard practice to present the posterior mean maps, we also develop a method to provide the joint posterior modal map (i.e., the map with the highest posterior probability over the ensemble). For the COPD data, collected 1988-1992 over 798 health service areas, we use Markov chain Monte Carlo methods to fit the model, and an output analysis is used to construct the new maps.

This thesis contributes to the field of Gaussian Graphical Models by exploring either numerically or theoretically various topics of Bayesian Methods in Gaussian Graphical Models and by providing a number of interesting results, the further exploration of which would be promising, pointing to numerous future research directions. Gaussian Graphical Models are statistical methods for the investigation and representation of interdependencies between components of continuous random vectors. This thesis aims to investigate some issues related to the application of Bayesian methods for Gaussian Graphical Models. We adopt the popular $G$-Wishart conjugate prior $W_G(\delta,D)$ for the precision matrix. We propose an efficient sampling method for the $G$-Wishart distribution based on the Metropolis Hastings algorithm and show its validity through a number of numerical experiments. We show that this method can be easily used to estimate the Deviance Information Criterion, providing a computationally inexpensive approach for model selection. In addition, we look at the marginal likelihood of a graphical model given a set of data. This is proportional to the ratio of the posterior over the prior normalizing constant. We explore methods for the estimation of this ratio, focusing primarily on applying the Monte Carlo simulation method of path sampling. We also explore numerically the effect of the completion of the incomplete matrix $D^{\mathcal{V}}$, hyperparameter of the $G$-Wishart distribution, for the estimation of the normalizing constant. We also derive a series of exact and approximate expressions for the Bayes Factor between two graphs that differ by one edge. A new theoretical result regarding the limit of the normalizing constant multiplied by the hyperparameter $\delta$ is given and its implications to the validity of an improper prior and of the subsequent Bayes Factor are discussed.

This thesis contributes to the field of Gaussian Graphical Models by exploring either numerically or theoretically various topics of Bayesian Methods in Gaussian Graphical Models and by providing a number of interesting results, the further exploration of which would be promising, pointing to numerous future research directions. Gaussian Graphical Models are statistical methods for the investigation and representation of interdependencies between components of continuous random vectors. This thesis aims to investigate some issues related to the application of Bayesian methods for Gaussian Graphical Models. We adopt the popular $G$-Wishart conjugate prior $W_G(\delta,D)$ for the precision matrix. We propose an efficient sampling method for the $G$-Wishart distribution based on the Metropolis Hastings algorithm and show its validity through a number of numerical experiments. We show that this method can be easily used to estimate the Deviance Information Criterion, providing a computationally inexpensive approach for model selection. In addition, we look at the marginal likelihood of a graphical model given a set of data. This is proportional to the ratio of the posterior over the prior normalizing constant. We explore methods for the estimation of this ratio, focusing primarily on applying the Monte Carlo simulation method of path sampling. We also explore numerically the effect of the completion of the incomplete matrix $D^{\mathcal{V}}$, hyperparameter of the $G$-Wishart distribution, for the estimation of the normalizing constant. We also derive a series of exact and approximate expressions for the Bayes Factor between two graphs that differ by one edge. A new theoretical result regarding the limit of the normalizing constant multiplied by the hyperparameter $\delta$ is given and its implications to the validity of an improper prior and of the subsequent Bayes Factor are discussed.

In order to personalize or tailor treatments to maximize impact among different subgroups, there is need to model not only the main effects of intervention but also the variation in intervention impact by baseline individual level risk characteristics. To this end a suitable statistical model will allow researchers to answer a major research question: who benefits or is harmed by this intervention program? Commonly in social and psychological research, the baseline risk may be unobservable and have to be estimated from observed indicators that are measured with errors; also it may have nonlinear relationship with the outcome. Most of the existing nonlinear structural equation models (SEM’s) developed to address such problems employ polynomial or fully parametric nonlinear functions to define the structural equations. These methods are limited because they require functional forms to be specified beforehand and even if the models include higher order polynomials there may be problems when the focus of interest relates to the function over its whole domain. To develop a more flexible statistical modeling technique for assessing complex relationships between a proximal/distal outcome and 1) baseline characteristics measured with errors, and 2) baseline-treatment interaction; such that the shapes of these relationships are data driven and there is no need for the shapes to be determined a priori. In the ALV model structure the nonlinear components of the regression equations are represented as generalized additive model (GAM), or generalized additive mixed-effects model (GAMM). Replication study results show that the ALV model estimates of underlying relationships in the data are sufficiently close to the true pattern. The ALV modeling technique allows researchers to assess how an intervention affects individuals differently as a function of baseline risk that is itself measured with error, and uncover complex relationships in the data that might otherwise be missed. Although the ALV approach is computationally intensive, it relieves its users from the need to decide functional forms before the model is run. It can be extended to examine complex nonlinearity between growth factors and distal outcomes in a longitudinal study.

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Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2018-03-21T12:37:29Z No. of bitstreams: 1 MarconioSilvaDosSantos_TESE.pdf: 17070388 bytes, checksum: 46702d837c8c304ffc379088625742aa (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2018-03-23T15:26:03Z (GMT) No. of bitstreams: 1 MarconioSilvaDosSantos_TESE.pdf: 17070388 bytes, checksum: 46702d837c8c304ffc379088625742aa (MD5) / Made available in DSpace on 2018-03-23T15:26:03Z (GMT). No. of bitstreams: 1 MarconioSilvaDosSantos_TESE.pdf: 17070388 bytes, checksum: 46702d837c8c304ffc379088625742aa (MD5) Previous issue date: 2017-11-24 / Neste trabalho, ? feita uma modelagem estat?stica da precipita??o pluvial. Este ? um trabalho metodol?gico que utiliza simula??es estoc?sticas para estimar as distribui??es de probabilidades envolvidas na modelagem dessa vari?vel atmosf?rica. A fim de estimar os par?metros dessas distribui??es, foram utilizados m?todos de Monte Carlo via cadeias de Markov para gerar amostras sint?ticas de tamanho grande a partir de dados observados. Os m?todos utilizados foram o algoritmo de Metropolis-Hastings e o amostrador de Gibbs. As simula??es foram feitas sob a hip?tese de que os dias de um mesmo per?odo do ano (m?s ou esta??o chuvosa) podem ser considerados como identicamente distribu?dos em rela??o ? probabilidade de ocorrer precipita??o. Essa pesquisa possibilitou a produ??o de quatro artigos. O primeiro artigo utilizou o algoritmo de Metropolis-Hastings para modelar a probabilidade de ocorr?ncia de precipita??o em um dia qualquer do m?s. As simula??es desse artigo foram feitas com dados observados de algumas cidades brasileiras. Os demais artigos utilizaram o amostrador de Gibbs e os m?todos propostos foram aplicados em cidades da regi?o Nordeste do Brasil. No segundo artigo, as distribui??es Beta e Binomial foram utilizadas para modelar o n?mero de dias do m?s com ocorr?ncia de precipita??o. No terceiro artigo, a distribui??o de Poisson foi utilizada para modelar o n?mero de dias com valores extremos de precipita??o na esta??o chuvosa. Um m?todo alternativo para estimar esses valores extremos e sua distribui??o ? apresentado no quarto artigo, utilizando a distribui??o Gama. De acordo com os resultados dessas pesquisas, amostrador de Gibbs foi considerado adequado para estimar as distribui??es na modelagem da precipita??o em cidades para as quais h? poucos dados hist?ricos. / In this work, it was made a statistical modeling of precipitation. This is a methodological work that uses stochastic simulations to estimate the probability distributions related to this atmospheric variable. In order to estimate the parameters of these distributions, Markov chain Monte Carlo methods were used to generate large size synthetic samples from observed data. The used methods were the Metropolis-Hastings algorithm and the Gibbs sampler. The simulations were performed under the hypothesis that the days of of the same period of the year (month or rainy season) can be considered to be identically distributed concernig the probability of precipitation. This research allowed the production of four papers. The first paper used the Metropolis-Hastings algorithm to model the probability of occurrence of precipitation on any day of the month. The simulations of this paper were perfomed with observed data of some Brazilian cities. The other papers used the Gibbs sampler and the proposed methods were applied to data from cities in the Northeast Brazil. In the second paper, Beta and Binomial distributions were used to model the number of days of the month with occurrence of precipitation. In the third paper, the Poisson distribution was used to model the number of days with precipitation extreme values in the rainy season. An alternative method for estimating these extreme values and their distribution is presented in the fourth paper, using the Gamma distribution. According to the results obtained by these researches, the Gibbs sampler was considered to be adequate to estimate distributions in the modeling of precipitation on cities for which there are few historical data.

In high range-resolution (HRR) radar systems, the returns from a single target may fall in multiple adjacent range bins which individually vary in amplitude. A target following this representation is commonly referred to as an extended target and results in more information about the target. However, extracting this information from the radar returns is challenging due to several complexities. These complexities include the single dimensional nature of the radar measurements, complexities associated with the scattering of electromagnetic waves, and complex environments in which radar systems are required to operate. There are several applications of HRR radar systems which extract target information with varying levels of success. A commonly used application is that of imaging referred to as synthetic aperture radar (SAR) and inverse SAR (ISAR) imaging. These techniques combine multiple single dimension measurements in order to obtain a single two dimensional image. These techniques rely on rotational motion between the target and the radar occurring during the collection of the single dimension measurements. In the case of ISAR, the radar is stationary while motion is induced by the target. There are several difficulties associated with the unknown motion of the target when standard Doppler processing techniques are used to synthesise ISAR images. In this dissertation, a non-standard Dop-pler approach, based on Bayesian inference techniques, was considered to address the difficulties. The target and observations were modelled with a non-linear state space model. Several different Bayesian techniques were implemented to infer the hidden states of the model, which coincide with the unknown characteristics of the target. A simulation platform was designed in order to analyse the performance of the implemented techniques. The implemented techniques were capable of successfully tracking a randomly generated target in a controlled environment. The influence of varying several parameters, related to the characteristics of the target and the implemented techniques, was explored. Finally, a comparison was made between standard Doppler processing and the Bayesian methods proposed. / Dissertation (MEng)--University of Pretoria, 2013. / gm2014 / Electrical, Electronic and Computer Engineering / unrestricted

Dans ce mémoire, nous nous intéressons à une nouvelle classe de distributions instrumentales informatives dans le cadre de l'algorithme Metropolis-Hastings. Ces distributions instrumentales, dites en équilibre, sont obtenues en ajoutant de l'information à propos de la distribution cible à une distribution instrumentale non informative. Une chaîne de Markov générée par une distribution instrumentale en équilibre est réversible par rapport à la densité cible sans devoir utiliser une probabilité d'acceptation dans deux cas extrêmes: le cas local lorsque la variance instrumentale tend vers 0 et le cas global lorsqu'elle tend vers l'infini. Il est nécessaire d'approximer les distributions instrumentales en équilibre afin de pouvoir les utiliser en pratique. Nous montrons que le cas local mène au Metropolis-adjusted Langevin algorithm (MALA), tandis que le cas global mène à une légère modification du MALA. Ces résultats permettent de concevoir un nouvel algorithme généralisant le MALA grâce à l'ajout d'un nouveau paramètre. En fonction de celui-ci, l'algorithme peut utiliser l'équilibre local ou global ou encore une interpolation entre ces deux cas. Nous étudions ensuite la paramétrisation optimale de cet algorithme en fonction de la dimension de la distribution cible sous deux régimes: le régime asymptotique puis le régime en dimensions finies. Diverses simulations permettent d'illustrer les résultats théoriques obtenus. De plus, une application du nouvel algorithme à un problème de régression logistique bayésienne permet de comparer son efficacité à des algorithmes existants. Les résultats obtenus sont satisfaisants autant d'un point de vue théorique que computationnel. / In this master's thesis, we are interested in a new class of informed proposal distributions for Metropolis-Hastings algorithms. These new proposals, called balanced proposals, are obtained by adding information about the target density to an uninformed proposal distribution. A Markov chain generated by a balanced proposal is reversible with respect to the target density without the need for an acceptance probability in two extreme cases: the local case, where the proposal variance tends to zero, and the global case, where it tends to infinity. The balanced proposals need to be approximated to be used in practice. We show that the local case leads to the Metropolis-adjusted Langevin algorithm (MALA), while the global case leads to a small modification of the MALA. These results are used to create a new algorithm that generalizes the MALA by adding a new parameter. Depending on the value of this parameter, the new algorithm will use a locally balanced proposal, a globally balanced proposal, or an interpolation between these two cases. We then study the optimal choice for this parameter as a function of the dimension of the target distribution under two regimes: the asymptotic regime and a finite-dimensional regime. Simulations are presented to illustrate the theoretical results. Finally, we apply the new algorithm to a Bayesian logistic regression problem and compare its efficiency to existing algorithms. The results are satisfying on a theoretical and computational standpoint.

Large and complex software that is frequently changed leads to testing challenges. It is well established that the later a fault is detected in software development, the more it costs to fix. This thesis aims to research and develop a method of generating relevant and non-redundant test cases for a regression test suite, to catch bugs as early in the development process as possible. The research was executed at Axis Communications AB with their products and systems in mind. The approach utilizes user data to dynamically generate a Markov chain model and with a Markov chain Monte Carlo method, strengthen that model. The model generates test case proposals, detects test gaps, and identifies redundant test cases based on the user data and data from a test suite. The sampling in the Markov chain Monte Carlo method can be modified to bias the model for test coverage or relevancy. The model is generated generically and can therefore be implemented in other API-driven systems. The model was designed with scalability in mind and further implementations can be made to increase the complexity and further specialize the model for individual needs.

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