In this thesis, the problem of probability density function estimation using finite mixture models is considered. Gaussian mixture modelling is used to provide a semi-parametric density estimate for a given data set. The fundamental problem with this approach is that the number of mixtures required to adequately describe the data is not known in advance. In this work, a predictive validation technique [91] is studied and developed as a useful, operational tool that automatically selects the number of components for Gaussian mixture models. The predictive validation test approves a candidate model if, for the set of events they try to predict, the predicted frequencies derived from the model match the empirical ones derived from the data set. A model selection algorithm, based on the validation test, is developed which prevents both problems of over-fitting and under-fitting. We investigate the influence of the various parameters in the model selection method in order to develop it into a robust operational tool. The capability of the proposed method in real world applications is examined on the problem of face image segmentation for automatic initialisation of lip tracking systems. A segmentation approach is proposed which is based on Gaussian mixture modelling of the pixels RGB values using the predictive validation technique. The lip region segmentation is based on the estimated model. First a grouping of the model components is performed using a novel approach. The resulting groups are then the basis of a Bayesian decision making system which labels the pixels in the mouth area as lip or non-lip. The experimental results demonstrate the superiority of the method over the conventional clustering approaches. In order to improve the method computationally an image sampling technique is applied which is based on Sobol sequences. Also, the image modelling process is strengthened by incorporating spatial contextual information using two different methods, a Neigh-bourhood Expectation Maximisation technique and a spatial clustering method based on a Gibbs/Markov random field modelling approach. Both methods are developed within the proposed modelling framework. The results obtained on the lip segmentation application suggest that spatial context is beneficial.

The theory of abelian and non-abelian L-functions is developed with a view to providing an understanding of the Langlands-Deligne local root number and local Galois Gauss sum. The relationship between the Galois Gauss sum of a character of a group and the Galois Gauss sum of the restriction of that character to a subgroup is examined. In particular a generalization of a theorem of Hasse-Davenport (1934) to the global, non-abelian case is seen to result from the relation between Galois Gauss sums and the adelic resolvents of Fröhlich. / Science, Faculty of / Mathematics, Department of / Unknown

Gaussian graphical model has been a popular tool to investigate conditional dependency between random variables by estimating sparse precision matrices. Two problems have been discussed. One is to learn multiple Gaussian graphical models at multilevel from unknown classes. Another one is to select Gaussian process in semiparametric multi-kernel machine regression. The first problem is approached by Gaussian graphical model. In this project, I consider learning multiple connected graphs among multilevel variables from unknown classes. I esti- mate the classes of the observations from the mixture distributions by evaluating the Bayes factor and learn the network structures by fitting a novel neighborhood selection algorithm. This approach is able to identify the class membership and to reveal network structures for multilevel variables simultaneously. Unlike most existing methods that solve this problem by frequentist approaches, I assess an alternative to a novel hierarchical Bayesian approach to incorporate prior knowledge. The second problem focuses on the analysis of correlated high-dimensional data which has been useful in many applications. In this work, I consider a problem of detecting signals with a semiparametric regression model which can study the effects of fixed covariates (e.g. clinical variables) and sets of elements (e.g. pathways of genes). I model the unknown high-dimension functions of multi-sets via multi-Gaussian kernel machines to consider the possibility that elements within the same set interact with each other. Hence, my variable selection can be considered as Gaussian process selection. I develop my Gaussian process selection under the Bayesian variable selection framework. / Doctor of Philosophy / A network can be represented by nodes and edges between nodes. Under the assumption of multivariate Gaussian distribution, a graphical model is called a Gaussian graphical model, where edges are undirected. Gaussian graphical model has been studied for years to understand conditional dependency structure between random variables. Two problems have been discussed. In the first project, I consider learning multiple connected graphs among multilevel variables from unknown classes. I estimate the classes of the observations from the mixture distributions. This approach is able to identify the class membership and to reveal network structures for multilevel variables simultaneously. Unlike most existing methods that solve this problem by frequentist approaches, I assess an alternative to a novel hierarchical Bayesian approach to incorporate prior knowledge. The second problem focuses on the analysis of correlated high-dimensional data which has been useful in many applications. In this work, I consider a problem of detecting signals with a semiparametric regression model which can study the effects of fixed covariates (e.g. clinical variables) and sets of elements (e.g. pathways of genes). I model the unknown high-dimension functions of multi-sets via multi-Gaussian kernel machines to consider the possibility that elements within the same set interact with each other. Hence, my variable selection can be considered as Gaussian process selection. I develop my Gaussian process selection under the Bayesian variable selection framework

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / In this thesis, we study distribution functions and distributional-related quantities for various stochastic processes and probability distributions, including Gaussian processes, inverse Levy subordinators, Poisson stochastic integrals, non-negative infinitely divisible distributions and the Rosenblatt distribution. We obtain analytical results for each case, and in instances where no closed form exists for the distribution, we provide numerical solutions. We mainly use two methods to analyze such distributions. In some cases, we characterize distribution functions by viewing them as solutions to differential equations. These are used to obtain moments and distributions functions of the underlying random variables. In other cases, we obtain results using inversion of Laplace or Fourier transforms. These methods include the Post-Widder inversion formula for Laplace transforms, and Edgeworth approximations. In Chapter 1, we consider differential equations related to Gaussian processes. It is well known that the heat equation together with appropriate initial conditions characterize the marginal distribution of Brownian motion. We generalize this connection to finite dimensional distributions of arbitrary Gaussian processes. In Chapter 2, we study the inverses of Levy subordinators. These processes are non-Markovian and their finite-dimensional distributions are not known in closed form. We derive a differential equation related to these processes and use it to find an expression for joint moments. We compute numerically these joint moments in Chapter 3 and include several examples. Chapter 4 considers Poisson stochastic integrals. We show that the distribution function of these random variables satisfies a Kolmogorov-Feller equation, and we describe the regularity of solutions and numerically solve this equation. Chapter 5 presents a technique for computing the density function or distribution function of any non-negative infinitely divisible distribution based on the Post-Widder method. In Chapter 6, we consider a distribution given by an infinite sum of weighted gamma distributions. We derive the Levy-Khintchine representation and show when the tail of this sum is asymptotically normal. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. Finally, in Chapter 7 we look at the Rosenblatt distribution, which can be expressed as a infinite sum of weighted chi-squared distributions. We apply the expansions in Chapter 6 to compute its distribution function. / 2999-01-01

Latent Gaussian models (LGMs) are perhaps the most commonly used class of statistical models with broad applications in various fields, including biostatistics, econometrics, and spatial modeling. LGMs assume that a set of unobserved or latent variables follow a Gaussian distribution, commonly used to model spatial and temporal dependence in the data. The availability of computational tools, such as R-INLA, that permit fast and accurate estimation of LGMs has made their use widespread. Nevertheless, it is easy to find datasets that contain inherently non-Gaussian features, such as sudden jumps or spikes, that adversely affect the inferences and predictions made from an LGM. These datasets require more general latent non-Gaussian models (LnGMs) that can automatically handle these non-Gaussian features by assuming more flexible and robust non-Gaussian distributions on the latent variables. However, fast implementation and easy-to-use software are lacking, which prevents LnGMs from becoming widely applicable. This dissertation aims to tackle these challenges and provide ready-to-use implementations for the R-INLA package. We view scientific learning as an iterative process involving model criticism followed by model improvement and robustification. Thus, the first step is to provide a framework that allows researchers to criticize and check the adequacy of an LGM without fitting the more expensive LnGM. We employ concepts from Bayesian sensitivity analysis to check the influence of the latent Gaussian assumption on the statistical answers and Bayesian predictive checking to check if the fitted LGM can predict important features in the data. In many applications, this procedure will suffice to justify using an LGM. For cases where this check fails, we provide fast and scalable implementations of LnGMs based on variational Bayes and Laplace approximations. The approximation leads to an LGM that downweights extreme events in the latent variables, reducing their impact and leading to more robust inferences. Each step, the first of LGM criticism and the second of LGM robustification, can be executed in R-INLA, requiring only the addition of a few lines of code. This results in a robust workflow that applied researchers can readily use.

International Telemetering Conference Proceedings / October 29-November 02, 1990 / Riviera Hotel and Convention Center, Las Vegas, Nevada / An innovative digital approach to analog noise synthesis is described. This method can be used to test bit synchronizers and other communications equipment over a wide range of data rates. A generator has been built which has a constant RMS output voltage and a well-defined, closely Gaussian amplitude distribution. Its frequency spectrum is flat within 0.3 dB from dc to an upper limit which can be varied from 1 Hz to over 100 MHz. Both simulation and practical measurement have confirmed that this generator can verify the performance of bit synchronizers with respect to the standard error rate curve.

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