Thesis (Ph. D.)--University of Wisconsin--Madison, 1973. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.A.)--University of Toronto, 2005.

Thesis (MSc(R)) - University of Glasgow, 2008. / MSc(R). thesis submitted to the Department of Mathematics, Faculty of Information and Mathematical Sciences, University of Glasgow, 2008. Includes bibliographical references.

The genome is a treasure trove of information about the history of an individual, his population, and his species. For as long as genomic data have been available, methods have been developed to retrieve this information and learn about population history. Over the last decade, large international genomic projects (e.g. the HapMap Project and the 1000 Genomes Project) have offered access to high quality data collected from thousands of individuals from a vast number of populations. Freely available to all, these databases offer the possibility to develop new methods to uncover the history of the peopling of the world by modern humans. Due to the complexity of the problem and the large amount of available data, all developed methods either simplify the model with strong assumptions or use an approximation; they also dramatically down-sample their data by either using fewer individuals or only portions of the genome. In this thesis, we present a novel method to jointly estimate the time of divergence of a pair of populations and their variable sizes, a previously unsolved problem. The method uses multiple regions of the genome with low recombination rate. For each region, we use an importance sampler to build a large number of possible genealogies, and from those we estimate the likelihood function of parameters of interest. By modelling the population sizes as piecewise constant within fixed time intervals, we aim to capture population size variation through time. We show via simulation studies that the method performs well in many situations, even when the model assumptions are not totally met. We apply the method to five populations from the 1000 Genomes Project, obtaining estimates of split times between European groups and among Europe, Africa and Asia. We also infer shared and non-shared bottlenecks in out-of- Africa groups, expansions following population separations, and the sizes of ancestral populations further back in time.

This thesis is concerned with graphical Gaussian models with equality constraints on the concentration or partial correlation matrix introduced by Højsgaard and Lauritzen (2008) as RCON and RCOR models. The models can be represented by vertex and edge coloured graphs G = (V,ε), where parameters associated with equally coloured vertices or edges are restricted to being identical. In the first part of this thesis we study the problem of estimability of a non-zero model mean μ if the covariance structure Σ is restricted to satisfy the constraints of an RCON or RCOR model but is otherwise unknown. Exploiting results in Kruskal (1968), we obtain a characterisation of suitable linear spaces Ω such that if Σ is restricted as above, the maximum likelihood estimator μ(with circumflex) and the least squares estimator μ* of μ coincide for μ ∈ Ω, thus allowing μ and Σ to be estimated independently. For the special case of Ω being specified by equality relations among the entries of μ according to a partition M of the model variables V, our characterisation translates into a necessary and sufficient regularity condition on M and (V,ε). In the second part we address model selection of RCON and RCOR models. Due to the large number of models, we study the structure of four model classes lying strictly within the sets of RCON and RCOR models, each of which is defined by desirable statistical properties corresponding to colouring regularity conditions. Two of these appear in Højsgaard and Lauritzen (2008), while the other two arise from the regularity condition ensuring equality of estimators μ(with circumflex) = μ* we find in the first part. We show each of the colouring classes to form complete lattices, which qualifies the corresponding model spaces for an Edwards-Havránek model selection procedure (Edwards and Havránek, 1987). We develop a coresponding algorithm for one of the model classes and give an algorithm for a systematic search in accordance with the Edwards-Havránek principles for a second class. Both are applied to data sets previously analysed in the literature, with very encouraging performances.

A recent application of point processes has emerged from the electronic trading of financial assets. Many securities are now traded on purely electronic exchanges where demand and supply are aggregated in limit order books. Buy and sell trades in the asset as well as quote additions and cancellations can then be interpreted as events that not only determine the shape of the order book, but also define point processes that exhibit a rich internal structure. A large class of such point processes are those driven by a diffusive intensity process. A flexible choice with favourable analytic properties is a Cox-Ingersoll-Ross (CIR) diffusion. We adopt a Bayesian perspective on the statistical inference for these doubly stochastic processes, and focus on filtering the latent intensity process. We derive analytic results for the moment generating function of its posterior distribution. This is achieved by solving a partial differential equation for a linearised version of the filtering equation. We also establish an efficient and simple numerical evaluation of the posterior mean and variance of the intensity process. This relies on extending an equivalence result between a point process with CIR-intensity and a partially observed population process. We apply these results to empirical datasets from foreign exchange trading. One objective is to assess whether a CIR-driven point process is a satisfactory model for the variations in trading activity. This is answered in the negative, as sudden bursts of activity impair the fit of any diffusive intensity model. Controlling for such spikes, we conclude with a discussion of the stochastic control of a market making strategy when the only information available are the times of buy and sell trades.

This thesis proposes a collection of analytical and computational methods for inferring an underlying social structure of a given population, observed only via timestamped occurrences of its members across a range of locations. It shows that such data streams have a modular and temporally-focused structure, neither fully ordered nor completely random, with individuals appearing in "gathering events". By exploiting such structure, the thesis proposes an appropriate mapping of those spatio-temporal data streams to a social network, based on the co-occurrences of agents across gathering events, while capturing the uncertainty over social ties via the use of probability distributions. Given the extracted graphs mentioned above, an approach is proposed for studying their community organisation. The method considers communities as explanatory variables for the observed interactions, producing overlapping partitions and node membership scores to groups. The aforementioned models are motivated by a large ongoing experiment at Wytham woods, Oxford, where a population of Parus major wild birds is tagged with RFID devices and a grid of feeding locations generates thousands of spatio-temporal records each year. The methods proposed are applied on such data set to demonstrate how they can be used to explore wild bird sociality, reveal its internal organisation across a variety of different scales and provide insights into important biological processes relating to mating pair formation.

The past five years has seen the discovery of a wealth of genetics variants associated with an incredible range of diseases and traits that have been identified in genome- wide association studies (GWAS). These GWAS have typically been performed in in- dividuals of European descent, prompting a call for such studies to be conducted over a more diverse range of populations. These include groups such as African Ameri- cans and Latinos as they are recognised as bearing a disproportionately large burden of disease in the U.S. population. The variation in ancestry among such groups must be correctly accounted for in association studies to avoid spurious hits arising due to differences in ancestry between cases and controls. Such ancestral variation is not all problematic as it may also be exploited to uncover loci associated with disease in an approach known as admixture mapping, or to estimate recombination rates in admixed individuals. Many models have been proposed to infer genetic ancestry and they differ in their accuracy, the type of data they employ, their computational efficiency, and whether or not they can handle multi-way admixture. Despite the number of existing models, there is an unfulfilled requirement for a model that performs well even when the ancestral populations are closely related, is extendible to multi-way admixture scenarios, and can handle whole- genome data while remaining computationally efficient. In this thesis we present a novel method of ancestry estimation named MULTIMIX that satisfies these criteria. The underlying model we propose uses a multivariate nor- mal to approximate the distribution of a haplotype at a window of contiguous SNPs given the ancestral origin of that part of the genome. The observed allele types and the ancestry states that we aim to infer are incorporated in to a hidden Markov model to capture the correlations in ancestry that we expect to exist between neighbouring sites. We show via simulation studies that its performance on two-way and three-way admixture is competitive with state-of-the-art methods, and apply it to several real admixed samples of the International HapMap Project and the 1000 Genomes Project.

Cette thèse concerne le développement, l'extension et l'application d'une formulation stochastique des équations de la mécanique des fluides introduite par Mémin (2014). La vitesse petite échelle, non-résolue, est modélisée au moyen d'un champ aléatoire décorrélé en temps. Cela modifie l'expression de la dérivée particulaire et donc les équations de la mécanique des fluides. Les modèles qui en découlent sont dénommés modèles sous incertitude de position. La thèse s'articulent autour de l'étude successive de modèles réduits, de versions stochastiques du transport et de l'advection à temps long d'un champ de traceur par une vitesse mal résolue. La POD est une méthode de réduction de dimension, pour EDP, rendue possible par l'utilisation d'observations. L'EDP régissant l'évolution de la vitesse du fluide est remplacée par un nombre fini d'EDOs couplées. Grâce à la modélisation sous incertitude de position et à de nouveaux estimateurs statistiques, nous avons dérivé et simulé des versions réduites, déterministe et aléatoire, de l'équation de Navier-Stokes. Après avoir obtenu des versions aléatoires de plusieurs modèles océaniques, nous avons montré numériquement que ces modèles permettaient de mieux prendre en compte les petites échelles des écoulements, tout en donnant accès à des estimés de bonne qualité des erreurs du modèle. Ils permettent par ailleurs de mieux rendre compte des évènements extrêmes, des bifurcations ainsi que des phénomènes physiques réalistes absents de certains modèles déterministes équivalents. Nous avons expliqué, démontré et quantifié mathématiquement l'apparition de petites échelles de traceur, lors de l'advection par une vitesse mal résolu. Cette quantification permet de fixer proprement des paramètres de la méthode d'advection Lagrangienne, de mieux le comprendre le phénomène de mélange et d'aider au paramétrage des simulations grande échelle en mécanique des fluides. / This thesis develops, analyzes and demonstrates several valuable applications of randomized fluid dynamics models referred to as under location uncertainty. The velocity is decomposed between large-scale components and random time-uncorrelated small-scale components. This assumption leads to a modification of the material derivative and hence of every fluid dynamics models. Through the thesis, the mixing induced by deterministic low-resolution flows is also investigated. We first applied that decomposition to reduced order models (ROM). The fluid velocity is expressed on a finite-dimensional basis and its evolution law is projected onto each of these modes. We derive two types of ROMs of Navier-Stokes equations. A deterministic LES-like model is able to stabilize ROMs and to better analyze the influence of the residual velocity on the resolved component. The random one additionally maintains the variability of stable modes and quantifies the model errors. We derive random versions of several geophysical models. We numerically study the transport under location uncertainty through a simplified one. A single realization of our model better retrieves the small-scale tracer structures than a deterministic simulation. Furthermore, a small ensemble of simulations accurately predicts and describes the extreme events, the bifurcations as well as the amplitude and the position of the ensemble errors. Another of our derived simplified model quantifies the frontolysis and the frontogenesis in the upper ocean. This thesis also studied the mixing of tracers generated by smooth fluid flows, after a finite time. We propose a simple model to describe the stretching as well as the spatial and spectral structures of advected tracers. With a toy flow but also with satellite images, we apply our model to locally and globally describe the mixing, specify the advection time and the filter width of the Lagrangian advection method, as well as the turbulent diffusivity in numerical simulations.