<p>In this thesis, we explore several stochastic models associated withthe two-parameter Poisson-Dirichlet distribution and population genetics.The impacts of mutation, selection and time onthe population evolutionary process will be studied by focusing on two aspects of the model:equilibrium and non-equilibrium. In the first chapter, we introduce relevant background on stochastic genetic models, andsummarize our main results and their motivations. In the second chapter, the two-parameter GEM distribution is constructedfrom a linear birth process with immigration. The derivationrelies on the limiting behavior of the age-ordered family frequencies. In the third chapter, to show the robustness of the sampling formula we derive the Laplace transform of the two-parameterPoisson-Dirichlet distribution from Pitman sampling formula. The correlationmeasure of the two-parameter point process is obtained in our proof. We also reverse this derivationby getting the sampling formula from the Laplace transform. Then,we establish a central limit theorem for the infinitely-many-neutral-alleles modelat a fixed time as the mutation rate goes to infinity.Lastly, we get the Laplace transform for the selectionmodel from its sampling formula. In the fourth chapter, we establisha central limit theorem for the homozygosity functions under overdominant selectionwith mutation approaching infinity. The selection intensity is given by a multiple of certain powerof the mutation rate. This result shows an asymptotic normality for the properly scaled homozygosities,resembling the neutral model without selection.This implies that the influence of selection can hardly be observed with large mutation. In the fifth chapter, the stochastic dynamics of the two-parameter extension of theinfinitely-many-neutral-alleles model is characterized by the derivation of its transition function,which is absolutely continuous with respect to the stationary distribution being the two-parameter Poisson-Dirichlet distribution.The transition density is obtained by the expansion of eigenfunctions.Combining this result with the correlation measure in Chapter 3, we obtain the probability generatingfunction of a random sampling from the two-parameter model at a fixed time. Finally, we obtain two results based on the quasi-invariance of the Gamma processwith respect to the multiplication transformation group.One is the quasi-invariance property of the two-parameter Poisson-Dirichletdistribution with respect to Markovian transformation group.The other one is the equivalence between the quasi-invarianceof the stationary distributions of aclass of branching processes and their reversibility.</p> / Doctor of Philosophy (PhD)

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The polytopic model (PM) structure is often used in the areas of automatic control and fault detection and isolation (FDI). It is an alternative to the multiple model approach which explicitly allows for interpolation among local models. This thesis proposes a novel approach to PM estimation by modeling the set of PM weights as a random vector with Dirichlet Distribution (DD). A new approximate (adaptive) PM estimator, referred to as a Quasi-Bayesian Adaptive Kalman Filter (QBAKF) is derived and implemented. The model weights and state estimation in the QBAKF is performed adaptively by a simple QB weights' estimator and a single KF on the PM with the estimated weights. Since PM estimation problem is nonlinear and non-Gaussian, a DD marginalized particle filter (DDMPF) is also developed and implemented similar to MPF. The simulation results show that the newly proposed algorithms have better estimation accuracy, design simplicity, and computational requirements for PM estimation.

針對具有多項分配(multinomial distribution)母體的類別資料，貝氏分析通常採取Dirichlet分配作為其先驗分配(prior distribution)，但在很多實際應用時，卻會遭遇困難；例如，當我們欲推估各年齡層佔總勞動力人口之比例時，母體除具多項分配外，其相鄰類別之比例亦相對接近；換言之，此時母體為具有平滑性(smooth)的多項分配，若依然採用Dirichlet分配作為其先驗分配，則會因為Dirichlet分配本身不具有平滑的特性，因而在做貝氏分析時會產生困擾。對這個難題Dickey and Jiang於1998年提出一個解決之道，他們的理論是對Dirichlet分配作適當之調整，將經過線性轉換後之Dirichlet分配稱為過濾後Dirichlet分配(filtered-variate Dirichlet distribution)，以過濾後Dirichlet分配作為調整後之先驗分配。對於Dickey and Jiang提出的方法，我們重新以蒙地卡羅法(Monte Carlo method)求出貝氏解，同時也嘗試以類似Makov and Smith (1977)和Smith and Makov (1978)對混合分配(mixture distribution)所用之準貝氏方法(quasi-Bayesian method)來逼近貝氏解。而本文將由電腦模擬的方式，探討貝氏方法與準貝氏方法之執行結果，並且考察準貝氏方法之收斂行為，對準貝氏方法的使用時機提出建議。

Software maintenance and the understanding of where in the source code features are implemented are two strongly coupled tasks that make up a large portion of the effort spent on developing applications. The concept of feature location investigated in this thesis can serve as a supporting factor in those tasks as it facilitates the automation of otherwise manual searches for source code artifacts. Challenges in this subject area include the aggregation and composition of a training corpus from historical codebase data for models as well as the integration and optimization of qualified topic modeling techniques. Building up on previous research, this thesis provides a comparison of two different techniques and introduces a toolkit that can be used to reproduce and extend on the results discussed. Specifically, in this thesis a changeset-based approach to feature location is pursued and applied to a large open-source Java project. The project is used to optimize and evaluate the performance of Latent Dirichlet Allocation models and Pachinko Allocation models, as well as to compare the accuracy of the two models with each other. As discussed at the end of the thesis, the results do not indicate a clear favorite between the models. Instead, the outcome of the comparison depends on the metric and viewpoint from which it is assessed.

Cette thèse est dédiée à l'étude des marches aléatoires en milieu aléatoire sur Zd. On s'intéresse tout particulièrement aux environnements qui sont elliptiques, mais pas uniformément elliptiques, et qui peuvent donc contenir des pièges sur lesquels la marche passe beaucoup de temps. Le premier résultat de cette thèse (chapitre 4) concerne les environnements de Dirichlet, qui forment une sous-classe de marches aléatoires en milieu aléatoire présentant des propriétés remarquables. On se place en dimension d≥ 3 et on étudie le cas où les pièges dus à la non-uniforme ellipticité sont prépondérants. Dans ce contexte, on montre l'équivalence des points de vue statique et dynamique pour une marche accélérée. Ceci permet de compléter les résultats de transience et récurrence directionnelles obtenus par Sabot, et de donner le degré polynomial de l'éloignement de la marche par rapport à l'origine dans le cas sous-balistique et transient. On se place ensuite (chapitre 5) dans le cas des marches transientes dans une direction, et on étudie les conditions sur la loi de l'environnement nécessaires pour assurer l'existence de moments pour les temps de renouvellement. On améliore ainsi les résultats obtenus par Campos et Ramírez. Dans la dernière partie (chapitre 6), on étudie les conditions d'application du théorème central limite quenched dans le cas des marches aléatoires balistiques. Sous la condition supplémentaire (T), on affaiblit les hypothèses sur l'intégrabilité des temps de renouvellement des travaux de Rassoul-Agha et Seppäläinen et de Berger et Zeitouni : on arrive à la condition E (τ12+ε) < +∞ (pour le théorème annealed la condition optimale est E (τ12) < +∞) / In this thesis we study random walks in random environment on Zd. We are particularly interested in environments that are elliptic, but not uniformly elliptic. Those environments can contain traps on which the walk spends a lot of time. The first results in this thesis (chapter 4) deal with the particular case of Dirichlet environments. Random walks in Dirichlet environment form a sub-class of random walks in random environment with specific properties. We consider dimensions d 3 and we study the behavior of the walk when the traps created by the non-uniform ellipticity play an important part. In this context, we show the equivalence between the static and dynamic points of view for an accelerated walk. This completes the results of directional transience and recurrence obtained by Sabot, and it allows to find the polynomial order of the magnitude of the walk’s displacement in the sub-ballistic transient case. Then (chapter 5) we consider the case of directionally transient walks, and we study the conditions on the law of the environment that ensure the existence of moments for the regeneration times. We thus improve the results obtained by Campos and Ramírez. In the last section (chapter 6), we consider the case of ballistic random walks and we study the conditions under which a quenched central limit theorem holds. Under the additional assumption (T), we weaken the integrability of the regeneration times necessary for the works of Rassoul- Agha and Seppäläinen, and Berger and Zeitouni. We obtain the condition E (τ12+ε) < +∞ (whereas for the annealed theorem, the optimal condition is E (τ12) < +∞)

Les marches aléatoires en milieu aléatoire ont suscité un vif intérêt au cours de ces dernières années, tant en sciences appliquées, comme moyen notamment d'affiner des modèles par une prise en compte des fluctuations de l'environnement, qu'en mathématiques, de par la multiplicité et la richesse des comportements qu'elles présentent. Cette thèse est dédiée à l'étude de divers aspects de la transience des marches aléatoires en milieu aléatoire. Elle est composée de deux parties, la première consacrée au cas des environnements de Dirichlet sur Z^d, la seconde au régime transient sous-diffusif sur Z. La loi de Dirichlet apparaît naturellement du fait de son lien avec les marches renforcées. Certaines de ses spécificités permettent de plus d'obtenir des résultats sensiblement plus précis qu'en général. On démontre ainsi tout d'abord une caractérisation de l'intégrabilité des temps de sortie de parties finies de graphes quelconques, qui permet de raffiner un critère de balisticité dans Z^d. On prouve également que les marches aléatoires en environnement de Dirichlet sont transientes directionnellement, avec probabilité positive, dès que les paramètres ne sont pas symétriques. En dimension 1, la thèse se focalise sur le rôle des vallées profondes de l'environnement, en fournissant une nouvelle preuve du théorème de Kesten-Kozlov-Spitzer dans le cas sous-diffusif basée sur l'étude fine du comportement de la marche. Outre une meilleure compréhension de l'émergence de la loi limite, cette preuve a l'avantage de fournir la valeur explicite de ses paramètres. / Random walks in random environment have raised a great interest in the last few years, both among applied scientists, notably as a way to refine models by taking fluctuations of the surrounding environment into account, and among mathematicians, because of the variety and wealth of behaviours they display. This thesis aims at the study of miscellaneous aspects of the transience of random walks in random environment. A first part is dedicated to Dirichlet environments on Z^d and a second one to the transient subdiffusive regime on Z. Random walks in Dirichlet environment arise naturally as an equivalent model for oriented-edge reinforced reinforced random walks. Its specificities also allow for sensibly sharper results than in the general case. We thus prove a characterization of the integrability of exit times out of finite subsets of arbitrary graphs, which enables us to refine a ballisticity criterion on Z^d. We also prove that these random walks are transient with positive probability as soon as the parameters are non-symmetric. In dimension 1, the thesis focuses on the role of the deep valleys of the environment. We give a new proof of Kesten-Kozlov-Spitzer theorem in the subdiffusive regime based on a fine study of the behaviour of the walk. Together with a better understanding of the origin of the limit law, this proof also provides its explicit parameters.

<p>This thesis studies non-nearest-neighbor random walks in random environments (RWRE) on the integers and on the d-dimensional integer lattic that are drawn in an i.i.d. way according to a Dirichlet distribution. We complete a characterization of recurrence and transience in a given direction for random walks in Dirichlet environments (RWDE) by proving directional recurrence in the case where the Dirichlet parameters are balanced and the annealed drift is zero. As a step toward this, we prove a 0-1 law for directional transience of i.i.d. RWRE on the 2-dimensional integer lattice with bounded jumps. Such a 0-1 law was proven by Zerner and Merkl for nearest-neighbor RWRE in 2001, and Zerner gave a simpler proof in 2007. We modify the latter argument to allow for bounded jumps. We then characterize ballisticity, or nonzero liiting velocity, of transienct RWDE on the integers. It turns out that ballisticity is controlled by two parameters, kappa0 and kappa1. The parameter kappa0, which controls finite traps, is known to characterize ballisticity for nearest-neighbor RWDE on the d-dimensional integer lattice for dimension d at least 3, where transient walks are ballistic if and only if kappa0 is greater than 1. The parameter kappa1, which controls large-scale backtracking, is known to characterize ballisticity for nearest-neighbor RWDE on the one-dimensional integer lattice, where transient walks are ballistic if and only if the absolute value of kappa1 is greater than 1. We show that in our model, transient walks are ballistic if and only if both parameters are greater than 1. Our characterization is thus a mixture of known characterizations of ballisticity for nearest-neighbor one-dimensional and higher-dimensional cases. We also prove more detailed theorems that help us better understand the phenomena affecting ballisticity.</p>

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

Dirichlet regression models can be used to analyze a set of variables lying in a bounded interval that sum up to a constant (e.g., proportions, rates, compositions, etc.) exhibiting skewness and heteroscedasticity, without having to transform the data. There are two parametrization for the presented model, one using the common Dirichlet distribution's alpha parameters, and a reparametrization of the alpha's to set up a mean-and-dispersion-like model. By applying appropriate link-functions, a GLM-like framework is set up that allows for the analysis of such data in a straightforward and familiar way, because interpretation is similar to multinomial logistic regression. This paper gives a brief theoretical foundation and describes the implementation as well as application (including worked examples) of Dirichlet regression methods implemented in the package DirichletReg (Maier, 2013) in the R language (R Core Team, 2013). (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics