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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
41

#### On Critical Points of Random Polynomials and Spectrum of Certain Products of Random Matrices

Annapareddy, Tulasi Ram Reddy January 2015 (has links) (PDF)
In the first part of this thesis, we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let X1, X2,..., Xk be i.i.d Ginibre matrices of size n ×n whose entries are standard complex Gaussian random variables. We derive eigenvalue density for matrices of the form X1 ε1 X2 ε2 ... Xk εk , where εi = ±1 for i =1,2,..., k. We show that the eigenvalues form a determinantal point process. The case where k =2, ε1 +ε2 =0 was derived earlier by Krishnapur. In the case where εi =1 for i =1,2,...,n was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.
42

#### Hattendorff’s theorem and Thiele’s differential equation generalized

Messerschmidt, Reinhardt 20 February 2006 (has links)
Hattendorff's theorem on the zero means and uncorrelatedness of losses in disjoint time periods on a life insurance policy is derived for payment streams, discount functions and time periods that are all stochastic. Thiele's differential equation, describing the development of life insurance policy reserves over the contract period, is derived for stochastic payment streams generated by point processes with intensities. The development follows that by Norberg. In pursuit of these aims, the basic properties of Lebesgue-Stieltjes integration are spelled out in detail. An axiomatic approach to the discounting of payment streams is presented, and a characterization in terms of the integral of a discount function is derived, again following the development by Norberg. The required concepts and tools from the theory of continuous time stochastic processes, in particular point processes, are surveyed. / Dissertation (MSc (Actuarial Science))--University of Pretoria, 2007. / Insurance and Actuarial Science / unrestricted
43

#### Modélisation de grands réseaux de neurones par processus de Hawkes / Modelling large neural networks via Hawkes processes

Chevallier, Julien 09 September 2016 (has links)
Comment fonctionne le cerveau ? Peut-on créer un cerveau artificiel ? Une étape essentielle en vue d'obtenir une réponse à ces questions est la modélisation mathématique des phénomènes à l'œuvre dans le cerveau. Ce manuscrit se focalise sur l'étude de modèles de réseaux de neurones inspirés de la réalité.Cette thèse se place à la rencontre entre trois grands domaines des mathématiques - l'étude des équations aux dérivées partielles (EDP), les probabilités et la statistique - et s'intéresse à leur application en neurobiologie. Dans un premier temps, nous établissons les liens qui existent entre deux échelles de modélisation neurobiologique. À un niveau microscopique, l'activité électrique de chaque neurone est représentée par un processus ponctuel. À une plus grande échelle, un système d'EDP structuré en âge décrit la dynamique moyenne de ces activités. Il est alors montré que le modèle macroscopique peut se retrouver de deux manières distinctes : en étudiant la dynamique moyenne d'un neurone typique ou bien en étudiant la dynamique d'un réseau de \$n\$ neurones en champ-moyen quand \$n\$ tend vers l’infini. Dans le second cas, la convergence vers une dynamique limite est démontrée et les fluctuations de la dynamique microscopique autour de cette limite sont examinées. Dans un second temps, nous construisons une procédure de test d'indépendance entre processus ponctuels, ces derniers étant destinés à modéliser l'activité de certains neurones. Ses performances sont contrôlées théoriquement et vérifiées d'un point de vue pratique par une étude par simulations. Pour finir, notre procédure est appliquée sur de vraies données / How does the brain compute complex tasks? Is it possible to create en artificial brain? In order to answer these questions, a key step is to build mathematical models for information processing in the brain. Hence this manuscript focuses on biological neural networks and their modelling. This thesis lies in between three domains of mathematics - the study of partial differential equations (PDE), probabilities and statistics - and deals with their application to neuroscience. On the one hand, the bridges between two neural network models, involving two different scales, are highlighted. At a microscopic scale, the electrical activity of each neuron is described by a temporal point process. At a larger scale, an age structured system of PDE gives the global activity. There are two ways to derive the macroscopic model (PDE system) starting from the microscopic one: by studying the mean dynamics of one typical neuron or by investigating the dynamics of a mean-field network of \$n\$ neurons when \$n\$ goes to infinity. In the second case, we furthermore prove the convergence towards an explicit limit dynamics and inspect the fluctuations of the microscopic dynamics around its limit. On the other hand, a method to detect synchronisations between two or more neurons is proposed. To do so, tests of independence between temporal point processes are constructed. The level of the tests are theoretically controlled and the practical validity of the method is illustrated by a simulation study. Finally, the method is applied on real data
44

#### Nonparametric Bayesian Clustering under Structural Restrictions

Hanxi Sun (11009154) 23 July 2021 (has links)
<div>Model-based clustering, with its flexibility and solid statistical foundations, is an important tool for unsupervised learning, and has numerous applications in a variety of fields. This dissertation focuses on nonparametric Bayesian approaches to model-based clustering under structural restrictions. These are additional constraints on the model that embody prior knowledge, either to regularize the model structure to encourage interpretability and parsimony or to encourage statistical sharing through underlying tree or network structure.</div><div><br></div><div>The first part in the dissertation focuses on the most commonly used model-based clustering models, mixture models. Current approaches typically model the parameters of the mixture components as independent variables, which can lead to overfitting that produces poorly separated clusters, and can also be sensitive to model misspecification. To address this problem, we propose a novel Bayesian mixture model with the structural restriction being that the clusters repel each other.The repulsion is induced by the generalized Matérn type-III repulsive point process. We derive an efficient Markov chain Monte Carlo (MCMC) algorithm for posterior inference, and demonstrate its utility on a number of synthetic and real-world problems. <br></div><div><br></div><div>The second part of the dissertation focuses on clustering populations with a hierarchical dependency structure that can be described by a tree. A classic example of such problems, which is also the focus of our work, is the phylogenetic tree with nodes often representing biological species. The structure of this problem refers to the hierarchical structure of the populations. Clustering of the populations in this problem is equivalent to identify branches in the tree where the populations at the parent and child node have significantly different distributions. We construct a nonparametric Bayesian model based on hierarchical Pitman-Yor and Poisson processes to exploit this, and develop an efficient particle MCMC algorithm to address this problem. We illustrate the efficacy of our proposed approach on both synthetic and real-world problems.</div>
45

#### Limit Theorems for Random Simplicial Complexes

Akinwande, Grace Itunuoluwa 22 October 2020 (has links)
We consider random simplicial complexes constructed on a Poisson point process within a convex set in a Euclidean space, especially the Vietoris-Rips complex and the Cech complex both of whose 1-skeleton is the Gilbert graph. We investigate at first the Vietoris-Rips complex by considering the volume-power functionals defined by summing powers of the volume of all k-dimensional faces in the complex. The asymptotic behaviour of these functionals is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. This behaviour is observed in different regimes. Univariate and multivariate central limit theorems are proven, and analogous results for the Cech complex are then given. Finally we provide a Poisson limit theorem for the components of the f-vector in the sparse regime.
46

#### Apprentissage statistique pour séquences d’évènements à l’aide de processus ponctuels / Learning from Sequences with Point Processes

Achab, Massil 09 October 2017 (has links)
Le but de cette thèse est de montrer que l'arsenal des nouvelles méthodes d'optimisation permet de résoudre des problèmes d'estimation difficile basés sur les modèles d'évènements.Alors que le cadre classique de l'apprentissage supervisé traite les observations comme une collection de couples de covariables et de label, les modèles d'évènements ne regardent que les temps d'arrivée d'évènements et cherchent alors à extraire de l'information sur la source de donnée.Ces évènements datés sont ordonnés de façon chronologique et ne peuvent dès lors être considérés comme indépendants.Ce simple fait justifie l'usage d'un outil mathématique particulier appelé processus ponctuel pour apprendre une certaine structure à partir de ces évènements.Deux exemples de processus ponctuels sont étudiés dans cette thèse.Le premier est le processus ponctuel derrière le modèle de Cox à risques proportionnels:son intensité conditionnelle permet de définir le ratio de risque, une quantité fondamentale dans la littérature de l'analyse de survie.Le modèle de régression de Cox relie la durée avant l'apparition d'un évènement, appelé défaillance, aux covariables d'un individu.Ce modèle peut être reformulé à l'aide du cadre des processus ponctuels.Le second est le processus de Hawkes qui modélise l'impact des évènements passés sur la probabilité d'apparition d'évènements futurs.Le cas multivarié permet d'encoder une notion de causalité entre les différentes dimensions considérées.Cette thèse est divisée en trois parties.La première s'intéresse à un nouvel algorithme d'optimisation que nous avons développé.Il permet d'estimer le vecteur de paramètre de la régression de Cox lorsque le nombre d'observations est très important.Notre algorithme est basé sur l'algorithme SVRG (Stochastic Variance Reduced Gradient) et utilise une méthode MCMC (Monte Carlo Markov Chain) pour approcher un terme de la direction de descente.Nous avons prouvé des vitesses de convergence pour notre algorithme et avons montré sa performance numérique sur des jeux de données simulés et issus de monde réel.La deuxième partie montre que la causalité au sens de Hawkes peut être estimée de manière non-paramétrique grâce aux cumulants intégrés du processus ponctuel multivarié.Nous avons développer deux méthodes d'estimation des intégrales des noyaux du processus de Hawkes, sans faire d'hypothèse sur la forme de ces noyaux. Nos méthodes sont plus rapides et plus robustes, vis-à-vis de la forme des noyaux, par rapport à l'état de l'art. Nous avons démontré la consistence statistique de la première méthode, et avons montré que la deuxième peut être réduite à un problème d'optimisation convexe.La dernière partie met en lumière les dynamiques de carnet d'ordre grâce à la première méthode d'estimation non-paramétrique introduite dans la partie précédente.Nous avons utilisé des données du marché à terme EUREX, défini de nouveaux modèles de carnet d'ordre (basés sur les précédents travaux de Bacry et al.) et appliqué la méthode d'estimation sur ces processus ponctuels.Les résultats obtenus sont très satisfaisants et cohérents avec une analysé économétrique.Un tel travail prouve que la méthode que nous avons développé permet d'extraire une structure à partir de données aussi complexes que celles issues de la finance haute-fréquence. / The guiding principle of this thesis is to show how the arsenal of recent optimization methods can help solving challenging new estimation problems on events models.While the classical framework of supervised learning treat the observations as a collection of independent couples of features and labels, events models focus on arrival timestamps to extract information from the source of data.These timestamped events are chronologically ordered and can't be regarded as independent.This mere statement motivates the use of a particular mathematical object called point process to learn some patterns from events.Two examples of point process are treated in this thesis.The first is the point process behind Cox proportional hazards model:its conditional intensity function allows to define the hazard ratio, a fundamental quantity in survival analysis literature.The Cox regression model relates the duration before an event called failure to some covariates.This model can be reformulated in the framework of point processes.The second is the Hawkes process which models how past events increase the probability of future events.Its multivariate version enables encoding a notion of causality between the different nodes.The thesis is divided into three parts.The first focuses on a new optimization algorithm we developed to estimate the parameter vector of the Cox regression in the large-scale setting.Our algorithm is based on stochastic variance reduced gradient descent (SVRG) and uses Monte Carlo Markov Chain to estimate one costly term in the descent direction.We proved the convergence rates and showed its numerical performance on both simulated and real-world datasets.The second part shows how the Hawkes causality can be retrieved in a nonparametric fashion from the integrated cumulants of the multivariate point process.We designed two methods to estimate the integrals of the Hawkes kernels without any assumption on the shape of the kernel functions. Our methods are faster and more robust towards the shape of the kernels compared to state-of-the-art methods. We proved the statistical consistency of the first method, and designed turned the second into a convex optimization problem.The last part provides new insights from order book data using the first nonparametric method developed in the second part.We used data from the EUREX exchange, designed new order book model (based on the previous works of Bacry et al.) and ran the estimation method on these point processes.The results are very insightful and consistent with an econometric analysis.Such work is a proof of concept that our estimation method can be used on complex data like high-frequency financial data.
47

#### Spatio-temporal Event Prediction via Deep Point Processes / 深層点過程を用いた時空間イベント予測

Okawa, Maya 23 March 2022 (has links)

48

#### Rates of Convergence and Microscopic Information in Random Matrix Theory

Taljan, Kyle 25 January 2022 (has links)
No description available.
49