Loi de Dirichlet

thumb|right|250px|Plusieurs images de la densité de la loi de Dirichlet lorsque K=3 pour différents vecteurs de paramètres α. Dans le sens horaire à partir du coin supérieur gauche : α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). En probabilité et statistiques, la loi de Dirichlet, souvent notée Dir(α), est une famille de lois de probabilité continues pour des variables aléatoires multinomiales. Cette loi (ou encore distribution) est paramétrée par le vecteur α de nombres réels positifs et tire son nom de Johann Peter Gustav Lejeune Dirichlet. Elle est vue comme la généralisation multinomiale de la loi bêta. Densité de probabilité thumb|right|250px|Illustration du changement de la log-densité avec K=3 lorsque le vecteur α varie de α=(0.3, 0.3, 0.3) à (2.0, 2.0, 2.0), en conservant toutes les composantes individuelles \alpha_i égales entre elles. La loi de Dirichlet d'ordre K ≥ 2 de paramètre

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (21)

Publications associées

Chargement

Personnes associées (4)

Personnes associées

Chargement

Unités associées (2)

Unités associées

Chargement

Concepts associés (21)

Concepts associés

Chargement

Cours associés (15)

Cours associés

Chargement

Séances de cours associées (23)

Séances de cours associées

Chargement

Regularity of solutions for a fourth-order elliptic system via Conservation law

Changyu Guo

In this paper, we obtain interior Holder continuity for solutions of the fourth-order elliptic system Delta(2)u = Delta(V center dot del u) + div(w del u) + W center dot del u formulated by Lamm and Riviere [Comm. Partial Differential Equations 33 (2008) 245-262]. Boundary continuity is also obtained under a standard Dirichlet or Navier boundary condition. We also use conservation law to establish a weak compactness result which generalizes a result of Riviere for the second-order problem.

2019

Chargement

Le carré de la fonction des diviseurs

Nicolas Bersier

Let d(n) denote Dirichlet's divisor function for positive integer numbers. This work is primarily concerned with the study of We are interested, in the error term where Ρ3 is a polynomial of degree 3 ; more precisely xΡ3(log x) is the residue of in s = 1. A. Ivić showed that E(x) = Ο(x1/2+ε) for all ε > 0 (cf. [9], p.394). We will prove that for all x > 0, we have With this intention, we apply Perron's formula to the generating function ζ4(s)/ ζ (2s) and Landau's finite difference method. It was conjectured that E(x) = Ο(x1/4+ε) for ε > 0. The existence of non-trivial zeros of the Riemann ζ function implies that we cannot do better, that is The study of the Riesz means for ρ sufficiently large shows that their error term, is an infinite series , on the zeros of the Riemann Zeta function added with a development , into a series of Hardy-Voronoï's type, both being convergent. To find the "meaning" of , one could consider the difference But the series (probably) doesn't converge.We will thus substract only a finite part of , weighted by a smooth function ω, the number of terms of the finite part depending on x. If we consider this new error term , we obtain, using a classical method due to Hardy, that for x ≥ 1.

EPFL2007

Chargement

Mixture models for multivariate extremes

Marc-Olivier Boldi

This thesis is a contribution to multivariate extreme value statistics. The tail of a multivariate distribution function is characterized by its spectral distribution, for which we propose a new semi-parametric model based on mixtures of Dirichlet distributions. To estimate the components of this model, reversible jump Monte Carlo Markov chain and EM algorithms are developed. Their performances are illustrated on real and simulated data, obtained using new representations of the extremal logistic and Dirichlet models. In parallel with the estimation of the spectral distribution, extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not. This selection is achieved by a new method based on heuristic arguments. It allows a selection independent of the dimension of the data. Its performance is illustrated on real and simulated data. Primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure, for which the estimation of the spectral measure is a means rather than an end. These two issues are addressed. For the first, a Monte Carlo method is developed based on simulation of extremes. It is compared with classical and new methods of the literature. For the second one, an original conditional dependence analysis is proposed, which enlightens various aspects of the dependence structure of the data. Examples using real data sets are given. In the last part, the semi-parametric model and the presented methods are extended to spatial extremes. It is made possible by considering the spectral distribution as the distribution of a random probability, an original viewpoint adopted throughout this thesis. Classical multivariate extremes are extended to extremes of random measures. The application is illustrated on rainfall data in China.

EPFL2004

Chargement

Loi bêta

Dans la théorie des probabilités et en statistiques, la loi bêta est une famille de lois de probabilités continues, définies sur , paramétrée par deux paramètres de forme, typiquement notés (alpha)

Categorical distribution

In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes th

Échantillonnage de Gibbs

L' est une méthode MCMC. Étant donné une distribution de probabilité sur un univers , cet algorithme définit une chaîne de Markov dont la distribution stationnaire est . Il permet ainsi de tirer a

DH-406: Machine learning for DH

This course aims to introduce the basic principles of machine learning in the context of the digital humanities. We will cover both supervised and unsupervised learning techniques, and study and implement methods to analyze diverse data types, such as images, music and social network data.

CS-431: Introduction to natural language processing

The objective of this course is to present the main models, formalisms and algorithms necessary for the development of applications in the field of natural language information processing. The concepts introduced during the lectures will be applied during practical sessions.

FIN-403: Econometrics

The course covers basic econometric models and methods that are routinely applied to obtain inference results in economic and financial applications.