Auswahl der wissenschaftlichen Literatur zum Thema „Modèle Poisson log-normal“

Cette thèse traite de la modélisation et de l’analyse de données de comptage de haute dimension dans le cadre des modèles à variables latentes, ainsi que de l’optimisation de tels modèles. Les modèles à variables latentes ont démontré leur efficacité dans la modélisation de structures de dépendance complexes pour les données de comptage, avec le modèle Poisson Log-Normal (PLN) comme exemple principal. Cependant, le modèle PLN ne répond pas aux caractéristiques des jeux de données de comptage réels, principalement en raison de son incapacité à produire un grand nombre de zéros. Nous proposons une extension, appelée PLN zéro inflaté (ZIPLN) pour répondre à ce problème. Ce dernier et d’autres variantes de PLN sont implémentés dans un package Python utilisant l’inférence variationnelle pour maximiser la log-vraisemblance. Dans la deuxième partie, nous nous concentrons sur le problème de maximisation d’une somme finie de fonctions, un problème couramment rencontré lors de l’optimisation d’une vaste catégorie de modèles à variables latentes. Nous introduisons une méthode adaptative nommée AdaLVR, qui évolue efficacement à la fois avec la dimensionnalité et la taille de l’échantillon du jeu de données, conçue explicitement pour ce problème d’optimisation. Une analyse théorique est menée, et une vitesse de convergence de O(T ⁻¹) est obtenue dans le cadre convexe, où T désigne le nombre d’itérations. Dans la troisième partie, nous discutons de l’optimisation des modèles à variables latentes par méthodes de Monte-Carlo, avec un accent particulier sur le modèle PLN. L’optimisation se fait dans un cadre non convexe et nécessite le calcul du gradient, qui est exprimé comme une intégrale intractable. Dans ce contexte, nous proposons un algorithme de premier ordre où le gradient est estimé par échantillonnage préférentiel auto-normalisé. Des garanties de convergence sont obtenues sous certaines hypothèses facilement vérifiables malgré le biais inhérent à l’estimateur du gradient. Il est important de noter que l’applicabilité du théorème de convergence va au-delà du cadre de l’optimisation dans les modèles à variables latentes. Dans la quatrième partie, nous nous concentrons sur la mise en œuvre de l’inférence pour les modèles PLN, avec un accent particulier sur les détails de l’inférence variationnelle conçue pour ces modèles. Dans l’annexe, nous dérivons des intervalles de confiance pour le modèle PLN et proposons une extension au modèle ZI-PLN intégrant l’Analyse en Composantes Principales. Une approche semi-paramétrique est également introduite. Parallèlement, une analyse d’un jeu de données génomiques réel est menée, révélant comment différents types de cellules dans les feuilles de plantes répondent à un pathogène bactérien
This thesis deals with the modeling and analysis of high-dimensional count data through the framework of latent variable models, as well as the optimization of such models. Latent variable models have demonstrated their efficacy in modeling count data with complex dependency structures, with the Poisson Log-Normal (PLN) model serving as a prime example. However, the PLN model does not meet the characteristics of real-world count datasets, primarily due to its inability to produce a high number of zeros. We propose the Zero-Inflated PLN (ZIPLN) extension to meet these characteristics. The latter and other variants of PLN are implemented in a Python package using variational inference to maximize the log-likelihood. In the second part, we focus on the finite-sum maximization problem, a common challenge when optimizing a wide range of latent variable models. We introduce an adaptive method named AdaLVR, scaling effectively with both the dimensionality and the sample size of the dataset, designed explicitly for this finite-sum optimization problem. A theoretical analysis of AdaLVR is conducted, and the convergence rate of O(T ⁻¹) is obtained in the convex setting, where T denotes the number of iterations. In the third part, we discuss the optimization of latent variable models using Monte Carlo methods, with a particular emphasis on the PLN model. The optimization occurs in a non-convex setting and necessitates the computation of the gradient, which is expressed as an intractable integral. In this context, we propose a first-order algorithm where the gradient is estimated using self-normalized importance sampling. Convergence guarantees are obtained under certain easily verifiable assumptions despite the inherent bias in the gradient estimator. Importantly, the applicability of the convergence theorem extends beyond the scope of optimization in latent variable models. In the fourth part, we focus on the implementation of the inference for PLN models, with a particular emphasis on the details of variational inference designed for these models. In the appendix, we derive confidence intervals for the PLN model, and an extension to the ZIPLN model, integrating Principal Component Analysis, is proposed. A semi-parametric approach is also introduced. Concurrently, an analysis of a real-world genomic dataset is conducted, revealing how different types of cells in plant leaves respond to a bacterial pathogen

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.

Le modèle Poisson log-normal multivarié propose une modélisation conjointe des abondances des espèces d’une communauté distinguant les effets environnementaux (abiotiques) des interactions entre espèces (biotiques). Ses différentes variantes permettent la visualisation par réduction de dimension ou l’inférence du réseau d’interactions directes entre les espèces. Ces approches sont utilisées pour analyser l’écosystème marin de la forêt de kelp de l’île d’Anacapa.

Abstract Count data analysed under a Poisson assumption often exhibit overdispersion. To accommodate the extra-Poisson variation, mixed Poisson models are frequently used. Inference using maximum likelihood techniques is possible assuming the Poisson mixture to be, for example, the negative binomial or the Poisson log normal distribution. However, interest here focuses on the use of estimating equations and in particular, quadratic and quasi-likelihood estimating equations. A general discussion of optimal quadratic estimation is provided by Crowder (1987) and Godambe and Thompson (1989). The estimators obtained from the use of optimal quadratic estimating equations are shown to be very efficient under a variety of distributions.

Micro-particle adhesion, detachment and resuspension from surfaces have attracted considerable attention due to their numerous applications in semiconductor, xerographic, and pharmaceutical industries, and, more recently, in understanding indoor air quality. However, most earlier studies have focused on idealized spherical particles and smooth surfaces, and the effects of particle irregularities and surface roughness on the rate of particle removal and resuspension are not well understood. In this work, a Monte Carlo simulation of particle resuspension from a surface under turbulent flow conditions was developed and resuspension of nearly spherical and irregular shaped particles with rough surfaces from substrates under turbulent flow condition was studied. Following our earlier approach, compact irregular shaped particles were modeled as spherical particles with a number of hemispherical bumps. It was assumed that the bump surfaces also have fine roughness. The extended Johnson-Kendall-Roberts (JKR) adhesion theory for rough surfaces was used to model the particle adhesion and detachment. A number of assumptions were made to apply the model. It was assumed that the particles have a Gaussian size distribution. The number of bumps of the irregular particles and surface roughness values of particle are assumed to be random, respectively, with Poisson and log-normal distributions. For particle detachment from the surface, the theory of critical moment was used. The effects of particle size, turbulent flow, particle irregularity and surface roughness on particle detachment and resuspension were studied for different cases. The Monte Carlo model predictions show probabilistic distributions of the particle resuspension. The simulation results are compared with the available experimental data and good agreement was found. The study provided information on the random nature of particle resuspension due to the randomness in the airflow, particle size distribution and surface roughness.