Academic literature on the topic 'Poisson log-normal model'

AbstractResearchers and regulatory agencies often make statistical inferences from insect count data using modelling approaches that assume homogeneous variance. Such models do not allow for formal appraisal of variability which in its different forms is the subject of interest in ecology. Therefore, the objectives of this paper were to (i) compare models suitable for handling variance heterogeneity and (ii) select optimal models to ensure valid statistical inferences from insect count data. The log-normal, standard Poisson, Poisson corrected for overdispersion, zero-inflated Poisson, the negative binomial distribution and zero-inflated negative binomial models were compared using six count datasets on foliage-dwelling insects and five families of soil-dwelling insects. Akaike's and Schwarz Bayesian information criteria were used for comparing the various models. Over 50% of the counts were zeros even in locally abundant species such as Ootheca bennigseni Weise, Mesoplatys ochroptera Stål and Diaecoderus spp. The Poisson model after correction for overdispersion and the standard negative binomial distribution model provided better description of the probability distribution of seven out of the 11 insects than the log-normal, standard Poisson, zero-inflated Poisson or zero-inflated negative binomial models. It is concluded that excess zeros and variance heterogeneity are common data phenomena in insect counts. If not properly modelled, these properties can invalidate the normal distribution assumptions resulting in biased estimation of ecological effects and jeopardizing the integrity of the scientific inferences. Therefore, it is recommended that statistical models appropriate for handling these data properties be selected using objective criteria to ensure efficient statistical inference.

Poisson Log-Normal Model is one of the hierarchical mixed models that can be used for count data. Several estimation methods can be used to estimate the model parameters. The first objective of this study was to examine the performance of the parameter estimator and model built using the Hierarchical Bayes method via Markov Chain Monte Carlo (MCMC) with simulation. The second objective was applied the Poisson Log-Normal model to the West Java illiteracy Cases data which is sourced from the Susenas data on March 2019. In 2019, the incidence of illiteracy is a very rare occurrence in West Java Province. So that, it is suitable as an application case in this study. The simulation results showed that the Hierarchical Bayes parameter estimator through MCMC has the smallest Root Mean Squared Error of Prediction (RMSEP) value and the absolute bias is relatively mostly similar when compared to the Maximum Likelihood (ML) and Penalized Quasi-Likelihood (PQL) methods. Meanwhile, the empirical results showed that the fixed variable is the number of respondents who have a maximum education of elementary school have the greatest risk of illiteracy. Also, the diversity of census blocks significantly affects illiteracy cases in West Java 2019.

AbstractIn this paper, we propose an approach for modeling claim dependence, with the assumption that the claim numbers and the aggregate claim amounts are mutually and serially dependent through an underlying hidden state and can be characterized by a hidden finite state Markov chain using bivariate Hidden Markov Model (BHMM). We construct three different BHMMs, namely Poisson–Normal HMM, Poisson–Gamma HMM, and Negative Binomial–Gamma HMM, stemming from the most commonly used distributions in insurance studies. Expectation Maximization algorithm is implemented and for the maximization of the state-dependent part of log-likelihood of BHMMs, the estimates are derived analytically. To illustrate the proposed model, motor third-party liability claims in Istanbul, Turkey, are employed in the frame of Poisson–Normal HMM under a different number of states. In addition, we derive the forecast distribution, calculate state predictions, and determine the most likely sequence of states. The results indicate that the dependence under indirect factors can be captured in terms of different states, namely low, medium, and high states.

Population-averaged and subject-specific models are available to evaluate count data when repeated observations per subject are present. The latter are also known in the literature as generalised linear mixed models (GLMM). In GLMM repeated measures are taken into account explicitly through random animal effects in the linear predictor. In this paper the relevant GLMMs are presented based on conditional Poisson or negative binomial distribution of the response variable for given random animal effects. Equations for the repeatability of count data are derived assuming normal distribution and logarithmic gamma distribution for the random animal effects. Using count data on aggressive behaviour events of pigs (barrows, sows and boars) in mixed-sex housing, we demonstrate the use of the Poisson »log-gamma intercept«, the Poisson »normal intercept« and the »normal intercept« model with negative binomial distribution. Since not all count data can definitely be seen as Poisson or negative-binomially distributed, questions of model selection and model checking are examined. Emanating from the example, we also interpret the least squares means, estimated on the link as well as the response scale. Options provided by the SAS procedure NLMIXED for estimating model parameters and for estimating marginal expected values are presented.

Abstract. Population-averaged and subject-specific models are available to evaluate count data when repeated observations per subject are present. The latter are also known in the literature as generalised linear mixed models (GLMM). In GLMM repeated measures are taken into account explicitly through random animal effects in the linear predictor. In this paper the relevant GLMMs are presented based on conditional Poisson or negative binomial distribution of the response variable for given random animal effects. Equations for the repeatability of count data are derived assuming normal distribution and logarithmic gamma distribution for the random animal effects. Using count data on aggressive behaviour events of pigs (barrows, sows and boars) in mixed-sex housing, we demonstrate the use of the Poisson »log-gamma intercept«, the Poisson »normal intercept« and the »normal intercept« model with negative binomial distribution. Since not all count data can definitely be seen as Poisson or negative-binomially distributed, questions of model selection and model checking are examined. Emanating from the example, we also interpret the least squares means, estimated on the link as well as the response scale. Options provided by the SAS procedure NLMIXED for estimating model parameters and for estimating marginal expected values are presented.

This paper summarizes the mechanisms behind the patterning of the intra-population abundance distribution of the land snail Vallonia pulchella (Müller, 1774). The molluscs were collected in recultivated soil formed on red-brown clays (Pokrov, Ukraine). Data obtained in this study reveal that V. pulchella population abundance ranges from 1 to 13 individuals per 100 g of soil sample. To obtain estimates of the mean, three models were used: the model of the arithmetic mean, the Poisson model and a log-normal model. The arithmetic mean of the occurrence of this species during the study period was 1.84 individuals/sample. Estimation of the average number of molluscs in one sample calculated using the Poisson model is lower and equals 1.40 individuals/sample. The distribution of the number of individuals in a population was described by the graphics "rank – abundance". The individual sample plot sites with molluscs may be regarded as equivalents of individual species in the community. For the analysis, the following models were used: broken sticks model, niche preemption model, log-normal model, Zipf model, and Zipf-Mandelbrot model. Applying the log-normal distribution gives a lower estimate of the mean density at 1.28 individuals/sample. Median value and mode is estimated at 1.00 individuals/sample. The Zipf-Mandelbrot model was shown as the most adequate to describe distribution of the V. pulchella population within the study area. The Zipf-Mandelbrot model belongs to the family of so-called non-Gaussian distributions. This means that the sample statistics do not possess asymptotic properties and by increasing the sample size, they tend to infinity, and are not close to the values of the general population. Therefore, the average value of the random variable that describes the non-Gaussian distribution has no statistical meaning. From an environmental point of view, this means that within the study area the capacity of the habitat is large, and for some combination of environmental conditions the rapid growth of the abundance of a given species is possible.

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.

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.