Poisson Mixture Model Research Articles

On the measure and the estimation of evenness and diversity

Modelling word or species frequency count data through zero truncated Poisson mixture models allows one to interpret the model mixing distribution as the distribution of the word or species frequencies of the vocabulary or population. As a consequence, estimates of their mixing density can be used as a fingerprint of the style of the author in his texts or of the ecosystem in its samples. Definitions of measure of the evenness and of measure of the diversity within a vocabulary or population are given, and the novelty of these definitions is explained. It is then proposed that the measures of the evenness and of the diversity of a vocabulary or population be approximated through the expectation of these measures under the word or species frequency distribution. That leads to the assessment of the lack of diversity through measures of the variability of the mixing frequency distribution estimates described above.

Collision statistics of clusters: from Poisson model to Poisson mixtures

Clusters traverse a gas and collide with gas particles. The gas particles are absorbed, and the clusters become hosts. If the clusters are size-selected, the number of guests will be Poisson distributed. We review this by showcasing four laboratory procedures that all rely on the validity of the Poisson model. The effects of a statistical distribution of the clusters' sizes in a beam of clusters are discussed. We derive the average collision rates. Additionally, we present Poisson mixture models that also involve standard deviations. We derive the collision statistics for common size distributions of hosts and also for some generalizations thereof. The models can be applied to large noble gas clusters traversing doping gas. While outlining how to fit a generalized Poisson to the statistics, we still find even these Poisson models to be often insufficient.

Estimation of Viral Infection and Replication in Cells by Using Convolution Models

Summary In some assays, a diluted suspension of infected cells is plated onto multiple wells. In each well the number of genome copies of virus, Y, is recorded, but interest focuses on the number of infected cells, X, and the number of genome copies in the infected cells, W1,…,WX. The statistical problem is to recover the distribution or at least moments of X and W on the basis of the convolution Y. We evaluate various parametric statistical models for this ‘mixture’- type problem and settle on a flexible robust approach where X follows a two-component Poisson mixture model and W is a shifted negative binomial distribution. Data analysis and simulations reveal that the means and occasionally variances of X and W can be reliably captured by the model proposed. We also identify the importance of selecting an appropriate dilution for a reliable assay.

Poisson mixture model for measurements using counting

Starting with the basic Poisson statistical model of a counting measurement process, 'extraPoisson' variance or 'overdispersion' are included by assuming that the Poisson parameter representing the mean number of counts itself comes from another distribution. The Poisson parameter is assumed to be given by the quantity of interest in the inference process multiplied by a lognormally distributed normalising coefficient plus an additional lognormal background that might be correlated with the normalising coefficient (shared uncertainty). The example of lognormal environmental background in uranium urine data is discussed. An additional uncorrelated background is also included. The uncorrelated background is estimated from a background count measurement using Bayesian arguments. The rather complex formulas are validated using Monte Carlo. An analytical expression is obtained for the probability distribution of gross counts coming from the uncorrelated background, which allows straightforward calculation of a classical decision level in the form of a gross-count alarm point with a desired false-positive rate. The main purpose of this paper is to derive formulas for exact likelihood calculations in the case of various kinds of backgrounds.

Extended truncated Inverse Gaussian–Poisson model

The inverse Gaussian–Poisson mixture model is very useful when modelling highly skewed non-negative integer data in fields as diverse as linguistics, ecology, market research, bibliometry, engineering and insurance. When using this statistical model on the frequency of word or species frequency data, one typically truncates its sample space at zero to accommodate for the ignorance about the number of words or species that are not observed. In this paper, we show that by truncating the sample space of the inverse Gaussian–Poisson model, one is allowed to extend its parameter space and in that way improve its fit when the frequency of one is larger and the right tail is heavier than is allowed by the unextended model. By fitting the extended model to word frequency count data, we find many instances where the maximum likelihood estimates fall in the extension of the parameter space.

Application of finite mixture models for vehicle crash data analysis

Developing sound or reliable statistical models for analyzing motor vehicle crashes is very important in highway safety studies. However, a significant difficulty associated with the model development is related to the fact that crash data often exhibit over-dispersion. Sources of dispersion can be varied and are usually unknown to the transportation analysts. These sources could potentially affect the development of negative binomial (NB) regression models, which are often the model of choice in highway safety. To help in this endeavor, this paper documents an alternative formulation that could be used for capturing heterogeneity in crash count models through the use of finite mixture regression models. The finite mixtures of Poisson or NB regression models are especially useful where count data were drawn from heterogeneous populations. These models can help determine sub-populations or groups in the data among others. To evaluate these models, Poisson and NB mixture models were estimated using data collected in Toronto, Ontario. These models were compared to standard NB regression model estimated using the same data. The results of this study show that the dataset seemed to be generated from two distinct sub-populations, each having different regression coefficients and degrees of over-dispersion. Although over-dispersion in crash data can be dealt with in a variety of ways, the mixture model can help provide the nature of the over-dispersion in the data. It is therefore recommended that transportation safety analysts use this type of model before the traditional NB model, especially when the data are suspected to belong to different groups.

A Poisson mixture model to identify changes in RNA polymerase II binding quantity using high-throughput sequencing technology

We present a mixture model-based analysis for identifying differences in the distribution of RNA polymerase II (Pol II) in transcribed regions, measured using ChIP-seq (chromatin immunoprecipitation following massively parallel sequencing technology). The statistical model assumes that the number of Pol II-targeted sequences contained within each genomic region follows a Poisson distribution. A Poisson mixture model was then developed to distinguish Pol II binding changes in transcribed region using an empirical approach and an expectation-maximization (EM) algorithm developed for estimation and inference. In order to achieve a global maximum in the M-step, a particle swarm optimization (PSO) was implemented. We applied this model to Pol II binding data generated from hormone-dependent MCF7 breast cancer cells and antiestrogen-resistant MCF7 breast cancer cells before and after treatment with 17β-estradiol (E2). We determined that in the hormone-dependent cells, ~9.9% (2527) genes showed significant changes in Pol II binding after E2 treatment. However, only ~0.7% (172) genes displayed significant Pol II binding changes in E2-treated antiestrogen-resistant cells. These results show that a Poisson mixture model can be used to analyze ChIP-seq data.

Translated Poisson Mixture Model for Stratification Learning

A framework for the regularized and robust estimation of non-uniform dimensionality and density in high dimensional noisy data is introduced in this work. This leads to learning stratifications, that is, mixture of manifolds representing different characteristics and complexities in the data set. The basic idea relies on modeling the high dimensional sample points as a process of translated Poisson mixtures, with regularizing restrictions, leading to a model which includes the presence of noise. The translated Poisson distribution is useful to model a noisy counting process, and it is derived from the noise-induced translation of a regular Poisson distribution. By maximizing the log-likelihood of the process counting the points falling into a local ball, we estimate the local dimension and density. We show that the sequence of all possible local countings in a point cloud formed by samples of a stratification can be modeled by a mixture of different translated Poisson distributions, thus allowing the presence of mixed dimensionality and densities in the same data set. With this statistical model, the parameters which best describe the data, estimated via expectation maximization, divide the points in different classes according to both dimensionality and density, together with an estimation of these quantities for each class. Theoretical asymptotic results for the model are presented as well. The presentation of the theoretical framework is complemented with artificial and real examples showing the importance of regularized stratification learning in high dimensional data analysis in general and computer vision and image analysis in particular.

Statistical analysis and significance testing of serial analysis of gene expression data using a Poisson mixture model

BackgroundSerial analysis of gene expression (SAGE) is used to obtain quantitative snapshots of the transcriptome. These profiles are count-based and are assumed to follow a Binomial or Poisson distribution. However, tag counts observed across multiple libraries (for example, one or more groups of biological replicates) have additional variance that cannot be accommodated by this assumption alone. Several models have been proposed to account for this effect, all of which utilize a continuous prior distribution to explain the excess variance. Here, a Poisson mixture model, which assumes excess variability arises from sampling a mixture of distinct components, is proposed and the merits of this model are discussed and evaluated.ResultsThe goodness of fit of the Poisson mixture model on 15 sets of biological SAGE replicates is compared to the previously proposed hierarchical gamma-Poisson (negative binomial) model, and a substantial improvement is seen. In further support of the mixture model, there is observed: 1) an increase in the number of mixture components needed to fit the expression of tags representing more than one transcript; and 2) a tendency for components to cluster libraries into the same groups. A confidence score is presented that can identify tags that are differentially expressed between groups of SAGE libraries. Several examples where this test outperforms those previously proposed are highlighted.ConclusionThe Poisson mixture model performs well as a) a method to represent SAGE data from biological replicates, and b) a basis to assign significance when testing for differential expression between multiple groups of replicates. Code for the R statistical software package is included to assist investigators in applying this model to their own data.

Estimating the number of classes

Estimating the unknown number of classes in a population has numerous important applications. In a Poisson mixture model, the problem is reduced to estimating the odds that a class is undetected in a sample. The discontinuity of the odds prevents the existence of locally unbiased and informative estimators and restricts confidence intervals to be one-sided. Confidence intervals for the number of classes are also necessarily one-sided. A sequence of lower bounds to the odds is developed and used to define pseudo maximum likelihood estimators for the number of classes.

A Nonstationary Negative Binomial Time Series With Time-Dependent Covariates

Boston Harbor has a history of poor water quality, including contamination by enteric pathogens. We conduct a statistical analysis of data collected by the Massachusetts Water Resources Authority (MWRA) between 1996 and 2002 to evaluate the effects of court-mandated improvements in sewage treatment. Motivated by the ineffectiveness of standard Poisson mixture models and their zero-inflated counterparts, we propose a new negative binomial model for time series of Enterococcus counts in Boston Harbor, where nonstationarity and autocorrelation are modeled using a nonparametric smooth function of time in the predictor. Without further restrictions, this function is not identifiable in the presence of time-dependent covariates; consequently, we use a basis orthogonal to the space spanned by the covariates and use penalized quasi-likelihood (PQL) for estimation. We conclude that Enterococcus counts were greatly reduced near the Nut Island Treatment Plant (NITP) outfalls following the transfer of wastewaters from NITP to the Deer Island Treatment Plant (DITP) and that the transfer of wastewaters from Boston Harbor to the offshore diffusers in Massachusetts Bay reduced the Enterococcus counts near the DITP outfalls.

Soft Counting Poisson Mixture Model-Based Polling Method for Speech/Nonspeech Classification

In this letter, a new segment-level speech/nonspeech classification method based on the Poisson polling technique is proposed. The proposed method makes two modifications from the baseline Poisson polling method to further improve the classification accuracy. One of them is to employ Poisson mixture models to more accurately represent various segmental patterns of the observed frequencies for frame-level input features. The other is the soft counting-based frequency estimation to improve the reliability of the observed frequencies. The effectiveness of the proposed method is confirmed by the experimental results showing the maximum error reduction of 39% compared to the segmentally accumulated log-likelihood ratio-based method.

Inference on the Number of Species Through Geometric Lower Bounds

Estimating the number of species in a population from a sample of individuals is investigated in a nonparametric Poisson mixture model. A sequence of lower bounds to the odds that a species is unseen in the sample are proposed from a geometric perspective. A lower bound and its representing mixing distribution can be computed by linear programming with guaranteed convergence. These lower bounds can be estimated by the maximum likelihood method and used to construct lower confidence limits for the number of species by the bootstrap method. Computing the nonparametric maximum likelihood estimator is discussed. Simulation is used to assess the performance of estimated lower bounds and compare them with several existing estimators. A genomic application is investigated.

Using Booking Data to Model Drug User Arrest Rates: A Preliminary to Estimating the Prevalence of Chronic Drug Use

Public policy is often concerned with the size and characteristics of special populations that are difficult to reach in household surveys. Chronic drug users, who often live outside conventional households, provide the illustration motivating this paper. An alternative to household surveys is to question chronic drug users where they congregate—jails, treatment programs, and shelters, for example. Using such opportunistic data for prevalence estimation raises difficult problems for statistical inference: Study subjects who arrive at the collection points cannot be deemed a random sample of the general population. However, if we could estimate the rates at which chronic drug users arrive at the collection points, then we could use those estimates to weight the sample to represent the population. This paper presents a modified Poisson mixture model used to estimate the stochastic process that accounts for how chronic drug users get arrested. It uses that model to estimate arrest rates for 38 counties using up to sixteen quarters of data from the Arrestee Drug Abuse Monitoring survey.

Robust estimation of mixture complexity for count data

For finite mixtures, consistent estimation of unknown number of components, called mixture complexity, is considered based on a random sample of counts, when the exact form of component probability mass functions are unknown but are postulated to belong to some parametric family. Following a recent approach of Woo and Sriram [2006. Robust estimation of mixture complexity. J. Amer. Statist. Assoc., to appear.], we develop an estimator of mixture complexity as a by-product of minimizing a Hellinger information criterion, when all the parameters associated with the mixture model are unknown. The estimator is shown to be consistent. Monte Carlo simulations illustrate the ability of our estimator to correctly determine the mixture complexity when the postulated Poisson mixture model is correct. When the postulated model is a Poisson mixture but the data comes from a negative binomial mixture with moderate to more extreme overdispersion in one of its components, simulation results show that our estimator continues to perform well. These confirm the efficiency of the estimator when the model is correctly specified and the robustness when the model is incorrectly specified. A count dataset with overdispersion and possible zero inflation is analyzed to further illustrate the ability of our estimator to determine the number of components.

Estimating the number of drug users in Bangkok 2001: A capture–recapture approach using repeated entries in one list

Conventionally, capture-recapture techniques involving different lists such as police or hospitals are used for quantifying populations which are difficult to count, such as illicit drug user populations. Here, a novel approach is suggested based upon repeated entries in one list, which is less dependent on matching entries from different sources as in the conventional approach. For this purpose, a population-based study was conducted that utilizes all data on treatment episodes of drug users from all 61 health treatment centers in the Bangkok metropolitan region to estimate the size of drug use in the Bangkok metropolitan region. The data stem from the drug treatment surveillance system of the Office of the Narcotics Control Board (ONCB) and cover the period from October 1 to December 31, 2001. Based upon the frequency of treatment episodes of each patient, a count distribution arose which could be modelled well by means of a Poisson mixture model. Using this count model, an estimate for the number of unobserved drug users could be constructed. From 11,222 drug users found during the period, 7063 (62.9%) were heroin users, 3346 (29.8%) metamphetamine users, and the remaining 813 (7.3%) distributed under 15 drug categories, none above 1%. The study concentrated on heroin and metamphetamine users who were predominantly male (96.2% for heroin and 91.8% for metamphetamine). Metamphetamine users were younger than heroin users (22.3 years 95% CI: 22.1-22.5 vs. 30.8 years 95% CI: 30.6-31.0). By using the truncated Poisson mixture model, an estimate of the unobserved frequency of drug users with zero treatment episodes could be constructed leading to an estimate of 11,296 (95% CI: 8,964-13,628) heroin users (completeness of identification: 38.42, 95% CI: 34.03-44.04%) and 32,105 (95% CI: 24,647-39,563) metamphetamine users (completeness of identification: 9.44, 95% CI: 7.79-11.97%) for the Bangkok metropolitan region. The proposed model showed excellent goodness-of-fit, unspecified for drug type and also if specified for the major drug types which allowed the prediction of the unobserved number of drug users in a realistic way, avoiding artefacts due to severe matching problems when using several, different sources. The technique is also easy to implement and can be used routinely to monitor drug user populations in space and time.

Two-way Poisson mixture models for simultaneous document classification and word clustering

An approach to simultaneous document classification and word clustering is developed using a two-way mixture model of Poisson distributions. Each document is represented by a vector with each dimension specifying the number of occurrences of a particular word in the document in question. As a collection of documents across several classes usually makes use of a large number of words, the document vectors are of high dimension. On the other hand, the number of distinct words in any single document is usually substantially smaller than the size of the vocabulary, leading to sparse document vectors. A mixture of Poisson distributions is used to model the multivariate distribution of the word counts in the documents within each class. To address the issues of high dimensionality and sparsity, the parameters in the mixture model are regularized by imposing a clustering structure on the set of words. An EM-style algorithm for the two-way mixture model will be derived for parameter estimation with the clustering of words part of the estimation process. The connection of the two-way mixture model with dimension reduction will also be elucidated. Experiments on the newsgroup data have demonstrated promising results.

A multivariate Poisson mixture model for marketing applications

This paper describes a multivariate Poisson mixture model for clustering supermarket shoppers based on their purchase frequency in a set of product categories. The multivariate nature of the model accounts for cross‐selling effects between the purchases made in different product categories. However, for computational reasons, most multivariate approaches limit the covariance structure by including just one common interaction term, or by not including any covariance at all. Although this reduces the number of parameters significantly, it is often too simplistic as typically multiple interactions exist on different levels. This paper proposes a theoretically more complete variance/covariance structure of the multivariate Poisson model, based on domain knowledge or preliminary statistical analysis of significant purchase interaction effects in the data. Consequently, the model does not contain more parameters than necessary, whilst still accounting for the existing covariance in the data. Practically, retail category managers can use the model to devise customized merchandising strategies.

Increasing incidence of Type 1 diabetes – role for genes?

BackgroundThe incidence of Type 1 diabetes (T1DM) is increasing fast in many populations. The reasons for this are not known, although an increase in the penetrance of the diabetes-associated alleles, through changes in the environment, might be the most plausible mechanism. After the introduction of insulin treatment in 1930s, an increase in the pool of genetically susceptible individuals has been suggested to contribute to the increase in the incidence of Type 1 diabetes.ResultsTo explore this hypothesis, the authors formulate a simple population genetic model for the incidence change driven by non-Mendelian transmission of a single susceptibility factor, either allele(s) or haplotype(s). A Poisson mixture model is used to model the observed number of cases. Model parameters were estimated by maximizing the log-likelihood function. Based on the Finnish incidence data 1965–1996 the point estimate of the transmission probability was 0.998. Given our current knowledge of the penetrance of the most diabetic gene variants in the HLA region and their transmission probabilities, this value is exceedingly unrealistic.ConclusionsAs a consequence, non-Mendelian transmission of diabetic allele(s)/haplotype(s) if present, could explain only a small part of the increase in incidence in Finland. Hence, the importance of other, probably environmental factors modifying the disease incidence is emphasized.

A hierarchical Poisson mixture regression model to analyse maternity length of hospital stay.

Inpatient length of stay (LOS) is often considered as a proxy of hospital resource consumption. Using statewide obstetrical delivery data, a two-component Poisson mixture model provides a reasonable fit to the heterogeneous LOS distribution. Adopting the generalized linear mixed model (GLMM) approach, random effects are introduced to the two-component Poisson mixture regression model to account for the inherent correlation of patients clustered within hospitals. An EM algorithm is developed for the joint estimation of regression coefficients and variance component parameters. Related diagnostic measures for assessing model adequacy are derived. When applying the method to analyse maternity LOS, appropriate risk factors for the short-stay and long-stay subgroups can be identified from the respective Poisson components. In addition, predicted random hospital effects enable the comparison of relative efficiencies among hospitals after adjustment for patient case-mix and health provision characteristics.