Poisson Mixture Model Research Articles

Poisson and multinomial mixture models for multivariate SIMS image segmentation.

Multivariate statistical methods have been advocated for analysis of spectral images, such as those obtained with imaging time-of-flight secondary ion mass spectrometry (TOF-SIMS). TOF-SIMS images using total secondary ion counts or secondary ion counts at individual masses often fail to reveal all salient chemical patterns on the surface. Multivariate methods simultaneously analyze peak intensities at all masses. We propose multivariate methods based on Poisson and multinomial mixture models to segment SIMS images into chemically homogeneous regions. The Poisson mixture model is derived from the assumption that secondary ion counts at any mass in a chemically homogeneous region vary according to the Poisson distribution. The multinomial model is derived as a standardized Poisson mixture model, which is analogous to standardizing the data by dividing by total secondary ion counts. The methods are adapted for contextual image segmentation, allowing for spatial correlation of neighboring pixels. The methods are applied to 52 mass units of a SIMS image with known chemical components. The spectral profile and relative prevalence for each chemical phase are obtained from estimates of model parameters.

A Poisson model for the coverage problem with a genomic application

SUMMARY Suppose a population has infinitely many individuals and is partitioned into unknown N disjoint classes. The sample coverage of a random sample from the population is the total proportion of the classes observed in the sample. This paper uses a nonparametric Poisson mixture model to give new understanding and results for inference on the sample coverage. The Poisson mixture model provides a simplified framework for inferring any general abundance-f coverage, the sum of the proportions of those classes that contribute exactly k individuals in the sample for some k in *, with * being a set of nonnegative integers. A new moment-based derivation of the well-known Turing estimators is presented. As an application, a gene-categorisation problem in genomic research is addressed. Since Turing's approach is a moment-based method, maximum likelihood estimation and minimum distance estimation are indicated as alternatives for the coverage problem. Finally, it will be shown that any Turing estimator is asymptotically fully efficient.

Comparison of Poisson Mixture Models for Count Data Clusterization

Five methods for count data clusterization based on Poisson mixture models are described. Two of them are parametric, the others are semi-parametric. The methods emlploy the plug-in Bayes classification rule. Their performance is investigated by making use of computer simulation and compared mainly by the clusterization error rate. We also apply the clusterization procedures to real count data and discuss the results.

On Bayesian analyses and finite mixtures for proportions

When the results of biological experiments are tested for a possible difference between treatment and control groups, the inference is only valid if based upon a model that fits the experimental results satisfactorily. In dominant-lethal testing, foetal death has previously been assumed to follow a variety of models, including a Poisson, Binomial, Beta-binomial and various mixture models. However, discriminating between models has always been a particularly difficult problem. In this paper, we consider the data from 6 separate dominant-lethal assay experiments and discriminate between the competing models which could be used to describe them. We adopt a Bayesian approach and illustrate how a variety of different models may be considered, using Markov chain Monte Carlo (MCMC) simulation techniques and comparing the results with the corresponding maximum likelihood analyses. We present an auxiliary variable method for determining the probability that any particular data cell is assigned to a given component in a mixture and we illustrate the value of this approach. Finally, we show how the Bayesian approach provides a natural and unique perspective on the model selection problem via reversible jump MCMC and illustrate how probabilities associated with each of the different models may be calculated for each data set. In terms of estimation we show how, by averaging over the different models, we obtain reliable and robust inference for any statistic of interest.

Standard Errors for EM Estimation

Summary The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.

Stochastic convexity of the Poisson mixture model

This paper is devoted to the study of the compound Poisson mixture model in an actuarial framework. Using the s-convex stochastic orderings and stochastic s-convexity, several problems involving an unknown mixing parameter with given moments are examined; namely, the specification of the number of support points in a finite mixture model, and the derivation of extremal mixture distributions. The theory is enhanced with the derivation of theoretical and numerical bounds on several quantities of actuarial interest.

Semiparametric estimation of count regression models

This paper develops a semiparametric estimation approach for mixed count regression models based on series expansion for the unknown density of the unobserved heterogeneity. We use the generalized Laguerre series expansion around a gamma baseline density to model unobserved heterogeneity in a Poisson mixture model. We establish the consistency of the estimator and present a computational strategy to implement the proposed estimation techniques in the standard count model as well as in truncated, censored, and zero-inflated count regression models. Monte Carlo evidence shows that the finite sample behavior of the estimator is quite good. The paper applies the method to a model of individual shopping behavior.

Fluctuation Properties of Precipitation. Part I: On Deviations of Single-Size Drop Counts from the Poisson Distribution

Abstract The traditional statistical description of the spatial and temporal distributions of cloud droplets and raindrops is the Poisson process, which tends to place the drops as uniformly as randomness allows. Yet, the “clumpy” nature of clouds and precipitation is apparent to most casual observers and well known to cloud physicists. Is such clumpiness consistent with the Poisson statistics? The authors explore the possibility of deviations from the Poisson distribution using temporal raindrop counting experiments. Disdrometer measurements during the passage of a squall line strongly indicate that a mixture of Poisson distributions (Poisson mixture) provides a better description of the frequency of drop arrivals per unit time in variable rain than does a simple Poisson model. Poisson mixture generally yields distributions different from Poissonian. While the validity of the Poisson mixture model to smaller scales requires much finer temporal resolution than available in this study, these results do sho...

Adaptive demixing in Poisson mixture models

Let $X_1, X_2, \dots, X_n$ be an i.i.d. sample from the Poisson mixture distribution $p(x) = (1/x!) \int_0^{\infty} s^x e^{-s}f(s) ds$. Rates of convergence in mean integrated squared error (MISE) of orthogonal series estimators for the mixing density f supported on $[a, b]$ are studied. For the Hölder class of densities whose rth derivative is Lipschitz $\alpha$, the MISE converges at the rate $(\log n/ \log \log n)^{-2(r +\alpha)}$. For Sobolev classes of densities whose rth derivative is square integrable, the MISE converges at the rate $(\log n/ \log \log n)^{-2r}$. The estimator is adaptive over both these classes. For the Sobolev class, a lower bound on the minimax rate of convergence is $(\log n/ \log \log n)^{-2r}$, and so the orthogonal polynomial estimator is rate optimal.

Mixed Poisson Regression Models with Covariate Dependent Rates

This paper studies a class of Poisson mixture models that includes covariates in rates. This model contains Poisson regression and independent Poisson mixtures as special cases. Estimation methods based on the EM and quasi-Newton algorithms, properties of these estimates, a model selection procedure, residual analysis, and goodness-of-fit test are discussed. A Monte Carlo study investigates implementation and model choice issues. This methodology is used to analyze seizure frequency and Ames salmonella assay data.

Bayesian Analysis of Two Overdispersed Poisson Models

In this paper, we consider the Bayesian analysis of two overdispersed Poisson models. The first is an overdispersed generalized Poisson model. The second is an ordinary Poisson and overdispersed generalized Poisson mixture model. Shoukri and Consul (1989, Comimintiicationis inz Statistics: Simnllationi anzd Comiiptutationi 18, 1465-1480) have previously considered a limited form of approximate Bayesian analysis for the first of these two models requiring the use of Pearson curves and the assumption that a certain model parameter has support on a finite number of values. By way of comparison, this paper demonstrates how a full Bayesian analysis of either model may proceed by making use of the Gibbs sampler and adaptive rejection sampling methods for log-concave densities. The methodology is illustrated with an application to a biological data set.

Bayesian analysis of two overdispersed poisson regression models

Shookri and Consul (1989) and Scollnik (1995) have previously considered the Bayesian analysis of an overdispersed generalized Poisson model. Scollnik (1995) also considered the Bayesian analysis of an ordinary Poisson and over-dispersed generalized Poisson mixture model. In this paper, we discuss the Bayesian analysis of these models when they are utilised in a regression context. Markov chain Monte Carlo methods are utilised, and an illustrative analysis is provided.

Maximum-Penalized-Likelihood Estimation for Independent and Markov- Dependent Mixture Models

This paper concerns the use and implementation of maximum-penalized-likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and Markov-dependent mixture models. Computation of the estimates is achieved via algorithms for the automatic generation of starting values for the EM algorithm. Computation of the information matrix is also discussed. Poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound. The resulting estimates are seen to provide plausible mechanisms for the physiological process.

A STATISTICAL METHOD FOR EVALUATING SUICIDE CLUSTERS AND IMPLEMENTING CLUSTER SURVEILLANCE

The absence of any standard definition of suicide cluster events hinders understanding of the prevalence of the problem, hinders the development of appropriate public health responses to observed clusters, and ultimately hinders investigation of the mechanisms underlying contagious communication of suicidal behavior. The authors introduce a Poisson mixture model for assessing potential clusters of adolescent suicide, apply that model to the monthly incidence rates of adolescent suicide for one populous US county over the last 11 years, and generate 99% tolerance limits with 95% confidence for the number of suicides which may occur by chance within specific intervals of time in that county. The suicide incidence data showed a remarkable fit to a single Poisson distribution, suggesting it is not unreasonable to consider the cases as randomly-distributed and independent events. The authors conclude that there is no evidence that adolescent suicides occurred in clusters in the place and in the time frame under study, and recommend the Poisson mixture model for ascertaining clusters as well as implementing cluster surveillance.

Relation between mean and variance in catch numbers of fish egg and larva

In each of six areas with various horizontal scale from 0.4 to 15.6 nautical miles, many surveys with vertical net hauls are made for sampling fish eggs and larvae. Though values ofCA are widely spread in each area, the variances2 and the meanm of catch numbers calculated from each survey follow approximately the relation from a Poisson mixture model (PMM). $$s^2 = m + C_A m^2 $$ . WhereCA (Kuno's aggegation index) is generally an index of magnitude of variability in catch numbers on the surveyed space (time or time-space). Moreover, this relation is equivalent to the relation between the variance and the mean in a negative binomial distribution model (NBM). However, NBM is a special case in the PMM.

反復ネット採集におけるカタクチイワシ卵子魚採集数の変動

Six series of ten repeated net hauls were made to examine the variation in catches of eggs and pre-larvae among the repeated hauls. The obtained results show that catches of eggs and pre-larvae of Engraulis japonicus were slightly over-dispersed or randomly distributed. A poisson mixture model is applied to explain these results. Suppose plankton organisms of some definite types, distributed at random with a mean M. Let the mean M be variable and set Q(M)dM be probability that M lies between M and M+dM. Then the probability of catching n organisms isP(n)=∫∞0Q(M)e-MMn/n!dM.Consequently, we can obtain following relation between the variance σn2 and mean μn of catch n; σn2=μn+μn2(σM/μM)2where σM2 and μM are the variance and mean of M respectively. This relation is equivalentto the one of the variance to the mean in a negative binomial model(NBM). But NBM is a special case that Q(M)is Gamma distribution in this model. Apparently, when σM≠0, then σn2>μn. But there is no reason why aggregation always occurred because variation of M is caused from many sources, for example, aggregation, prey-predator relation, transport by current, etc. Additional researches should be directed to-ward the causal factors inducing variation of M.