Showing papers on "Poisson distribution published in 2004"

A Modified Poisson Regression Approach to Prospective Studies with Binary Data

Abstract: Relative risk is usually the parameter of interest in epidemiologic and medical studies. In this paper, the author proposes a modified Poisson regression approach (i.e., Poisson regression with a robust error variance) to estimate this effect measure directly. A simple 2-by-2 table is used to justify the validity of this approach. Results from a limited simulation study indicate that this approach is very reliable even with total sample sizes as small as 100. The method is illustrated with two data sets.

N‐Mixture Models for Estimating Population Size from Spatially Replicated Counts

Abstract: Spatial replication is a common theme in count surveys of animals. Such surveys often generate sparse count data from which it is difficult to estimate population size while formally accounting for detection probability. In this article, I describe a class of models (N-mixture models) which allow for estimation of population size from such data. The key idea is to view site-specific population sizes, N, as independent random variables distributed according to some mixing distribution (e.g., Poisson). Prior parameters are estimated from the marginal likelihood of the data, having integrated over the prior distribution for N. Carroll and Lombard (1985, Journal of American Statistical Association 80, 423-426) proposed a class of estimators based on mixing over a prior distribution for detection probability. Their estimator can be applied in limited settings, but is sensitive to prior parameter values that are fixed a priori. Spatial replication provides additional information regarding the parameters of the prior distribution on N that is exploited by the N-mixture models and which leads to reasonable estimates of abundance from sparse data. A simulation study demonstrates superior operating characteristics (bias, confidence interval coverage) of the N-mixture estimator compared to the Caroll and Lombard estimator. Both estimators are applied to point count data on six species of birds illustrating the sensitivity to choice of prior on p and substantially different estimates of abundance as a consequence.

Regression models for relative survival.

Abstract: Four approaches to estimating a regression model for relative survival using the method of maximum likelihood are described and compared. The underlying model is an additive hazards model where the total hazard is written as the sum of the known baseline hazard and the excess hazard associated with a diagnosis of cancer. The excess hazards are assumed to be constant within pre-specified bands of follow-up. The likelihood can be maximized directly or in the framework of generalized linear models. Minor differences exist due to, for example, the way the data are presented (individual, aggregated or grouped), and in some assumptions (e.g. distributional assumptions). The four approaches are applied to two real data sets and produce very similar estimates even when the assumption of proportional excess hazards is violated. The choice of approach to use in practice can, therefore, be guided by ease of use and availability of software. We recommend using a generalized linear model with a Poisson error structure based on collapsed data using exact survival times. The model can be estimated in any software package that estimates GLMs with user-defined link functions (including SAS, Stata, S-plus, and R) and utilizes the theory of generalized linear models for assessing goodness-of-fit and studying regression diagnostics.

Density estimation in live‐trapping studies

Abstract: Unbiased estimation of population density is a major and unsolved problem in animal trapping studies. This paper describes a new and general method for estimating density from closed-population capture-recapture data. Many estimators exist for the size (N) and mean capture probability (p) of a closed population. These statistics suffer from an unknown bias due to edge effect that varies with trap layout and home range size. The mean distance between successive captures of an individual (d) provides information on the scale of individual movements, but is itself a function of trap spacing and grid size. Our aim is to define and estimate parameters that do not depend on the trap layout. In the new method, simulation and inverse prediction are used to estimate jointly the population density (D) and two parameters of individual capture probability, magnitude (go) and spatial scale (σ), from the information in N, p and d. The method uses any configuration of traps (e.g. grid, web or line) and any choice of closed-population estimator. It is assumed that home ranges have a stationary distribution in two dimensions, and that capture events may be simulated as the outcome of competing Poisson processes in time. The method is applied to simulated and field data. The estimator appears unusually robust and free from bias.

A nonstationary Poisson view of Internet traffic

Abstract: Since the identification of long-range dependence in network traffic ten years ago, its consistent appearance across numerous measurement studies has largely discredited Poisson-based models. However, since that original data set was collected, both link speeds and the number of Internet-connected hosts have increased by more than three orders of magnitude. Thus, we now revisit the Poisson assumption, by studying a combination of historical traces and new measurements obtained from a major backbone link belonging to a Tier 1 ISP. We show that unlike the older data sets, current network traffic can be well represented by the Poisson model for sub-second time scales. At multisecond scales, we find a distinctive piecewise-linear nonstationarity, together with evidence of long-range dependence. Combining our observations across both time scales leads to a time-dependent Poisson characterization of network traffic that, when viewed across very long time scales, exhibits the observed long-range dependence. This traffic characterization reconciliates the seemingly contradicting observations of Poisson and long-memory traffic characteristics. It also seems to be in general agreement with recent theoretical models for large-scale traffic aggregation

Binomial leap methods for simulating stochastic chemical kinetics.

Abstract: This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the τ-leap and midpoint τ-leap methods of Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches.

Recalculated probability of M >= 7 earthquakes beneath the Sea of Marmara, Turkey

Abstract: [1] New earthquake probability calculations are made for the Sea of Marmara region and the city of Istanbul, providing a revised forecast and an evaluation of time-dependent interaction techniques. Calculations incorporate newly obtained bathymetric images of the North Anatolian fault beneath the Sea of Marmara [Le Pichon et al., 2001; Armijo et al., 2002]. Newly interpreted fault segmentation enables an improved regional A.D. 1500– 2000 earthquake catalog and interevent model, which form the basis for time-dependent probability estimates. Calculations presented here also employ detailed models of coseismic and postseismic slip associated with the 17 August 1999 M = 7.4 Izmit earthquake to investigate effects of stress transfer on seismic hazard. Probability changes caused by the 1999 shock depend on Marmara Sea fault-stressing rates, which are calculated with a new finite element model. The combined 2004–2034 regional Poisson probability of M � 7 earthquakes is � 38%, the regional time-dependent probability is 44 ± 18%, and incorporation of stress transfer raises it to 53 ± 18%. The most important effect of adding time dependence and stress transfer to the calculations is an increase in the 30 year probability of a M � 7 earthquake affecting Istanbul. The 30 year Poisson probability at Istanbul is 21%, and the addition of time dependence and stress transfer raises it to 41 ± 14%. The ranges given on probability values are sensitivities of the calculations to input parameters determined by Monte Carlo analysis; 1000 calculations are made using parameters drawn at random from distributions. Sensitivities are large relative to mean probability values and enhancements caused by stress transfer, reflecting a poor understanding of large-earthquake aperiodicity. INDEX TERMS: 7223 Seismology: Seismic hazard assessment and prediction; 7230 Seismology: Seismicity and seismotectonics; 8150 Tectonophysics: Plate boundary—general (3040); KEYWORDS: earthquake probability, Sea of Marmara, seismic hazard, Turkey, stress interaction, North Anatolian fault

Accounting for Poisson noise in the multivariate analysis of ToF‐SIMS spectrum images

Abstract: Recent years have seen the introduction of many surface characterization instruments and other spectral imaging systems that are capable of generating data in truly prodigious quantities. The challenge faced by the analyst, then, is to extract the essential chemical information from this overwhelming volume of spectral data. Multivariate statistical techniques such as principal component analysis (PCA) and other forms of factor analysis promise to be among the most important and powerful tools for accomplishing this task. In order to benefit fully from multivariate methods, the nature of the noise specific to each measurement technique must be taken into account. For spectroscopic techniques that rely upon counting particles (photons, electrons, etc.), the observed noise is typically dominated by ‘counting statistics’ and is Poisson in nature. This implies that the absolute uncertainty in any given data point is not constant, rather, it increases with the number of counts represented by that point. Performing PCA, for instance, directly on the raw data leads to less than satisfactory results in such cases. This paper will present a simple method for weighting the data to account for Poisson noise. Using a simple time-of-flight secondary ion mass spectrometry spectrum image as an example, it will be demonstrated that PCA, when applied to the weighted data, leads to results that are more interpretable, provide greater noise rejection and are more robust than standard PCA. The weighting presented here is also shown to be an optimal approach to scaling data as a pretreatment prior to multivariate statistical analysis. Published in 2004 by John Wiley & Sons, Ltd.

A study of negative Poisson's ratios in auxetic honeycombs based on a large deflection model

Abstract: Materials or structures that contract in the transverse direction under uniaxial compression, or expand laterally when stretched are called to have negative Poisson's ratios. A theoretical approach to predict negative Poisson's ratios of auxetic honeycombs has been developed, which is based on the large deflection model. The equations of the deflection curves of the inclined member of the re-entrant cell, strains and Poisson's ratios of auxetic honeycombs in two orthogonal directions have been derived. The deformed shapes of the inclined member of the re-entrant cell are calculated. The negative Poisson's ratios of auxetic honeycombs are no longer a constant at large deformation. They vary significantly with the strain. The effect of the geometric parameters of the cell on the Poisson's ratios is analyzed.

A Haar-Fisz Algorithm for Poisson Intensity Estimation

Abstract: This article introduces a new method for the estimation of the intensity of an inhomogeneous one-dimensional Poisson process. The Haar-Fisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the Haar-Fisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the Haar-Fisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed state-of-the-art competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current state-of-the-art.

Convolution equivalence and infinite divisibility

Abstract: Known results relating the tail behaviour of a compound Poisson distribution function to that of its Levy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

Statistical Distributions of Poisson Voronoi Cells in Two and Three Dimensions

Abstract: Statistical distributions of geometrical characteristics concerning the Poisson Voronoi cells, namely, Voronoi cells for the homogeneous Poisson point processes, are numerically obtained in two- and three-dimensional spaces based on the computer experiments. In this paper, ten million and five million independent samples of Voronoi cells in two- and three-dimensional spaces, respectively, are generated. Geometrical characteristics such as the cell volume, cell surface area and so on, are fitted to the generalized gamma distribution. Then, maximum likelihood estimates of parameters of the generalized gamma distribution are given.

Analysis of low count time series data by poisson autoregression

Abstract: . This study provides new methods of assessing the adequacy of the Poisson autoregressive time-series model for count data. New expressions are given for the score function and the information matrix and these lead to the construction of new types of residuals for this model. However, these residuals often need to be supplemented by formal statistical procedures and an overall test of the model adequacy is given via the information matrix equality that holds for correctly specified models. The techniques are applied to a monthly count data set of claimants for wage loss benefit, in order to estimate the the expected duration of claimants in the system.

Use of exchangeable pairs in the analysis of simulations

Abstract: The method of exchangeable pairs has emerged as an important tool in proving limit theorems for Poisson, normal and other classical approximations. Here the method is used in a simulation context. We estimate transition probabilitites from the simulations and use these to reduce variances. Exchangeable pairs are used as control variates. Finally, a general approximation theorem is developed that can be complemented by simulations to provide actual estimates of approximation errors.

A fractional generalization of the Poisson processes

Abstract: It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.

OSEM-3D reconstruction strategies for the ECAT HRRT

Abstract: High spatial resolution dynamic brain PET imaging with the ECAT HRRT scanner with short frame durations is characterized by very few counts per sinogram bin due to the small size of the crystal surface The use of various weighting schemes for OSEM-3D can result in significantly different results In particular, the correction for random and scattered coincidences prior to the reconstruction can lead to a systematic positive bias in the reconstructed image We show that the use of Ordinary Poisson OSEM-3D, where all corrections are applied during the iterative steps, allows to avoid this bias without compromising spatial resolution, at a price of a lower convergence rate

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Abstract: General boundary conditions (‘branes’) for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization.

Occurrence and quantity of precipitation can be modelled simultaneously

Abstract: Many statistical models exist for modelling precipitation. One difficulty is that two issues need to be addressed: the probability of precipitation occurring, and then the quantity of precipitation recorded. This paper considers a family of distributions for modelling the quantity of precipitation, including those observations in which exactly no precipitation is recorded. Two examples are then discussed showing the distributions model the precipitation patterns well.

Computational methods for multiplicative intensity models using weighted gamma processes: Proportional hazards, marked point processes, and panel count data

Abstract: We develop computational procedures for a class of Bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures. A key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of Bayesian hierarchical models using the Dirichlet process. Using this fact, along with an approximation for the weighted gamma process, we show that with some care, one can adapt efficient algorithms used for the Dirichlet process to this setting. We discuss blocked Gibbs sampling procedures and Polya urn Gibbs samplers. We illustrate our methods with applications to proportional hazard models, Poisson spatial regression models, recurrent events, and panel count data.

Clustering analysis of SAGE data using a Poisson approach

Abstract: Serial analysis of gene expression (SAGE) data have been poorly exploited by clustering analysis owing to the lack of appropriate statistical methods that consider their specific properties. We modeled SAGE data by Poisson statistics and developed two Poisson-based distances. Their application to simulated and experimental mouse retina data show that the Poisson-based distances are more appropriate and reliable for analyzing SAGE data compared to other commonly used distances or similarity measures such as Pearson correlation or Euclidean distance.

An empirical model for underdispersed count data

Abstract: We present a novel distribution for modelling count data that are underdispersed relative to the Poisson distribution. The distribution is a form of weighted Poisson distribution and is shown to have advantages over other weighted Poisson distributions that have been proposed to model underdispersion. One key difference is that the weights in our distribution are centred on the mean of the underlying Poisson distribution. Several illustrative examples are presented that illustrate the consistently good performance of the distribution.

Poisson Statistics for the Largest Eigenvalues of Wigner Random Matrices with Heavy Tails

Abstract: We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson

Telegraph processes with random velocities

Abstract: We study a one-dimensional telegraph process (M t ) t≥0 describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution of M t and its first two moments are derived. We characterize the level hitting times of M t in terms of integro-differential equations which can be solved in special cases.

Exchangeable pairs and Poisson approximation

Abstract: This is a survey paper on Poisson approximation using Stein's method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector's problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.

Stein's Method: Expository Lectures and Applications

Abstract: This article presents a review of Stein’s method applied to the case of discrete random variables. We attempt to complete one of Stein’s open problems, that of providing a discrete version for chapter 6 of his book. This is illustrated by first studying the mechanics of comparison between two distributions whose characterizing operators are known, for example the binomial and the Poisson. Then the case where one of the distributions has an unknown characterizing operator is tackled. This is done for the hypergeometric which is then compared to a binomial. Finally the general case of the comparison of two probability distributions that can be seen as the stationary distributions of two birth and death chains is treated and conditions of the validity of the method are conjectured.