Showing papers on "Poisson distribution published in 1997"
TL;DR: In this article, a theoretical and experimental investigation of a two-dimensional chiral honeycomb was conducted, and the honeycomb exhibits a Poisson's ratio of 1 for deformations in plane.
787 citations
TL;DR: This work offers an approximation to central confidence intervals for directly standardized rates, where it is assumed that the rates are distributed as a weighted sum of independent Poisson random variables.
Abstract: We offer an approximation to central confidence intervals for directly standardized rates, where we assume that the rates are distributed as a weighted sum of independent Poisson random variables. Like a recent method proposed by Dobson, Kuulasmaa, Eberle and Scherer, our method gives exact intervals whenever the standard population is proportional to the study population. In cases where the two populations differ non-proportionally, we show through simulation that our method is conservative while other methods (the Dobson et al. method and the approximate bootstrap confidence method) can be liberal.
515 citations
Book•
01 May 1997
TL;DR: In this paper, the Mixed Poisson Distributions (MPD) is defined as a mixture of Cox Processes, Gauss-Poisson Processes and Mixed Renewal Processes.
Abstract: Preface Introduction The Mixed Poisson Distributions Some Basic Concepts The Mixed Poisson Process Some Related Processes Cox Processes Gauss-Poisson Processes Mixed Renewal Processes Characterization of Mixed Poisson Processes Reliability Properties of Mixed Poisson Processes Characterization within Birth Processes Characterization within Stationary Point Processes Characterization within General Point Processes Compound Mixed Poisson Distributions Compound Distributions Exponential Bounds Asymptotic Behaviour Recursive Evaluation The Risk Business The Claim Process Ruin Probabilities
360 citations
Proceedings Article•
01 Jan 1997TL;DR: In this paper, the authors describe four different methods for selecting thresholds that work on very different principles: either the noise or the signal is modeled, and the model covers either the spatial or intensity distribution characteristics.
Abstract: Image differencing is used for many applications involving change detection. Although it is usually followed by a thresholding operation to isolate regions of change there are few methods available in the literature specific to (and appropriate for) change detection. We describe four different methods for selecting thresholds that work on very different principles. Either the noise or the signal is modeled, and the model covers either the spatial or intensity distribution characteristics. The methods are as follows: (1) a Normal model is used for the noise intensity distribution, (2) signal intensities are tested by making local intensity distribution comparisons in the two image frames (i.e., the difference map is not used), (3) the spatial properties of the noise are modeled by a Poisson distribution, and (4) the spatial properties of the signal are modeled as a stable number of regions (or stable Euler number).
313 citations
TL;DR: In this paper, an alternative methodology for extreme values of univariate time series was developed, by assuming that the time series is Markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions.
Abstract: In recent research on extreme value statistics, there has been an extensive development of threshold methods, first in the univariate case and subsequently in the multivariate case as well. In this paper, an alternative methodology for extreme values of univariate time series is developed, by assuming that the time series is Markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions. A new likelihood representation for threshold methods is presented which we apply to a Markovian time series. An important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events. We address this issue by showing how a theory of compound Poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest. The methods are illustrated by application to temperature data.
227 citations
TL;DR: The results provide a uniform framework of perturbation realization for infinitesimal perturbations analysis (IPA) and non-IPA approaches to the sensitivity analysis of steady-state performance; they also provide a theoretical background for the PA algorithms developed in recent years.
Abstract: Two fundamental concepts and quantities, realization factors and performance potentials, are introduced for Markov processes. The relations among these two quantities and the group inverse of the infinitesimal generator are studied. It is shown that the sensitivity of the steady-state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a Markov process. Based on these results, algorithms for estimating performance sensitivities on a single sample path of a Markov process can be proposed. The potentials in this paper are defined through realization factors and are shown to be the same as those defined by Poisson equations. The results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis (IPA) and non-IPA approaches to the sensitivity analysis of steady-state performance; they also provide a theoretical background for the PA algorithms developed in recent years.
226 citations
01 Jan 1997
TL;DR: In this article, the authors provide a general theory about the Poisson-Binomial distribution concerning its computation and applications, and as by-products, they propose new weighted sampling schemes for finite population, a new method for hypothesis testing in logistic regression, and a new algorithm for finding the maximum conditional likelihood estimate (MCLE) in case-control studies.
Abstract: The distribution of Z1 +···+ZN is called Poisson-Binomial if the Zi are independent Bernoulli random variables with not-all-equal probabilities of success. It is noted that such a distribution and its computation play an important role in a number of seemingly unrelated research areas such as survey sampling, case-control studies, and survival analysis. In this article, we provide a general theory about the Poisson-Binomial distribution concerning its computation and applications, and as by-products, we propose new weighted sampling schemes for finite population, a new method for hypothesis testing in logistic regression, and a new algorithm for finding the maximum conditional likelihood estimate (MCLE) in case-control studies. Two of our weighted sampling schemes are direct generalizations of the "sequential" and "reservoir" methods of Fan, Muller and Rezucha (1962) for simple random sampling, which are of interest to computer scientists. Our new algorithm for finding the MCLE in case-control studies is an iterative weighted least squares method, which naturally bridges prospective and retrospective GLMs.
205 citations
TL;DR: In this paper, a hierarchical model coupled to geostatistics is proposed to deal with a non-gaussian data distribution and take explicitly into account complex spatial structures (i.e. trends, patchiness and random fluctuations).
Abstract: We propose a hierarchical model coupled to geostatistics to deal with a non-gaussian data distribution and take explicitly into account complex spatial structures (i.e. trends, patchiness and random fluctuations). A common characteristic of animal count data is a distribution that is both zero-inflated and heavy tailed. In such cases, empirical variograms are no more robust and most structural analyses result in poor and noisy estimated spatial variogram structures. Thus kriged maps feature a broad variance of prediction. Moreover, due to the heterogeneity of wildlife population habitats, a nonstationary model is often required. To avoid these difficulties, we propose a hierarchical model that assumes that the count data follow a Poisson distribution given a theoretical sighting density which is a latent variable to be estimate. This density is modelled as the product of a positive long range trend by a positive stationary random field, characterized by a unit mean and a variogram function. A first estimate of the drift is used to obtain an estimate of the variogram of residuals including a correction term for variance coming from the Poisson distribution and weights due to the non-constant spatial mean. Then a kriging procedure similar to a modified universal kriging is implemented to directly map the latent density from raw count data. An application on fin whale data illustrates the effectiveness of the method in mapping animal density in a context that is presumably non-stationary. E. Bellier and P. Monestiez Biostatistique et Processus Spatiaux, INRA, Domaine Saint-Paul, Site Agroparc, 84914 Avignon cedex 9, France E. Bellier ( ) Norwegian Institute for Nature Research NINA, NO-7485 Trondheim, NORWAY e-mail: edwige.bellier@nina.no C. Guinet Centre d’Etudes Biologiques de Chize, CNRS, 79360 Villiers-en-Bois, France P.M. Atkinson and C.D. Lloyd (eds.), geoENV VII – Geostatistics for Environmental Applications, Quantitative Geology and Geostatistics 16, DOI 10.1007/978-90-481-2322-3 1, c Springer Science+Business Media B.V. 2010 1
196 citations
TL;DR: The generalized Poisson regression model has statistical advantages over both standard Poisson and negative binomial regression models, and is suitable for analysis of count data that exhibit either over- Dispersion or under-dispersion.
Abstract: This paper models household fertility decisions by using a generalized Poisson regression model. Since the fertility data used in the paper exhibit under-dispersion, the generalized Poisson regression model has statistical advantages over both standard Poisson and negative binomial regression models, and is suitable for analysis of count data that exhibit either over-dispersion or under-dispersion. The model is estimated by the method of maximum likelihood. Approximate tests for the dispersion and goodness-of-fit measures for comparing alternative models are discussed. Based on observations from the Panel Study of Income Dynamics of 1989 interviewing year, the empirical results support the fertility hypothesis of Becker and Lewis (1973).
190 citations
TL;DR: In this paper, an introduction to inhomogeneous Poisson groups is given, and the generalized classical Yang-Baxter equation has only a one-dimensional right-hand side.
Abstract: An introduction to inhomogeneous Poisson groups is given. Poisson inhomogeneous O(p,q) are shown to be coboundary, the generalized classical Yang-Baxter equation having only a one-dimensional right-hand side. Normal forms of the classical r-matrices for the Poincare group (inhomogeneous O(1,3)) are calculated.
136 citations
Posted Content•
TL;DR: In this paper, the authors established an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algeses, and proved that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.
Abstract: The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylinski operator of a Poisson manifold and its modular class. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.
TL;DR: It is demonstrated that the inverse Gaussian mixture distribution gives a significantly better fit for a data set on the frequency of epileptic seizures than the traditional Poisson distribution.
Abstract: Count data often show overdispersion compared to the Poisson distribution. Overdispersion is typically modeled by a random effect for the mean, based on the gamma distribution, leading to the negative binomial distribution for the count. This paper considers a larger family of mixture distributions, including the inverse Gaussian mixture distribution. It is demonstrated that it gives a significantly better fit for a data set on the frequency of epileptic seizures. The same approach can be used to generate counting processes from Poisson processes, where the rate or the time is random. A random rate corresponds to variation between patients, whereas a random time corresponds to variation within patients.
TL;DR: In this paper, a model that allows positive dependencies in multivariate count data by specifying conditional distributions as Winsorized Poisson probability mass functions was developed, which may be used to incorporate either positive or negative dependencies among the variables.
TL;DR: A compact high-order difference approximation with multigrid V-cycle algorithm to solve the two-dimensional Poisson equation with Dirichlet boundary conditions and is compared with the five-point formula to show the dramatic improvement in computed accuracy.
TL;DR: In this article, the authors considered the first crossing between a compound Poisson trajectory and an upper increasing boundary and showed that the distribution of the ruin time can be expressed in terms of generalized Appell polynomials.
Abstract: The ruin time T is considered as the time of first crossing between a compound Poisson trajectory and an upper increasing boundary. Under the assumption that the claim sizes are integer-valued, we show that the distribution of T can be expressed in terms of generalized Appell polynomials. Using the algebraic properties of these polynomials elegant expressions are obtained for P(T > x).
TL;DR: In this article, the assumption that the normal approximation to the binomial and Poisson distributions will be adequate was made. But this assumption was not always the case when using attribute control charts with 3 sigma limits.
Abstract: Attributes control charts have historically been used with 3-sigma limits. When such an approach is used there is the implicit assumption that the normal approximation to the binomial and Poisson distributions will be adequate. Control chart properties ..
TL;DR: In this article, the product-limit estimators of F, G and K were proposed and studied based on the analogy with censored survival data: the distance from a fixed point to the nearest point of the process is right-censored by its distance to the boundary of the window.
Abstract: When a spatial point process is observed through a bounded window, edge effects hamper the estimation of characteristics such as the empty space function F, the nearest neighbor distance distribution G and the reduced second-order moment function K. Here we propose and study product-limit type estimators of F, G and K based on the analogy with censored survival data: the distance from a fixed point to the nearest point of the process is right-censored by its distance to the boundary of the window. The resulting estimators have a ratio-unbiasedness property that is standard in spatial statistics. We show that the empty space function F of any stationary point process is absolutely continuous, and so is the product-limit estimator of F. The estimators are strongly consistent when there are independent replications or when the sampling window becomes large. We sketch a CLT for independent replications within a fixed observation window and asymptotic theory for independent replications of sparse Poisson processes. In simulations the new estimators are generally more efficient than the "border method" estimator but (for estimators of K), somewhat less efficient than sophisticated edge corrections.
TL;DR: In this article, the strain-dependent behavior characteristic of auxetic polymers has been modelled using a simple geometric model which consists of rectangular nodules intecronnected by fibrils.
Abstract: The strain-dependent behaviour characteristic of auxetic (i.e. having a negative Poisson's ratio) polymers has been modelled using a simple geometric model which consists of rectangular nodules intecronnected by fibrils. Careful consideration of the correct form of the model to use depending on the experimental method employed to test samples of auxetic ultra high molecular weight polyethylene (UHMWPE) has resulted in very good agreement between the experimental and theoretical Poisson's ratios and total engineering strain ratios when the deformation is predominantly due to hinging of the fibrils.Auxetic UHMWPE has been processed to yield a very wide range of Poisson's ratios depending on its microstructural parameters (i.e. nodule shape and size, fibril length and the angle between the fibril and nodule). These can be predicted using the model, allowing the possibility of tailoring Poisson's ratio of the material.
TL;DR: In this article, the authors explore the possibility of deviations from the Poisson distribution using temporal raindrop counting experiments and find 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.
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...
TL;DR: In this article, the authors developed techniques for the determination of increasing failure rate and decreasing failure rate (DFR) property for a wide class of discrete distributions, instead of using the failure rate, they make use of the ratio of two consecutive probabilities.
TL;DR: Goodness of fit and predictive ability generally were better when models included permanent environmental effects, and both outperformed Poisson and negative binomial models.
Abstract: The performance of linear and nonlinear sire and animal models in the analyses of reproductive traits (fertility, litter size, and ovulation rate) in two sheep populations (Rambouillet and Finnsheep) was compared in terms of goodness of fit and predictive ability. Linear sire (LSM) and animal (LAM) models were used with all traits. Nonlinear models were the threshold, Poisson, and negative binomial. Threshold sire (TSM) and animal (TAM) models were also used with all traits. Litter size and ovulation rate were analyzed also with Poisson and negative binomial sire (PSM and NBSM, respectively) and animal (PAM and NBAM, respectively) models. Variance components were those reported in the companion article. For PAM a new set of variance components derived from estimates found with the linear animal model also was used (PAM-L). Mean squares error (MSE) and correlations between fitted and observed values were used to assess goodness of fit. Predictive ability was assessed by partitioning the data sets for the different traits into two subsets with the restriction that all levels of fixed effects were represented in each subset. Parameters from one subset were employed to predict observations in the other, and then MSE and correlations between observed and predicted values were used as criteria for model comparison. Within estimation procedure, breed, and trait, goodness of fit of sire and animal models was similar. Linear and threshold models resulted in similar fit, and both outperformed Poisson and negative binomial models. In terms of predictive ability, linear and threshold models performed only slightly better than Poisson and negative binomial models. Goodness of fit and predictive ability generally were better when models included permanent environmental effects.
TL;DR: In this article, a unified setting for generalized Poisson and Nambu-Poisson brackets is discussed, and it is proved that a Nambus-poisson bracket of even order is a generalized poisson bracket, and characterizations of generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.
Abstract: A unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.
TL;DR: In this paper, the Chen-Stein method was used to obtain Poisson approximations for two different counts of the number of occurrences of a word with unexpected frequencies in DNA sequences.
Abstract: Identifying words with unexpected frequencies is an important problem in the analysis of long DNA sequences. To solve it, we need an approximation of the distribution of the number of
occurrences N(W) of a word W. Modeling DNA sequences with m-order Markov chains, we use the Chen-Stein method to obtain Poisson approximations for two different counts. We approximate the “declumped” count of W by a Poisson variable and the number of occurrences N(W) by a compound Poisson variable. Combinatorial results are used to solve the general case of overlapping words and to calculate the parameters of these distributions.
TL;DR: In this paper, the authors give a version of integration by parts on the level of local martingales, combined with the optional sampling theorem, which allows them to obtain differentiation formulae for Poisson integr...
Abstract: We give a version of integration by parts on the level of local martingales; combined with the optional sampling theorem, this method allows us to obtain differentiation formulae for Poisson integr...
TL;DR: Goodness-of-fit tests based on the Cramer-von Mises statistics are given for the Poisson distribution as discussed by the authors, and power comparison shows that these statistics, particularly A2, give good overall tests of fit.
Abstract: Goodness-of-fit tests based on the Cramer-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A2, give good overall tests of fit. The statistic A2 will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.
Nous presentons ici des tests de validite de l'ajustement fondes sur les statistiques de Cramer-von Mises pour la distribution Poisson. Des comparaisons de pouvoir demontrent que ces statistiques, et particulierement A2 donnent en general de bons tests de l'ajustement. La statistique A2 sera particulierement utile pour detecter des distributions ou la variance est proche de la moyenne, mais qui ne sont pas Poisson.
TL;DR: In this paper, it was shown that any discrete distribution with non-negative support has a representation in terms of an extended Poisson process (or pure birth process), which admits a variety of distributions; the equations for such processes may be readily solved numerically.
Abstract: It is shown that any discrete distribution with non-negative support has a representation in terms of an extended Poisson process (or pure birth process). A particular extension of the simple Poisson process is proposed: one that admits a variety of distributions; the equations for such processes may be readily solved numerically. An analytical approximation for the solution is given, leading to approximate mean-variance relationships. The resulting distributions are then applied to analyses of some biological data-sets.
TL;DR: In this paper, the authors developed and evaluated procedures for estimating and simulating nonhomogeneous Poisson processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components or both.
Abstract: We develop and evaluate procedures for estimating and simulating nonhomogeneous Poisson processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components or both. Maximum likelihood estimates of the unknown continuous parameters of the rate function are obtained numerically, and the degree of the polynomial rate component is determined by a likelihood ratio test. The experimental performance evaluation for this estimation procedure involves applying the procedure to 100 independent replications of nine selected point processes that possess up to four trigonometric rate components together with a polynomial rate component whose degree ranges from zero to three. On each replication of each process, the fitting procedure is applied to estimate the parameters of the process; and then the corresponding estimates of the rate and mean-value functions are computed over the observation interval. Evaluation of the fitting procedure is based on plotted tolerance bands for the rate and mean-value functions together with summary statistics for the maximum and average absolute estimation errors in these functions computed over the observation interval. The experimental results provide substantial evidence of the numerical stability and usefulness of the fitting procedure in simulation applications.
TL;DR: In this article, the authors explore the boundary between these two kinds of behavior and show that maxima of independent Poisson variables cannot be normalized to converge to a nondegenerate limit distribution.
Abstract: It is known that maxima of independent Poisson variables cannot be normalized to converge to a nondegenerate limit distribution. On the other hand, the Normal distribution approximates the Poisson distribution for large values of the Poisson mean, and maxima of random samples of Normal variables may be linearly scaled to converge to a classical extreme value distribution. We here explore the boundary between these two kinds of behavior. Motivation comes from the wish to construct models for the statistical analysis of extremes of background gamma radiation over the United Kingdom. The methods extend to row-wise maxima of certain triangular arrays, for which limiting distributions are also derived.
TL;DR: A simple model is proposed for incidence prediction that preserves in the period of prediction the age pattern of incidence rates existing in the data, following models in environmental cancer epidemiology.
Abstract: A simple model is proposed for incidence prediction. The model is non-linear in parameters but linear in time, following models in environmental cancer epidemiology. Assuming a Poisson distribution for the age and period specific numbers of incident cases approximate confidence and prediction intervals are calculated. The major advantage of this model over current models is that age-specific predictions can be made with greater accuracy. The model also preserves in the period of prediction the age pattern of incidence rates existing in the data. It may be fitted with any package which includes an iteratively reweighted least squares algorithm, for example GLIM. Cancer incidence predictions for the Stockholm-Gotland Oncological Region in Sweden are presented as an example.