Showing papers on "Poisson distribution published in 1996"

Poisson's ratio and crustal seismology

Abstract: New measurements of compressional and shear wave velocities to hydrostatic pressures of 1 GPa are summarized for 678 rocks. Emphasis was placed on obtaining high-accuracy velocity measurements, which are shown to be critical in calculating Poisson's ratios from velocities. The rocks have been divided into 29 major groups for which velocities, velocity ratios, and Poisson's ratios are presented at several pressures. Observed Poisson's ratios for the monomineralic rocks compare favorably with theoretical Poisson's ratios calculated from single-crystal elastic constants. Plagioclase feldspar composition is important in understanding rock Poisson's ratios, since Poisson's ratio of albite increases from 0.28 to a predicted value of 0.31 for anorthite. Fe substitution for Mg in pyroxene and olivine also increases Poisson's ratio. Plotting rock compressional wave velocities versus Poisson's ratios reveals a triangular distribution bounded by quartzite with low compressional wave velocity and low Poisson's ratio, dunite with high compressional wave velocity and intermediate Poisson's ratio, and serpentinite with low compressional wave velocity and high Poisson's ratio. For common plutonic igneous rocks, there is a clear trend relating Poisson's ratio to composition, in which Poisson's ratio for granitic rocks increases from 0.24 to 0.29 as composition changes to gabbro and then decreases with decreasing plagioclase and increasing olivine contents to 0.25 in dunite. Changes in Poisson's ratio with progressive metamorphism of mafic and pelitic rocks correlate reasonably well with mineral reactions. There is no simple correlation between Poisson's ratio and felsic and mafic rock compositions; however, a linear correlation of increasing Poisson's ratio with decreasing SiO2 content is observed for rocks with 55 to 75 wt % SiO2. Average Poisson's ratios for continental and oceanic crusts are estimated to be 0.265 and 0.30, respectively.

A convergent adaptive algorithm for Poisson's equation

Abstract: We construct a converging adaptive algorithm for linear elements applied to Poisson’s equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extension to higher-order elements in two space dimension and numerical results are included.

Analysis of Frequency Count Data Using the Negative Binomial Distribution

Abstract: The statistical distributions of the counts of organisms are generally skewed, and hence not normally distributed, nor are variances constant across treatments. We present a likelihood—ratio testing framework based on the negative binomial distribution that tests for the goodness of fit of this distribution to the observed counts, and then tests for differences in the mean and/or aggregation of the counts among treatments. Inferences about differences in means among treatments as well as the dispersion of the counts are possible. Simulations demonstrated that the statistical power of ANOVA is about the same as the likelihood—ratio testing procedure for testing equality of means, but our proposed testing procedure also provides information on dispersion. Type I error rates of Poisson regression exceeded the expected 5%, even when corrected for overdispersion. Count data on Orange—crowned Warblers (Vermivora celata) are used to demonstrate the procedure.

Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography

Abstract: Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a "plug-in" approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems.

Probabilistic Interpretation of Population Codes

Abstract: We present a theoretical framework for population codes which generalizes naturally to the important case where the population provides information about a whole probability distribution over an underlying quantity rather than just a single value. We use the framework to analyze two existing models, and to suggest and evaluate a third model for encoding such probability distributions.

A Comparison of Poisson, Negative Binomial, and Semiparametric Mixed Poisson Regression Models: With Empirical Applications to Criminal Careers Data

Abstract: Specifications and moment properties of the univariate Poisson and negative binomial distributions are briefly reviewed and illustrated. Properties and limitations of the corresponding poisson and negative binomial (gamma mixtures of Poissons) regression models are described. It is shown how a misspecification of the mixing distribution of a mixed Poisson model to accommodate hidden heterogeneity ascribable to unobserved variables—although not affecting the consistency of maximum likelihood estimators of the Poisson mean rate parameter or its regression parameterization—can lead to inflated t ratios of regression coefficients and associated incorrect inferences. Then the recently developed semiparametric maximum likelihood estimator for regression models composed of arbitrary mixtures of Poisson processes is specified and further developed. It is concluded that the semiparametric mixed Poisson regression model adds considerable flexibility to Poisson-family regression models and provides opportunities for...

Excess Zeros in Count Models for Recreational Trips

Abstract: This article develops a modeling approach for a count dataset for recreational boating trips that shows a frequency of zero counts significantly higher than that expected for Poisson-distributed data. We consider several parametric and semiparametric mixed and modified Poisson models as alternatives to the Poisson regression. The analysis suggests that the negative binomial hurdles model, which allows for overdispersion and also accommodates the presence of excess zeros, is the most satisfactory of all those considered

Mixed Poisson regression models with covariate dependent rates.

Abstract: 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.

On a Voronoi Aggregative Process Related to a Bivariate Poisson Process

Abstract: We consider two independent homogeneous Poisson processes Π 0 and Π 1 in the plane with intensities λ 0 and λ 1 , respectively. We study additive functionals of the set of Π 0 -particles within a typical Voronoi Π 1 -cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π 0 -particles to the nucleus within a typical Voronoi Π 1 -cell.

Intensity, Duration, and Frequency of Residential Water Demands

Abstract: Water demands at four single-family residences were monitored for a 1-yr period. About 8,000 demands per capita were recorded and converted to single equivalent rectangular pulses. Each pulse was classified by type (deterministic or random), location (indoor or outdoor) and day (weekday or weekend). Basic exploratory data analyses were performed to estimate sample statistics, cumulative distributions, and hourly occurrences of the intensity, duration, and volumes of the rectangular pulses according to type, location, and day. In addition, the hypothesis that residential water demands occur as a nonhomogeneous Poisson rectangular pulse process was examined. Results show that the simple rectangular pulse provided a very reasonable approximation of indoor water demands. However, the variance of the daily pulse count appears to be too high for Poisson process. Potential causes of the high variance in the daily pulse count are discussed. Findings and data from this study can be used to develop and verify fine-resolution network models designed to predict the fate and travel time distribution of substances moving to remote points of consumption in the water distribution system.

Bayesian Computation for Nonhomogeneous Poisson Processes in Software Reliability

Abstract: A unified approach to the nonhomogeneous Poisson process in software reliability models is given. This approach models the epochs of failures according to a general order statistics model or to a record value statistics model. Their corresponding point processes can be related to the nonhomogeneous Poisson processes, for example, the Goel—Okumoto, the Musa—Okumoto, the Duane, and the Cox—Lewis processes. Bayesian inference for the nonhomogeneous Poisson processes is studied. The Gibbs sampling approach, sometimes with data augmentation and with the Metropolis algorithm, is used to compute the Bayes estimates of credible sets, mean time between failures, and the current system reliability. Model selection based on a predictive likelihood is studied. A numerical example with a real software failure data set is given.

Analysis of aggregated parasite distributions: A comparison of methods

Abstract: 1. Empirically, parasite distributions are often best described by the negative binomial distribution; some hosts have many parasites while most have just a few. Thus identifying heterogeneities in parasite burdens using conventional parametric methods is problematical. In an attempt to conform to the assumptions of parametric analyses, parasitologists and ecologists frequently log-transform their overdispersed data prior to analysis. In this paper, we compare this method of analysis with an alternative, generalized linear modelling (GLM), approach. 2. We compare the classical linear model using log-transformed data (Model 1) with two GLMs: one with Poisson errors and an empirical scale parameter (Model 2), and one in which negative binomial errors are explicitly defined (Model 3). We use simulated datasets and empirical data from a long-term study of parasitism in Soay Sheep on St Kilda to test the efficacies of these three statistical models. 3. We conclude that Model 1 is much more likely to produce type I errors than either of the two GLMs, and that it also tends to produce more type II errors. Model 3 is only marginally more successful than Model 2, indicating that the use of an empirical scale parameter is only slightly more likely to generate errors than using an explicitly defined negative binomial distribution. Thus, while we strongly recommend the use of GLMs over conventional parametric analyses, either GLM method will serve equally well.

A Simple Method for Generating Correlated Binary Variates

Abstract: Correlated binary data are frequently analyzed in studies of repeated measurements, reliability analysis, and others. In such studies correlations among binary variables are usually nonnegative. This article provides a simple algorithm for generating an arbitrary dimensional random vector of non-negatively correlated binary variables. In some frequently encountered situations the algorithm reduces to explicit expressions. The correlated binary variables are generated from correlated Poisson variables. The key idea lies in the property that any Poisson random variable can be expressed as a convolution of other independent Poisson random variables. The binary variables have desired correlations by sharing common independent Poisson variables.

Population dynamic models generating the lognormal species abundance distribution.

Abstract: This paper deals with a new class of stochastic species abundance models where the abundances are the points of an inhomogeneous Poisson process. These models are the result of a dynamic approach in which the changes in abundances through time are described by a multivariate diffusion and speciation constitutes a homogeneous Poisson process. In particular, the lognormal model is generated by assuming that the density regulation within each species is given by the Gompertz curve and that the environmental variances are constant. A substantial generalization is obtained by introducing a general type of interspecfic density regulation and correlated environmental noise. This more general mechanism also generates the lognormal species abundance distribution.

Second-order regular variation and rates of convergence in extreme-value theory

Abstract: Rates of convergence of the distribution of the extreme order statistic to its limit distribution are given in the uniform metric and the total variation metric. A second-order regular variation condition is imposed by supposing a von Mises type condition which allows a unified treatment. Rates are constructed from the parameters of the second-order regular variation condition. Some connections with Poisson processes are discussed.

Waveform inversion of marine reflection seismograms for P impedance and Poisson's ratio

Abstract: SUMMARY The estimation of the elastic properties o the crust from surface seismic recordings is of great importance for an understanding of the lithology and the detection of mineral resources. Although in marine reflection experiments only P waves are recorded, information on shear properties o the medium si contained in the reflection amplitudes recorded at different distances from the source. Being able to estimate both dialatational and shear properties gives stronger constraints on the lithology. It is therefore desirable to recover both types of elastic parameters from multi-offset seismograms. In the real-data example presented here, the amplitude of the reflections cannot be explained by one parameter related to the dilatational properties (P impedance) only, when, trying to minimize the least-squares fit between synthetic and real multi-offset seismograms. When adding an additional parameter related to the shear properties (Poisson's ratio), the fit between synthetic and real seismograms improves. Synthetic-wavefrom-fitting experiments underline the possibility of recovering Poisson's ration when the P-impedance model is well known from fitting waeveforms recorded at small offsetts. In the real-data example, the resulting models for P impedance and Poisson's ratio are anticorrelated in most depth regions, but are correlated in particular depth region, indicating a sudden change in lithology.

Time series models with univariate margins in the convolution-closed infinitely divisible class

Abstract: A unified way of obtaining stationary time series models with the univariate margins in the convolution-closed infinitely divisible class is presented. Special cases include gamma, inverse Gaussian, Poisson, negative binomial, and generalized Poisson margins. ARMA time series models obtain in the special case of normal margins, sometimes in a different stochastic representation. For the gamma and Poisson margins, some previously defined time series models are included, but for the negative binomial margin, the time series models are different and, in several ways, better than previously defined time series models. The models are related to multivariate distributions that extend a univariate

Empirical‐distribution‐function goodness‐of‐fit tests for discrete models

Abstract: We present a simple framework for studying empirical-distribution-function goodness-of-fit tests for discrete models. A key tool is a weak-convergence result for an estimated discrete empirical process, regarded as a random element in some suitable sequence space. Special emphasis is given to the problem of testing for a Poisson model and for the geometric distribution. Simulations show that parametric bootstrap versions of the tests maintain a nominal level of significance very closely even for small samples where reliance upon asymptotic critical values is doubtful.

Hurdle count-data models in recreation demand analysis

Abstract: When a sample of recreators is drawn from the general population using a survey, many in the sample will not recreate at a recreation site of interest. This study focuses on nonparticipation in recreation demand modeling and the use of modified countdata models. We clarify the meaning of the single-hurdle Poisson (SHP) model and derive the double-hurdle Poisson (DHP) model. The latter is contrasted with the SHP and we show how the DHP is consistent with Johnson and Kotz's zero-modified Poisson model.

On maximum likelihood estimation for a general non-homogeneous Poisson process

Abstract: Non-homogeneous Poisson processes (NHPPs) have been widely used in the study of software reliability. The statistical analysis for NHPPs is of interest to both theoreticians and practitioners. In this paper, maximum likelihood estimation under time-truncated sampling is studied for parametric NHPP software reliability models with bounded mean value functions. It is shown that the maximum likelihood estimators need not be consistent or asymptotically normal. The asymptotic distribution is derived for a specific NHPP model.

New generalized Poisson structures

Abstract: New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

Prediction of the Poisson's ratio of porous materials

Abstract: An equation is presented for the prediction of the Poisson's ratio of porous materials. The equation is strictly derived for spherical porosity and isotropic materials and it is valid for the whole porosity range. For low porosity, the equation coincides with a published equation, which has been verified in the past by comparison with extensive experimental data. For the high-porosity range, the theoretical variation of the Poisson's ratio exhibits a trend converging to a value νP=0.5, when the porosity increases to P=1, A similar converging trend has been found in other theoretical studies, but a rigorous experimental verification of such variations has still to be carried out.

On the comparison of multinomial and Poisson log-linear models

Abstract: SUMMARY We introduce a method for comparing multinomial and Poisson log-linear models which affords an explicit description of their equivalences and differences. The method involves specifying the model in terms of constraint equations, rather than the more common freedom equations. The Poisson and multinomial large sample distributions of log-linear model parameter estimators are derived and compared within this constraint equation context; reparameterizations are thereby avoided. As a by-product, the method provides the practitioner with the adjustment that is necessary to make valid inferences about all multinomial log-linear parameters when, as a matter of convenience, the Poisson log-linear model is fitted. This implies that valid large sample inferences about the multinomial cell probabilities can be made directly by using the Poisson log-linear model. To illustrate the utility of this approach, several examples are considered.

Major earthquakes in Mexicali Valley, Mexico, and fluid extraction at Cerro Prieto Geothermal Field

Abstract: Episodes of increased seismic moment release around the Cerro Prieto Geothermal Field (CPGF) show a significant correlation with three large increases of sustained fluid extraction, with delays of about 1 yr. Increased seismic activity involves three large strike-slip earthquakes: Imperial Valley ( ML = 6.6, 15 October 1979) and Victoria ( ML = 6.1, 9 June 1980), which occurred after a production increase in 1979, and Cerro Prieto ( ML = 5.4, 7 February 1987), which occurred after another production increase in 1986. The probabilities of the observed correlation between production increase and increased seismic activity occurring by chance (for binomial or Poisson seismicity distributions) are rather small, although their significance is not very large due to the small number of data. High probability of triggering by fluid extraction at times of high probability of earthquake occurrence from tectonic loading suggests a possible connection between production and occurrence of ML ≧ 5.4 earthquakes near CPGF. The strong earthquakes occurred at distances from CPGF within ranges over which induced seismicity has been observed for other engineering activities worldwide. Seismic diffusivity, calculated from their hypocentral distances and delay times from production increases, are in good agreement with values estimated worldwide for reservoirs, and estimated pressure changes at the earthquake sites, induced by production in CPGF, can be large enough for triggering. The observed time correlation, plus supporting statistical and spatio-temporal observations, although not conclusive, suggest the possibility of large tectonic earthquake triggering by extraction activity at the Cerro Prieto Geothermal Field. This possibility should be considered for estimation of seismic hazard and earthquake prediction in the Mexicali-Imperial Valley region.

Stein's method for geometric approximation

Abstract: The Stein-Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.