Extended poisson process modelling

Prawn trawling is a major fishing activity in the Great Barrier Reef Marine Park, generating millions of dollars for the Australian economy. It is important that appropriate management decisions are made to ensure the sustainability of the industry. An experimental study was conducted to quantify the recovery of seabed fauna after repeated trawling. Data from the study were composed of counts of observed benthic organisms on the seabed. The data were analysed using models based on a generalization of the simple Poisson process, which allows for over‐dispersion and under‐dispersion relative to the Poisson distribution. It was found that these models more faithfully described the data than more standard models such as the Poisson or negative binomial. It is argued that the use of inappropriate models can result in unreliable inferences, which in turn could lead to poor management decisions. Copyright © 2004 John Wiley & Sons, Ltd.

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.

A large scale BACI experiment to test the effects of prawn trawling on seabed biota in a closed area of the Great Barrier Reef Marine Park, Australia

Revisiting Mackay Online

If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework [Schnieper, R. (1991). Separating true ibnr and ibner claims. ASTIN Bulletin, 21(1), 111–127.] for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach.

Effectiveness of sampling plans by attributes based on mixture distributions characterising microbial clustering in food

This paper introduced a new mixed distribution, namely the negative binomial-generalized exponential (NB-GE) distribution, which it obtained by mixing the negative binomial (NB) distribution with a generalized exponential (GE) distribution. The NB-GE distribution offers the advantage of being able to handle this kind of data sets, while still maintaining similar characteristics as the traditional negative binomial. The closed form and the factorial moment of the NB-GE distribution are derived. The NB-GE distribution appropriate when we find the one related to count data which have a large number of zeros and overdispersion in Poisson distribution. In addition, we present the basic properties of the new distribution such as mean, variance, skewness and kurtosis. Including, the negative binomial-exponential distributions are presented as special cases of this NB-GE distribution. Parameters estimation is also implemented using maximum likelihood method and the usefulness of the NB-GE distribution is illustrated by real data set. When using a goodness of fit test, the results show that the NB-GE distribution can provide a better fit than the NB distribution and Poisson distribution for count data that contain a large number of zeros.

Limited information is available about the abundance of Atlantic Cod Gadus morhua and Cusk Brosme brosme in the coastal Gulf of Maine because the presence of lobster traps limits commercial fishing and surveys for these species. We developed abundance indices for Atlantic Cod and Cusk from bycatch data obtained in a lobster sea sampling program. We applied generalized linear models (GLMs) to standardize Atlantic Cod and Cusk bycatch rates. The CPUE data, measured as the count of Atlantic Cod and Cusk observed per trap haul, are characterized by an extremely skewed distribution with a high percentage of zero observations. Two general approaches were applied to tackle the zero-dominated data: modeling with different error distributions and aggregating data over spatial scales. We evaluated eight models: binomial, Poisson, negative binomial, Tweedie model, hurdle model (also referred to as the delta approach) with Poisson and negative binomial distribution, and zero-inflated model with Poisson and negative binomial distribution. The data were aggregated using six grids ranging from 0.01 to 0.5 geological degrees (0.6–30 nautical miles). The standardized CPUE showed a gradual decline from 2006 to 2011, except for 2009 when a slight increase occurred for Atlantic Cod, and a general decline from 2006 to 2010 followed by an increase in 2011 for Cusk. The standardized CPUEs were consistent, in general, among the models and among different spatial aggregation scenarios, suggesting that the CPUE standardization is robust with respect to choices of data aggregation and statistical models. This study highlights the feasibility of developing abundance indices based on bycatch data for monitoring fish stock dynamics in data-limited regions. Received September 17, 2014; accepted April 16, 2015

Methodologic Challenges in the Analysis of Count Data in Radiology Health Services Research

Estimating Retail Demand with Poisson Mixtures and Out-of-Sample Likelihood

The Poisson-lognormal distribution was suggested as a model for commonness of species by Preston. The analysis leading him to suggest this distribution was as follows: Suppose that the number of each species caught in a trap is assumed to be a single sample from a Poisson distribution with mean λ. R. S. Uhler and P. G. Bradley proposed two models to describe a phenomenon in petroleum exploration. These two models used the sum of lognormal random variables as a compounder of Poisson and negative binomial distributions. P. Holgate noted that a clear distinction between the P-LN and the lognormal has not always been maintained. R. M. Cassie indicated that for most plankton populations, samples with μ = 100 would not depart significantly from a lognormal distribution.

Prey selection can influence interactions among species, the composition and abundance of prey, and ultimately the movement of energy within the ecosystem. Different species of the exploited coral trout Plectropomus spp. often co-occur in reef environments, but their foraging behaviour and ecological niches are largely unknown. To explore niche overlap and resource use of sympatric adult coral trout, stable isotopes (δ13C and δ15N) were quantified for 3 tissues (muscle, red blood cells, and plasma) collected between August 2013 and February 2014 from P. leopardus (n = 117) and P. laevis (n = 36) at 4 reefs in eastern Australia. Bayesian standard ellipses were used to show that prey selection of P. leopardus varied considerably from P. laevis, particularly from P. laevis in the blue-spot colour phase. Size of adult individuals had little influence on δ13C and δ15N values for P. leo pardus and both footballer and blue-spot colour phases of P. laevis. Spatiotemporal comparisons of P. leopardus trophic positions, made by adjusting baseline algae and planktonic δ15N at each reef and sampling period, demonstrated that trophic positions varied in time and space, and warrants further investigation. This study highlights that sympatric species of coral trout have distinct ecological roles and will likely react differently to environmental disturbances and/or changes in habitat or prey composition.

The foliage in a fully-leafed deciduous forest canopy is clumped. Consequently, theory indicates that the probability of beam penetration will be estimated more accurately with a model based on the negative binomial distribution than with a model based on the Poisson distribution, incorporating an assumption of a spherical leaf inclination angle distribution. Flux densities of photosynthetically active radiation (PAR) were measured in and above a deciduous forest canopy and were computed with the canopy radiative transfer models based on the negative binomial and Poisson distributions. These radiation values were used to compute canopy photosynthesis and stomatal conductance. Canopy photosynthesis and stomatal conductance, based on the negative binomial model, overestimated values computed from measured PAR profiles by 8 and 9%. respectively. The canopy photosynthesis and stomatal conductance values computed with the spherical Poisson model under-estimated measured values by 17 and 10%, respectively. Thus, the negative binomial radiative transfer model improves estimates of canopy photosynthesis and, to a lesser extent, stomatal conductance, inside a deciduous forest.

BackgroundRNA sequencing (RNA-seq) is a powerful tool for genome-wide expression profiling of biological samples with the advantage of high-throughput and high resolution. There are many existing algorithms nowadays for quantifying expression levels and detecting differential gene expression, but none of them takes the misaligned reads that are mapped to non-exonic regions into account. We developed a novel algorithm, XBSeq, where a statistical model was established based on the assumption that observed signals are the convolution of true expression signals and sequencing noises. The mapped reads in non-exonic regions are considered as sequencing noises, which follows a Poisson distribution. Given measureable observed and noise signals from RNA-seq data, true expression signals, assuming governed by the negative binomial distribution, can be delineated and thus the accurate detection of differential expressed genes.ResultsWe implemented our novel XBSeq algorithm and evaluated it by using a set of simulated expression datasets under different conditions, using a combination of negative binomial and Poisson distributions with parameters derived from real RNA-seq data. We compared the performance of our method with other commonly used differential expression analysis algorithms. We also evaluated the changes in true and false positive rates with variations in biological replicates, differential fold changes, and expression levels in non-exonic regions. We also tested the algorithm on a set of real RNA-seq data where the common and different detection results from different algorithms were reported.ConclusionsIn this paper, we proposed a novel XBSeq, a differential expression analysis algorithm for RNA-seq data that takes non-exonic mapped reads into consideration. When background noise is at baseline level, the performance of XBSeq and DESeq are mostly equivalent. However, our method surpasses DESeq and other algorithms with the increase of non-exonic mapped reads. Only in very low read count condition XBSeq had a slightly higher false discovery rate, which may be improved by adjusting the background noise effect in this situation. Taken together, by considering non-exonic mapped reads, XBSeq can provide accurate expression measurement and thus detect differential expressed genes even in noisy conditions.