Poisson Lognormal Distribution Research Articles

Species diversity and community similarity in fluctuating environments: parametric approaches using species abundance distributions

Here we review recent advances in characterizing pattern of variation in community structure in space and time based on parametric approaches utilizing the full distribution of abundances of species rather than some summary indices. Assessment of biodiversity based on the structure of rank-abundance plots or simple species diversity indices, which describe properties of the sample of individuals, may reveal limited information about the underlying species abundance distribution of the community because the number of individuals counted are dependent on the sampling intensity. For instance, assuming Poisson sampling and an underlying lognormal species abundance distribution implies that observed abundances (counts) are a sample from a Poisson lognormal distribution. A convenient property of this distribution is that the estimate of σ(2) can be used as an inverse measure of species diversity in a community as well as the number of unobserved species can be estimated approximately without bias for unknown sampling intensities. If two communities can be described by a bivariate lognormal species abundance model, then the correlation between the log abundances of species in the two communities is an index of similarity that can be estimated without knowledge of sampling intensities using the bivariate Poisson lognormal distribution. This method is even applicable as an approximation when the abundance distribution deviates from the lognormal. An analysis of the interrelationship between the parameters of the lognormal species abundance distribution in communities of species from a wide variety of taxa shows that the canonical hypothesis of Preston in general, for a given number of species, gives far too large variances in the distribution of log abundances. A general feature in community dynamics is that a large component of the variance in the species abundance distribution is caused by heterogeneity among species in the population dynamics as well as environmental noise. This pattern is in contrast to the assumptions of the neutral theory of community dynamics. The choice of species abundance distribution should be a consequence of specific assumptions about the dynamics of the species. We suggest that such specific assumptions for the choice of species abundance model will facilitate more robust comparisons of changes in community structure in time and space.

Efficiency of the sampling plan for Cronobacter spp. assuming a Poisson lognormal distribution of the bacteria in powder infant formula and the implications of assuming a fixed within and between-lot variability

The introduction of Microbiological Criteria (MC) by the European Regulators (European Commission, 2005) has required the food industry to implement sampling plans in order to monitor food pathogens such as Cronobacter spp. in powder infant formula (PIF). In order to determine the performance of the sampling plans, Operating Characteristic curves (OC curves) were developed based on the assumption that Cronobacter spp. follow a Poisson lognormal (PLN) distribution in powder infant formula (PIF). The limitations of the mathematical models introduced by other authors based on the assumption of a fixed within-lot (S) and between-lot variability (Sb) for any sample weight (w) adopted are mathematically explored. In this study an alternative approach is suggested based on the assumption that for any sample of mass w employed, the variance-to-mean ratio of the PLN distribution (IPLN) is fixed and the within-lot variability changes accordingly. The implication of relating the probabilities of accepting/rejecting a lot to the mean expressed in the log10 scale (M) rather than to the arithmetic mean (μ) is also developed. A case study based on published data was also assessed and presented as example.

Combining counts and incidence data: an efficient approach for estimating the log-normal species abundance distribution and diversity indices

Obtaining accurate estimates of diversity indices is difficult because the number of species encountered in a sample increases with sampling intensity. We introduce a novel method that requires that the presence of species in a sample to be assessed while the counts of the number of individuals per species are only required for just a small part of the sample. To account for species included as incidence data in the species abundance distribution, we modify the likelihood function of the classical Poisson log-normal distribution. Using simulated community assemblages, we contrast diversity estimates based on a community sample, a subsample randomly extracted from the community sample, and a mixture sample where incidence data are added to a subsample. We show that the mixture sampling approach provides more accurate estimates than the subsample and at little extra cost. Diversity indices estimated from a freshwater zooplankton community sampled using the mixture approach show the same pattern of results as the simulation study. Our method efficiently increases the accuracy of diversity estimates and comprehension of the left tail of the species abundance distribution. We show how to choose the scale of sample size needed for a compromise between information gained, accuracy of the estimates and cost expended when assessing biological diversity. The sample size estimates are obtained from key community characteristics, such as the expected number of species in the community, the expected number of individuals in a sample and the evenness of the community.

Modelling homogeneous and heterogeneous microbial contaminations in a powdered food product

The actual physical distribution of microorganisms within a batch of food influences quantification of microorganisms in the batch, resulting from sampling and enumeration by microbiological tests. Quantification may be most accurate for batches in which microorganisms are distributed homogeneously. However, when the distribution is non-homogeneous, quantification may result in an under-, or overestimation. In the case of pathogens being non-homogeneously distributed, this heterogeneity will impact on public health. Enumeration data are commonly modelled by the Lognormal distribution. Although the Lognormal distribution can model heterogeneity, it does not allow for complete absence of microorganisms. Studies that validate the appropriateness of using Lognormal or other statistical distributions are scarce.This study systematically investigated laboratory and industrial scale batches of powdered infant formula, modelled the enumeration data using a range of statistical distributions, and assessed the appropriateness of individual models. For laboratory scale experiments, batches of milk powder were contaminated by distributing similar numbers of cells of Cronobacter sakazakii either homogeneously throughout a batch of milk powder or by distributing the cells in a localised part of the batch. Each batch was then systematically sampled and the distribution determined by enumerating the samples. By also enumerating the remainder of the batch, a balance could be made of the total number of microorganisms added and of the number retrieved from a batch. Discrete, as well as continuous statistical distributions, were fitted to enumeration data and the parameters estimated by Maximum Likelihood. The data were fitted both as censored and uncensored data. Enumeration data obtained for an industrial batch of powdered infant formula were investigated in this way as well. It was found that Normal, Poisson and Zero-Inflated Poisson distributions fitted the data sets very poorly. In case of homogeneous contamination, there was not a notable difference between the ability of Negative Binomial, Poisson-Lognormal, Weibull, Gamma, and Lognormal distributions to model the data. Overall, either the Negative Binomial distribution or the Poisson-Lognormal distribution fitted the data best in the 10 batches studied, especially when part of a data set contained zeros and/or the numbers were low. The Negative Binomial fitted the laboratory batches best and the Poisson-Lognormal fitted the industrial batch best.

Methods for fitting the Poisson-lognormal distribution to microbial testing data

The Poisson distribution can be used to describe the number of microorganisms in a serving of food, but in most food-safety applications, the variability of the Poisson distribution is insufficient to describe the heterogeneity in microbial contamination across the population of all servings. To model how contamination varies across the population of all possible servings, the lognormal distribution can be paired with the Poisson distribution to create an over-dispersed distribution, which is referred to as the Poisson-lognormal distribution. An advantage of this distribution is that random draws from the distribution are integer-valued. This is beneficial for some food-safety risk assessments because a modeled serving is either contaminated or not. This distribution is also appropriate when the results of a laboratory test are integer-valued, such as when direct plating is used to enumerate samples. While some surveys perform only absence/presence screenings tests, surveys that are more thorough can pair a screening test with enumeration of screen-test positive samples via direct plating. For this application, statistical methods that accommodate censored data are required to fit the data to a Poisson-lognormal distribution. This study compares a Bayesian hierarchical model to a maximum likelihood estimation approach for fitting data to the Poisson-lognormal distribution. Across a range of datasets, the Bayesian method demonstrates superior performance. OpenBUGS and R code are provided to implement both methods.

A comparative analysis of alternative approaches to fitting species-abundance models

Fits of species-abundance distributions to empirical data are increasingly used to evaluate models of diversity maintenance and community structure and to infer properties of communities, such as species richness. Two distributions predicted by several models are the Poisson lognormal (PLN) and the negative binomial (NB) distribution; however, at least three different ways to parameterize the PLN have been proposed, which differ in whether unobserved species contribute to the likelihood and in whether the likelihood is conditional upon the total number of individuals in the sample. Each of these has an analogue for the NB. Here, we propose a new formulation of the PLN and NB that includes the number of unobserved species as one of the estimated parameters. We investigate the performance of parameter estimates obtained from this reformulation, as well as the existing alternatives, for drawing inferences about the shape of species abundance distributions and estimation of species richness. We simulate the random sampling of a fixed number of individuals from lognormal and gamma community relative abundance distributions, using a previously developed 'individual-based' bootstrap algorithm. We use a range of sample sizes, community species richness levels and shape parameters for the species abundance distributions that span much of the realistic range for empirical data, generating 1 000 simulated data sets for each parameter combination. We then fit each of the alternative likelihoods to each of the simulated data sets, and we assess the bias, sampling variance and estimation error for each method. Parameter estimates behave reasonably well for most parameter values, exhibiting modest levels of median error. However, for the NB, median error becomes extremely large as the NB approaches either of two limiting cases. For both the NB and PLN, >90% of the variation in the error in model parameters across parameter sets is explained by three quantities that corresponded to the proportion of species not observed in the sample, the expected number of species observed in the sample and the discrepancy between the true NB or PLN distribution and a Poisson distribution with the same mean. There are relatively few systematic differences between the four alternative likelihoods. In particular, failing to condition the likelihood on the total sample sizes does not appear to systematically increase the bias in parameter estimates. Indeed, overall, the classical likelihood performs slightly better than the alternatives. However, our reparameterized likelihood, for which species richness is a fitted parameter, has important advantages over existing approaches for estimating species richness from fitted species-abundance models.

Impact of microbial distributions on food safety I. Factors influencing microbial distributions and modelling aspects

Relatively little is known about exactly how microorganisms are physically distributed in foods, yet these distributions determine the likelihood that a foodstuff will cause illness and the consequential public health burden. When a batch of food is sampled to determine the microbiological status of the batch, the effectiveness of the sampling programme is also related to the spatial distribution of the microorganisms that are being sampled for. In the absence of exact knowledge, generalising assumptions are often made as to the nature of the distributions. Better insight into the actual microbiological distributions may help to improve designing sampling plans and food safety management decision-making. This study discusses mechanisms influencing the spatial distributions of microorganisms in foods, three types of spatial distributions, i.e. regular, random, and clustered, the relationship between spatial distribution and frequency distributions, and the suitability of statistical distributions employed to model microbial distributions in foods. Commonly used statistical distributions, namely the Normal distribution, various types of the Poisson distribution, the Lognormal distribution, the Gamma distribution, the Negative Binomial distribution, and the Poisson-Lognormal distribution are examined and their strengths and weaknesses evaluated. Five specific criteria are proposed to assess the suitability of statistical distributions to model microbial distributions. These criteria require model outcomes to be non-negative, to allow zeros, to be discrete, to approximate Poisson and to approximate Lognormal. Especially the ability to model spatial clustering is investigated. It is concluded that the Poisson-Lognormal and the Negative Binomial are the most suitable statistical distributions given the suitability criteria proposed. However, the ultimate choice of the most suitable one should also depend on how well they fit actual observations.

Estimating similarity of communities: a parametric approach to spatio-temporal analysis of species diversity

Several stochastic models with environmental noise generate spatio-temporal Gaussian fields of log densities for the species in a community. Combinations of such models for many species often lead to lognormal species abundance distributions. In spatio-temporal analysis it is often realistic to assume that the same species are expected to occur at different times and/or locations because extinctions are rare events. Spatial and temporal b-diversity can then be analyzed by studying pairs of communities at different times or locations defined by a bivariate lognormal species abundance model in which a single correlation occurs. This correlation, which is a measure of similarity between two communities, can be estimated from samples even if the sampling intensities vary and are unknown, using the bivariate Poisson lognormal distribution. The estimators are approximately unbiased, although each specific correlation may be rather uncertain when the sampling effort is low with only a small fraction of the species represented in the samples. An important characteristic of this community correlation is that it relates to the classical Jaccard- or the Sorensen-indices of similarity based on the number of species present or absent in two communities. However, these indices calculated from samples of species in a community do not necessarily reflect similarity of the communities because the observed number of species depends strongly on the sampling intensities. Thus, we propose that our community correlation should be considered as an alternative to these indices when comparing similarity of communities. We illustrate the application of the correlation method by computing the similarity between temperate bird communities.

Estimation of Hurdle Models for Overdispersed Count Data

Hurdle models based on the zero-truncated Poisson-lognormal distribution are rarely used in applied work, although they incorporate some advantages compared with their negative binomial alternatives. I present a command that enables Stata users to estimate Poisson-lognormal hurdle models. I use adaptive Gauss–Hermite quadrature to approximate the likelihood function, and I evaluate the performance of the estimator in Monte Carlo experiments. The model is applied to the number of doctor visits in a sample of the U.S. Medical Expenditure Panel Survey.

A comparison between the discrete Poisson-gamma and Poisson-lognormal distributions to characterise microbial counts in foods

The choice of statistical distributions characterising microbial counts is essential in risk assessment and risk management. While the lognormal distribution has been long used to directly model the microbial data obtained from food samples, it does not allow for complete absence of microorganisms in a sample. Within a heterogeneous Poisson theoretical interpretation, a gamma or a lognormal population distribution for the unknown microbial concentration and a Poisson measurement distribution produces a discrete Poisson-gamma (λ, 1/k) or a Poisson-lognormal (μ,σ) distribution of observed plate counts. The capability of both distributions to deal with clustering was compared using six data sets of variable proportion of zero counts: total viable counts, coliforms and Escherichia coli on pre-chill and post-chill beef carcasses. Whereas the Poisson-lognormal distribution fitted better to the high counts data sets, the Poisson-gamma distribution represented the low counts data sets (13–81% zero counts) by far better than the Poisson-lognormal – which invariably tended to have a longer tail, an overestimated mean log and a lower predicted probability of zero counts. The inverse close relationship between the observed proportion of zero counts in the data set and the fitted dispersion factor 1/k suggested the possibility of obtaining a first approximation of 1/k by this means. Finally, in absence of zero counts, it was demonstrated that fitting a Poisson-lognormal to the observed plate count data can be closely approximated by the common practice of fitting a simple normal distribution to the back-calculated ‘unobserved’ mean concentrations in log CFU/g.

Model for comparative analysis of antigen receptor repertoires

In modern molecular biology one of the standard ways of analyzing a vertebrate immune system is to sequence and compare the counts of specific antigen receptor clones (either immunoglobulins or T-cell receptors) derived from various tissues under different experimental or clinical conditions. The resulting statistical challenges are difficult and do not fit readily into the standard statistical framework of contingency tables primarily due to the serious under-sampling of the receptor populations. This under-sampling is caused, on one hand, by the extreme diversity of antigen receptor repertoires maintained by the immune system and, on the other, by the high cost and labor intensity of the receptor data collection process. In most of the recent immunological literature the differences across antigen receptor populations are examined via non-parametric statistical measures of the species overlap and diversity borrowed from ecological studies. While this approach is robust in a wide range of situations, it seems to provide little insight into the underlying clonal size distribution and the overall mechanism differentiating the receptor populations. As a possible alternative, the current paper presents a parametric method that adjusts for the data under-sampling as well as provides a unifying approach to a simultaneous comparison of multiple receptor groups by means of the modern statistical tools of unsupervised learning. The parametric model is based on a flexible multivariate Poisson-lognormal distribution and is seen to be a natural generalization of the univariate Poisson-lognormal models used in the ecological studies of biodiversity patterns. The procedure for evaluating a model's fit is described along with the public domain software developed to perform the necessary diagnostics. The model-driven analysis is seen to compare favorably vis a vis traditional methods when applied to the data from T-cell receptors in transgenic mice populations.

A comparative Full Bayesian before-and-after analysis and application to urban road safety countermeasures in New Jersey

This paper develops a step-by-step methodology for the application of Full Bayes (FB) approach for before-and-after analysis of road safety countermeasures. As part of this methodology, it studies the posterior prediction capability of Bayesian approaches and their use in crash reduction factor (CRF) estimation. A collection of candidate models are developed to investigate the impacts of different countermeasures on road safety when limited data are available. The candidate models include traditional, random effects, non-hierarchical and hierarchical Poisson-Gamma and Poisson-Lognormal (P-LN) distributions. The use of random effects and hierarchical model structures allows treatment of the data in a time-series cross-section panel, and deal with the spatial and temporal effects in the data. Next, the proposed FB estimation methodology is applied to urban roads in New Jersey to investigate the impacts of different treatment measures on the safety of “urban collectors and arterial roads” with speed limits less than 45 mph. The treatment types include (1) increase in lane width, (2) installation of median barriers, (3) vertical and horizontal improvements in the road alignment; and (4) installation of guide rails. The safety performance functions developed via different model structures show that random effects hierarchical P-LN models with informative hyper-priors perform better compared with other model structures for each treatment type. The individual CRF values are also found to be consistent across the road sections, with all showing a decrease in crash rates after the specific treatment except guide rail installation treatment. The highest decrease in the crash rate is observed after the improvement in vertical and horizontal alignment followed by increase in lane width and installation of median barriers. Overall statistical analyses of the results obtained from different candidate models show that when limited data are available, P-LN model structure combined with higher levels of hierarchy and informative priors may reduce the biases in model parameters resulting in more robust estimates.

Testing species abundance models: a new bootstrap approach applied to Indo‐Pacific coral reefs

Patterns in the commonness and rarity of species are a fundamental characteristic of ecological assemblages; however, testing between alternative models for such patterns remains an important challenge. Conventional approaches to fitting or testing species abundance models often assume that species, not individuals, are the units that are sampled and that species' abundances are independent of one another. Here we test three different models (the Poisson lognormal, the negative binomial, and the neutral, "zero-sum multinomial" [ZSM]) against species abundance distributions of Indo-Pacific corals and reef fishes. We derive and apply several alternative bootstrap analyses of model fit, each of which makes different assumptions about how species abundance data are sampled, and we assess the extent to which tests of model fit are sensitive to such assumptions. For all models, goodness of fit is remarkably consistent, regardless of whether one assumes that species or individuals are the units that are sampled or whether or not one assumes that species' abundances are statistically independent of one another. However, goodness-of-fit estimates are approximately twice as precise and detect lack of model fit more frequently, when based on sampling of individuals, rather than species. Bootstrap analyses indicate that the Poisson lognormal distribution exhibits substantially better fit to species abundance patterns, consistent with model selection analyses. In particular, heterogeneity in species abundances (many rare and few highly abundant species) is too great to be captured by the ZSM model or the negative binomial model and is best explained by models that predict species abundance patterns that are much closer, but not identical, to the lognormal distribution. More broadly, our bootstrap analyses suggest that estimates of model fit are likely to be robust to assumptions about the statistical interdependence of species abundances, but that tests of model fit are more powerful when they assume sampling of individuals, rather than species. Such individual-based tests therefore may be able to identify lack of model fit where previous tests have been inconclusive.

A New Theoretical Discrete Growth Distribution with Verification for Microbial Counts in Water

Living microbes are discrete, not homogeneously distributed in environmental media, and the form of the distribution of their counts in drinking water has not been well established. However, this count may "scale" or range over orders of magnitude over time, in which case data representing the tail of the distribution, and governing the mean, would be represented only in impractically long data records. In the absence of such data, knowledge of the general form of the full distribution could be used to estimate the true mean accounting for low-probability, high-consequence count events and provide a basis for a general environmental dose-response function. In this article, a new theoretical discrete growth distribution (DGD) is proposed for discrete counts in environmental media and other discrete growth systems. The term growth refers not to microbial growth but to a general abiotic first-order growth/decay of outcome sizes in many complex systems. The emergence and stability of the DGD in such systems, defined in simultaneous work, are also described. The DGD is then initially verified versus 12 of 12 simulated long-term drinking water and short-term treated and untreated water microbial count data sets. The alternative Poisson lognormal (PLN) distribution was rejected for 2 (17%) of the 12 data sets with 95% confidence and, like other competitive distributions, was not found stable (in simultaneous work). Sample averages are compared with means assessed from the fitted DGD, with varying results. Broader validation of the DGD for discrete counts arising as outcomes of mathematical growth systems is suggested.

Maximum likelihood fitting of the Poisson lognormal distribution

In this paper some properties and analytic expressions regarding the Poisson lognormal distribution such as moments, maximum likelihood function and related derivatives are discussed. The author provides a sharp approximation of the integrals related to the Poisson lognormal probabilities and analyzes the choice of the initial values in the fitting procedure. Based on these he describes a new procedure for carrying out the maximum likelihood fitting of the truncated Poisson lognormal distribution. The method and results are illustrated on real data. The computer program for calculations is freely available.

Heterogeneous communities with lognormal species abundance distribution: Species–area curves and sustainability

Heterogeneous species abundance models are models in which the dynamics differ between species, described by variation among parameters defining the dynamics. Using a dynamic and heterogeneous species abundance model generating the lognormal species abundance distribution it is first shown that different degrees of heterogeneity may result in equivalent species abundance distributions. An alternative to Preston's canonical lognormal model is defined by assuming that reduction in resources, for example reduction in available area, increases the density regulation of each species. This leads to species–individual curves and species–area curves that are approximately linear in a double logarithmic plot. Preston's canonical parameter γ varies little along these curves and takes values in the neighborhood of one. Quite remarkably, the curves, which define the sensitivity of the community to area reductions, are independent of the heterogeneity among species for this model. As a consequence, the curves can be estimated from a single sample from the community using the Poisson lognormal distribution. It is shown how to perform sensitivity analysis with respect to over-dispersion in sampling relative to the Poisson distribution as well as sampling intensity, that is, the fraction of the community sampled. The method is exemplified by analyzing three simulated data sets.

Effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter of Poisson-gamma models for modeling motor vehicle crashes: A Bayesian perspective

There has been considerable research conducted on the development of statistical models for predicting motor vehicle crashes on highway facilities. Over the last few years, there has been a significant increase in the application hierarchical Bayes methods for modeling motor vehicle crash data. Whether the inferences are estimated using classical or Bayesian methods, the most common probabilistic structure used for modeling this type of data remains the traditional Poisson-gamma (or Negative Binomial) model. Crash data collected for highway safety studies often have the unusual attributes of being characterized by low sample mean values and, due to the prohibitive costs of collecting data, small sample sizes. Previous studies have shown that the dispersion parameter of Poisson-gamma models can be seriously mis-estimated when the models are estimated using the maximum likelihood estimation (MLE) method for these extreme conditions. Despite important work done on this topic for the MLE, nobody has so far examined how low sample mean values and small sample sizes affect the posterior mean of the dispersion parameter of Poisson-gamma models estimated using the hierarchical Bayes method. The inverse dispersion parameter plays an important role in various types of highway safety studies. It is therefore vital to determine the conditions in which the inverse dispersion parameter may be mis-estimated for this category of models. To accomplish the objectives of this study, a simulation framework is developed to generate data from the Poisson-gamma distributions using different values describing the mean, the dispersion parameter, the sample size, and the prior specification. Vague and non-vague prior specifications are tested for determining the magnitude of the biases introduced by low sample mean values and small sample sizes. A series of datasets are also simulated from the Poisson-lognormal distributions, in the light of recent work done by statisticians on this mixed distribution. The study shows that a dataset characterized by a low sample mean combined with a small sample size can seriously affect the estimation of the posterior mean of the dispersion parameter when a vague prior specification is used to characterize the gamma hyper-parameter. The risk of a mis-estimated posterior mean can be greatly minimized when an appropriate non-vague prior distribution is used. Finally, the study shows that Poisson-lognormal models are recommended over Poisson-gamma models when assuming vague priors and whenever crash data characterized by low sample mean values are used for developing crash prediction models.

Quantitative Risk Assessment of Noroviruses in Drinking Water Based on Qualitative Data in Japan

Noroviruses are one of the major causes of viral gastroenteritis in Japan. A quantitative risk assessment was conducted to evaluate the health risk caused by this virus in drinking water. A Monte Carlo analysis was used to calculate both the probability of infection and the disease burden using disability-adjusted life years (DALYs). The concentration of noroviruses in tap water was estimated based on qualitative data and a most probable number (MPN) method with an assumed Poisson lognormal distribution. This numerical method was evaluated using two sets of available count data of Cryptosporidium: that collected from a river and that found in tap water in Japan. The dose-response relationships for noroviruses were estimated using assumed ID50 (10 or 100). The annual risk was higher than the US-EPA acceptable level (10(-4) [infection/ person-year]) but around the WHO level (10(-6) [DALYs/ person-year]). As suggested by others, since microbial concentrations are generally lognormally distributed, the arithmetic mean was directly related to the annual risk, suggesting that the arithmetic mean is more useful in representing the degree of microbial contamination than the geometric mean.

Species Abundance in a Forest Community in South China: A Case of Poisson Lognormal Distribution

Abstract: Case studies on Poisson lognormal distribution of species abundance have been rare, especially in forest communities. We propose a numerical method to fit the Poisson lognormal to the species abundance data at an evergreen mixed forest in the Dinghushan Biosphere Reserve, South China. Plants in the tree, shrub and herb layers in 25 quadrats of 20 m× 20 m, 5 m× 5 m, and 1 m× 1 m were surveyed. Results indicated that: (i) for each layer, the observed species abundance with a similarly small median, mode, and a variance larger than the mean was reverse J-shaped and followed well the zero-truncated Poisson lognormal; (ii) the coefficient of variation, skewness and kurtosis of abundance, and two Poisson lognormal parameters (& and μ) for shrub layer were closer to those for the herb layer than those for the tree layer; and (iii) from the tree to the shrub to the herb layer, the α and the coefficient of variation decreased, whereas diversity increased. We suggest that: (i) the species abundance distributions in the three layers reflects the overall community characteristics; (ii) the Poisson lognormal can describe the species abundance distribution in diverse communities with a few abundant species but many rare species; and (iii) 1/α should be an alternative measure of diversity. (Managing editor: Ya-Qin HAN)

Electrochemical horizons for the Poisson-lognormal distribution of probability theory

The potential application of the basic theory of Poisson-lognormal distribution to electrochemical systems is illustrated by the specific examples of ionic-species identification, contamination of cathode deposits, and membrane failure-time.