dpoilog: Poisson-lognormal distribution

Description

Density, distribution function and quantile function for Poisson-lognormal distribution with parameters mu and sigma.

Usage

dpoilog( x, mu, sig, log=FALSE)
ppoilog( q, mu, sig, lower.tail=TRUE, log.p=FALSE)
qpoilog( p, mu, sig, lower.tail = TRUE, log.p = FALSE)

Arguments

vector of (non-negative integer) quantiles. Usually a vector of abundances of species in a sample.

vector of probabilities.

mu,sig

parameters of the compounding lognormal distribution (see details).

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X <= x],="" otherwise,="" p[x=""> x].

Value

'dpoilog' gives the (log) density of the density, 'ppoilog' gives the (log) distribution function, 'qpoilog' gives the quantile function.

source

The three functions were built from dpoilog function from poilog package (Vidar Grøtan{Grotan} and Steinar Engen). dpoilog is just a wrapper of poilog::dpoilog with an additional log argument.

ppoilog does the cumulative sum of poilog::dpoilog.

qpoilog uses modified bisection method to find numerically quantiles using ppoilog, and is extremely slow for a large number of quantiles.

encoding

UTF-8

Details

A compound Poisson-lognormal distribution is a Poisson probability distribution where its single parameter lambda is a random variable with lognormal distribution. The density function is

$$p(x) = \frac{e^{x \mu + x^2 \sigma/2} (2 \pi \sigma)^{-1/2}}{x!} \, g(y)$$

where $$g(y) = \int_{-\infty}^\infty \, e^{-e^y} \frac{e^{(-y-\mu-x \sigma)^2}}{2 \sigma} \, dy$$

(Bulmer 1974 eq.5). For x = 0, 1, 2, ... .

In ecology, this distribution gives the probability that a species has an abundance of x individuals in a random sample of a fraction 'f' of the community. In the community, the species abundances are independent random variables that follow a lognormal density function, with parameters (mu + ln(f), sigma) (Engen et al. 2002).

Hence, a Poisson-lognormal distribution is a model for species abundances distributions (SAD) in a sample taken from a community under the assumptions: (a) species abundances in the community are independent identically distributed lognormal variables, (b) sampling is a Poisson process with expected value E[x]= f*n where n is the abundance in the community and f the fraction of individuals sampled, (c) individuals are sampled with replacement, or the fraction of total individuals sampled is small enough to approximate a sample with replacement. See Engen (1977) and Alonso et al. (2008) for critical evaluations.

References

Alonso, D. and Ostling, A., and Etienne, R. S. 2008 The implicit assumption of symmetry and the species abundance distribution. Ecology Letters, 11: 93-105.

Bulmer,M. G. 1974. On Fitting the Poisson Lognormal Distribution to Species-Abundance Data. Biometrics, 30: 101-110.

Grøtan{Grotan} V. and Engen S. 2008. poilog: Poisson lognormal and bivariate Poisson lognormal distribution. R package version 0.4.

Engen, S. 1977. Comments on two different approaches to the analysis of species frequency data. Biometrics, 33: 205-213. Engen, S., R. Lande, T. Walla & P. J. DeVries. 2002. Analyzing spatial structure of communities using the two-dimensional Poisson lognormal species abundance model. American Naturalist 160: 60-73.