# 1dpoilog: Poisson lognormal distribution In poilog: Poisson lognormal and bivariate Poisson lognormal distribution

## Description

Density and random generation for the Poisson lognormal distribution with parameters mu and sig.

## Usage

 1 2 dpoilog(n, mu, sig) rpoilog(S, mu, sig, nu=1, condS=FALSE, keep0=FALSE) 

## Arguments

 n vector of observed individuals for each species S number of species in the community mu mean of lognormal distribution sig standard deviation of lognormal distribution nu sampling intensity, defaults to 1 condS logical; if TRUE random deviates are conditonal on S keep0 logical; if TRUE species with count 0 are included in the random deviates

## Details

The following is written from the perspective of using the Poisson lognormal distribution to describe community structure (the distribution of species when sampling individuals from a community of several species).

Under the assumption of random sampling, the number of individuals sampled from a given species with abundance y, say N, is Poisson distributed with mean \code{nu}\emph{y} where the parameter nu expresses the sampling intensity. If ln y is normally distributed with mean mu and standard deviaton sig among species, then the vector of individuals sampled from all S species then constitutes a sample from the Poisson lognormal distribution with parameters (mu + ln nu, sig), where mu and sig are the mean and standard deviaton of the log abundances. For nu = 1, this is the Poisson lognormal distribution with parameters (mu,sig) which may be written in the form

P(N=\code{n}; \code{mu},\code{sig}) = q(\code{n}; \code{mu},\code{sig}) = \int_-infty^infty g_\code{n}(\code{mu},\code{sig},u) phi(u) du,

where φ(u) is the standard normal distribution and

g_n(\code{mu},\code{sig},u) = exp(\code{mu} \code{sig} \code{n} + \code{mu} \code{n} + exp(-\code{mu} \code{sig} + \code{mu})) / \code{n}!

Since S is usually unknown, we only consider the observed number of individuals for the observed species. With a general sampling intensity nu, the distribution of the number of individuals then follows the zero-truncated Poisson lognormal distribution

q(\code{n}; \code{mu},\code{sig})/(1 - q(0; \code{mu},\code{sig}))

## Value

dpoilog returns the density
rpoilog returns random deviates

## Author(s)

Vidar Grøtan vidar.grotan@bio.ntnu.no and Steinar Engen

## References

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.

poilogMLE for ML estimation
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ### plot density for given parameters barplot(dpoilog(n=0:20,mu=2,sig=1),names.arg=0:20) ### draw random deviates from a community of 50 species rpoilog(S=50,mu=2,sig=1) ### draw random deviates including zeros rpoilog(S=50,mu=2,sig=1,keep0=TRUE) ### draw random deviates with sampling intensity = 0.5 rpoilog(S=50,mu=2,sig=1,nu=0.5) ### how many species are likely to be observed ### (given S,mu,sig2 and nu)? hist(replicate(1000,length(rpoilog(S=30,mu=0,sig=3,nu=0.7)))) ### how many individuals are likely to be observed ### (given S,mu,sig2 and nu)? hist(replicate(1000,sum(rpoilog(S=30,mu=0,sig=3,nu=0.7))))