R: Poisson lognormal distribution

Poisson lognormal {poilog} | R Documentation

## Poisson lognormal distribution

### Description

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

### Usage

    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.

### See Also

`poilogMLE` for ML estimation

### Examples

    ### 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))))

[Package poilog version 0.4 Index]