Poisson lognormal distribution

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Description

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

Usage

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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

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### 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))))