dpoig: Compound Poisson-gamma distribution
In sads: Maximum Likelihood Models for Species Abundance Distributions

Description Usage Arguments Details Value Author(s) References See Also

Density, distribution function, quantile function and random generation for for the Poisson-gamma compound probability distribution with parameters frac, rate and rate.

dpoig(x, frac, rate, shape, log=FALSE)
ppoig(q, frac, rate, shape, lower.tail=TRUE, log.p=FALSE)
qpoig(p, frac, rate, shape, lower.tail=TRUE, log.p=FALSE)
rpoig(n, frac, rate, shape)

`x`	vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundances of species in a sample.
`q`	vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundances of species in a sample.
`n`	number of random values to return.
`p`	vector of probabilities.
`frac`	single numeric '0<frac<=1'; fraction of the population or community sampled (see details)
`rate`	vector of (non-negative) rates of the gamma distribution of the sampled population (see details). Must be strictly positive.
`shape`	the shape parameter of the gamma distribution of the sampled population (see details). Must be positive.
`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].

A compound Poisson-gamma distribution is a Poisson probability distribution where its single parameter, the process mean rate, is frac*n, at which n is a random variable with gamma distribution. The density function is given by Green & Plotkin (2007).

In ecology, this distribution gives the probability that a species has an abundance of x individuals in a random sample of a fraction frac of the community. In the community the species abundances are independent random variables that follow a gamma density function.

Hence, a Poisson-gamma distribution is a model for species abundances distributions (SAD) under the assumptions: (a) species abundances in the community are independent identically distributed gamma variables, (b) sampling is a Poisson process with expected value frac*n, (c) the sampling is done with replacement, or the fraction sampled is small enough to approximate a sample with replacement.

The Poisson-gamma distribution is also known as the Negative Binomial distribution. The function dpoig is provided to express the Negative Binomial explicitly as a compound distribution. The Fisher log-series (Fisher 1943) is a limiting case where the dispersion parameter of the Negative Binomial tends to zero. As in the case of the Poisson-exponential, the user should not fit to the Poisson-gamma directly, and should use instead the fitls function.

(log) density of the (zero-truncated) density

Paulo I Prado prado@ib.usp.br, Cristiano Strieder and Andre Chalom.

Fisher, R.A, Corbert, A.S. and Williams, C.B. (1943) The Relation between the number of species and the number of individuals in a random sample of an animal population. The Journal of Animal Ecology, 12:42–58.

Green,J. and Plotkin, J.B. 2007 A statistical theory for sampling species abundances. Ecology Letters 10:1037–1045

Pielou, E.C. 1977. Mathematical Ecology. New York: John Wiley and Sons.

dgamma, dpois, dnbinom for related distributions, dpoix for the special case of Poisson-exponential

sads documentation built on May 2, 2019, 1:56 p.m.