R: Compound Poisson-gamma distribution

dpoig {sads} | R Documentation

## Compound Poisson-gamma distribution

### Description

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

### Usage

    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)

### Arguments

`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].

### Details

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.

### Value

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

### Author(s)

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

### References

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.

### See Also

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

[Package sads version 0.4.2 Index]