dpoig {sads} | R Documentation |

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

[Package *sads* version 0.4.2 Index]