Density, distribution function, quantile function and random generation for
Poisson-lognormal distribution with parameters `mu`

and `sigma`

.

1 2 3 4 |

`x` |
vector of (non-negative integer) quantiles. Usually a vector of abundances of species in a sample. |

`q` |
vector of (non-negative integer) quantiles. Usually a vector of abundances of species in a sample. |

`n` |
number of random values to return. |

`p` |
vector of probabilities. |

`mu,sig` |
parameters of the compounding lognormal distribution (see details). |

`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-lognormal distribution is a Poisson probability distribution where its single parameter lambda is a random variable with lognormal distribution. The density function is

*p(x) = (exp(x*mu + x^2*sig/2)*(2*pi*sig)^(-1/2))/x! * g(y)*

where

*g(y) = int_-infty^infty exp(-exp(y))*exp(((-y-mu-x*sig)^2)/(2*sig)) dy*

(Bulmer 1974 eq.5). For x = 0, 1, 2, ... .

In ecology, this distribution gives the probability that a species has an abundance of x individuals in a random sample of a fraction 'f' of the community. In the community, the species abundances are independent random variables that follow a lognormal density function, with parameters (mu + ln(f), sigma) (Engen et al. 2002).

Hence, a Poisson-lognormal distribution is a model for species abundances distributions (SAD) in a sample taken from a community under the assumptions: (a) species abundances in the community are independent identically distributed lognormal variables, (b) sampling is a Poisson process with expected value E[x]= f*n where n is the abundance in the community and f the fraction of individuals sampled, (c) individuals are sampled with replacement, or the fraction of total individuals sampled is small enough to approximate a sample with replacement. See Engen (1977) and Alonso et al. (2008) for critical evaluations.

'dpoilog' gives the (log) density of the density, 'ppoilog' gives the (log) distribution function, 'qpoilog' gives the quantile function.

Paulo I. Prado prado@ib.usp.br, Andre Chalom and Murilo Dantas Miranda

These functions were built from `dpoilog`

function from poilog
package (Vidar Grøtan and Steinar Engen).

`dpoilog`

is just a wrapper of `poilog::dpoilog`

with an additional `log`

argument.

`ppoilog`

does the cumulative sum of `poilog::dpoilog`

.

`qpoilog`

uses modified bisection method to find numerically quantiles using
`ppoilog`

.

`rpoilog`

selects random values from a poilog distribution. It is unrelated to the
function of the same name in poilog.

Alonso, D. and Ostling, A., and Etienne, R. S. 2008 The implicit
assumption of symmetry and the species abundance
distribution. *Ecology Letters, 11*: 93-105.

Bulmer,M. G. 1974. On Fitting the Poisson Lognormal Distribution to
Species-Abundance Data. *Biometrics, 30*: 101-110.

Grøtan V. and Engen S. 2008. poilog: Poisson lognormal and bivariate Poisson lognormal distribution. R package version 0.4.

Engen, S. 1977. Comments on two different approaches to the analysis
of species frequency data. *Biometrics, 33*: 205-213.

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.

dpois, dlnorm; dpoilog and rpoilog in poilog package; `rsad`

for random
generation, `fitpoilog`

for maximum likelihood estimation.

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