dpoix: Compound Poisson-Exponential distribution

Description

Density, distribution function and quantile function for the Poisson-exponential compound probability distribution with parameters fraction and rate.

Usage

dpoix(x, frac, rate, log=FALSE)
ppoix(q, frac, rate, lower.tail=TRUE, log.p=FALSE)
qpoix(p, frac, rate, S=30, lower.tail=TRUE, log.p=FALSE)

Arguments

vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundances of species in a sample.

vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundances of species in a sample.

vector of probabilities.

frac

single numeric 0 < frac

rate

vector of (non-negative) rates of the exponential distribution of the sampled population (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].

positive integer; initial value for bisection method. (see note)

Value

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

Details

A compound Poisson-exponential distribution is a Poisson probability distribution where its single parameter lambda, is frac*n, at which n is a random variable with exponential distribution. Thus, the expected value and variance are E[X] = Var[X] = frac*n . The density function is

p(y) = rate*frac^y / (frac + rate)^(y+1)

for x = 0, 1, 2, ...(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 an exponential density function.

Hence, a Poisson-exponential 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 exponential variables, (b) sampling is a Poisson process with expected value 'frac*n', (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 critic evaluations.

Notice that the Poisson-exponential can be seen as a different form for the MacArthur's Broken stick model (Baczkowski, 2000), so instead of fitting to a Poisson-exponential distribution directly, the user should use fitbs.

References

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.

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