PEkit (version 1.0.0.1000)

dPD: The Poisson-Dirichlet distribution

Description

Distribution function for the Poisson-Dirichlet distribution.

Usage

dPD(abund, psi)

Arguments

abund

An abundance vector.

psi

Dispersal parameter \(\psi\). Accepted input values are positive real numbers, "a" for absolute value \(\psi\)=1 by default, or "r" for relative value \(\psi=n\), where \(n\) is the size of the input sample.

Value

The probability of the Poisson-Dirichlet distribution for the input abundance vector, e.g. an exchangeable random partition, and a dispersal parameter \(\psi\).

Details

Given an abundance vector abunds, calculates the probability of a data vector x given by the Poisson-Dirichlet distribution. The higher the dispersal parameter \(\psi\), the higher the amount of distinct observed species. In terms of the paintbox process, a high \(\psi\) increases the size of the continuous part \(p_0\) of the process, while a low \(\psi\) will increase the size of the discrete parts \(p_{\neq 0}\).

References

W.J. Ewens, The sampling theory of selectively neutral alleles, Theoretical Population Biology, Volume 3, Issue 1, 1972, Pages 87-112, ISSN 0040-5809, <10.1016/0040-5809(72)90035-4>.

Examples

# NOT RUN {
## Get a random sample from the Poisson Dirichlet distribution, and
## find the probability of such a sample with psi=5:
set.seed(111)
s <- rPD(n=100,psi=5)
a=abundance(s)
dPD(a, psi=5)

# }