R: Power discrete distribution

dpower {sads} | R Documentation

## Power discrete distribution

### Description

Density, distribution function, quantile function and random generation for discrete version of power distribution with parameter `s`.

### Usage

    dpower( x, s, log=FALSE)
    ppower( q, s, lower.tail=TRUE, log.p=FALSE)
    qpower( p, s, lower.tail= TRUE, log.p=FALSE)
    rpower( n, s)

### Arguments

`x` | vector of (integer x>0) quantiles. In the context of species abundance distributions, this is a vector of abundances of species in a sample.
`q` | vector of (integer x>0) 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.
`s` | positive real s > 1; exponent of the power distribution.
`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

The power density is a discrete probability distribution defined for integer x > 0:

p(x) = x^(-s) / zeta (s)

Hence p(x) is proportional to a negative power of 'x', given by the 's' exponent. The Riemann's zeta function is the integration constant.

The power distribution can be used as a species abundance distribution (sad) model, which describes the probability of the abundance 'x' of a given species in a sample or assemblage of species.

### Value

`dpower` gives the (log) density of the density, `ppower` gives the (log) distribution function, `qpower` gives the quantile function.

Invalid values for parameter `s` will result in return values `NaN`, with a warning.

### Author(s)

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

### References

Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley. Section 11.2.20.

### See Also

`dzeta` in VGAM package; `fitpower` for maximum likelihood estimation in the context of species abundance distributions.

### Examples

    x <- 1:20
    PDF <- dpower(x=x, s=2)
    CDF <- ppower(q=x, s=2)
    par(mfrow=c(1,2))
    plot(x,CDF, ylab="Cumulative Probability", type="b",
         main="Power distribution, CDF")
    plot(x,PDF, ylab="Probability", type="h",
         main="Power distribution, PDF")
    par(mfrow=c(1,1))

    ## The power distribution is a discrete PDF, hence:
    all.equal( ppower(10, s=2), sum(dpower(1:10, s=2)) ) # should be TRUE

    ## quantile is the inverse of CDF
    all.equal(qpower(CDF, s=2), x)

[Package sads version 0.4.2 Index]