sads (version 0.2.2)

dpower: Power discrete distribution

Description

Density, distribution function and quantile function 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)

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.

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].

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.

Details

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

$$p(x) = \frac{x^{-s}}{\zeta (s)}$$

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.

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)