dpower: Power discrete distribution

In sads: Maximum Likelihood Models for Species Abundance Distributions

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, bissection = 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.
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].
bissection	logical; uses the bissection method to generate random numbers? If FALSE calls poweRlaw::rpldis, which provides a faster, continuous approximation for large numbers (Gilliespie 2015).
...	further arguments to be passed to poweRlaw::rpldis when bissection = TRUE

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.

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 [e-mail] 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.

Colin S. Gillespie (2015) Fitting Heavy Tailed Distributions: The poweRlaw Package. Journal of Statistical Software, 64(2), 1-16. URL http://www.jstatsoft.org/v64/i02/.

See Also

dzeta in VGAM package; poweRlaw 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) 

sads documentation built on June 22, 2024, 12:18 p.m.