k {kdist}

The K-distribution.

Description

Density, distribution function, quantile function and random generation for the K-distribution with parameters shape and scale.

Usage

dk(x, shape = 1, scale = 1, intensity = FALSE, log = FALSE)

pk(q, shape = 1, scale = 1, intensity = FALSE, log.p = FALSE,
  lower.tail = TRUE)

qk(p, shape = 1, scale = 1, intensity = FALSE, log.p = FALSE)

rk(n, shape = 1, scale = 1, intensity = FALSE)

Arguments

x, q

vector of quantiles

shape, scale

shape and scale parameters both defaulting to 1.

intensity

logical; if TRUE, quantiles are intensities not amplitudes.

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].

p

vector of probabilities

n

number of observations

Details

The K-distribution with shape parameter \nu and scale parameter b has amplitude density given by f(x) = [4 x^\nu / \Gamma(\nu)] [(\nu / b)^(1+\nu/2)] K(2 x \sqrt(\nu/b),\nu-1). Where K is a modified Bessel function of the second kind. For \nu -> Inf, the K-distrubution tends to a Rayleigh distribution, and for \nu = 1 it is the Exponential distribution. The function base::besselK is used in the calculation, and care should be taken with large input arguements to this function, e.g. b very small or x, \nu very large. The cumulative distribution function for the amplitude, x is given by F(x) = 1 - 2 x^\nu (\nu/b)^(\nu/2) K(2 x \sqrt(\nu/b), \nu). The K-Distribution is a compound distribution, with Rayleigh distributed amplitudes (exponential intensities) modulated by another underlying process whose amplitude is chi-distributed and whose intensity is Gamma distributed. An Exponential distributed number multiplied by a Gamma distributed random number is used to generate the random variates. The mth moments are given by \mu_m = (b/\nu)^(m/2) \Gamma(0.5m + 1) \Gamma(0.5m + \nu) / \Gamma(\nu), so that the root mean square value of x is the scale factor, <x^2> = b.

Value

The function dk gives the density, pk gives the distribution function, qk gives the quantile function, and rk generates random variates.

References

E Jakeman and R J A Tough, "Non-Gaussian models for the statistics of scattered waves", Adv. Phys., 1988, vol. 37, No. 5, pp471-529

See Also

Distributions for other standard distributions, including dweibull for the Weibull distribution and dexp for the exponential distribution.

Examples

#=====
r <- rk(10000, shape = 3, scale = 5, intensity = FALSE)
fn <- stats::ecdf(r)
x <- seq(0, 10, length = 100)
plot(x, fn(x))
lines(x, pk(x, shape = 3, scale = 5, intensity = FALSE))
#======
r <- rk(10000, shape = 3, scale = 5, intensity = FALSE)
d <- density(r)
x <- seq(0, 10, length = 100)
plot(d, xlim=c(0,10))
lines(x, dk(x, shape = 3, scale = 5, intensity = FALSE))