k() R function from [kdist]

The K-distribution.

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


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

Returns

The function dk gives the density, pk gives the distribution function, qk gives the quantile function, and rk

generates random variates.

Details

The K-distribution with shape parameter $\nu$ and scale parameter $b$ has amplitude density given by c(" $f(x) = [4 x^\\nu / \\Gamma(\\nu)]\n$ ", " $[(\\nu / b)^(1+\\nu/2)]\n$ ", " $K(2 x \\sqrt(\\nu/b),\\nu-1)$ "). 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 $m$ th moments are given by c(" $\\mu_m = (b/\\nu)^(m/2) \\Gamma(0.5m + 1)\n$ ", " $\\Gamma(0.5m + \\nu) / \\Gamma(\\nu)$ "), so that the root mean square value of x is the scale factor, $<x^2> = b$ .

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))

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

k function