Package 'kdist'

Title: K-Distribution and Weibull Paper
Description: Density, distribution function, quantile function and random generation for the K-distribution. A plotting function that plots data on Weibull paper and another function to draw additional lines. See results from package in T Lamont-Smith (2018), submitted J. R. Stat. Soc.
Authors: Tim Lamont-Smith [aut, cre]
Maintainer: Tim Lamont-Smith <[email protected]>
License: GPL-3
Version: 0.2
Built: 2025-02-16 05:15:54 UTC
Source: https://github.com/cran/kdist

Help Index

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 ν\nu and scale parameter bb has amplitude density given by f(x)=[4xν/Γ(ν)][(ν/b)(1+ν/2)]K(2x(ν/b),ν−1)f(x) = [4 x^\nu / \Gamma(\nu)] [(\nu / b)^(1+\nu/2)] K(2 x \sqrt(\nu/b),\nu-1). Where KK is a modified Bessel function of the second kind. For ν−>Inf\nu -> Inf, the K-distrubution tends to a Rayleigh distribution, and for ν=1\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. bb very small or x,νx, \nu very large. The cumulative distribution function for the amplitude, xx is given by F(x)=1−2xν(ν/b)(ν/2)K(2x(ν/b),ν)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 mmth moments are given by μm=(b/ν)(m/2)Γ(0.5m+1)Γ(0.5m+ν)/Γ(ν)\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, <x2>=b<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))

kdist: A package for calculating and plotting non-Gaussian distributions

Description

The kdist package provides two categories of important functions: dk etc, and weiplot.

dk functions

The kdist functions dk, pk, qk and rk, calculates the K-distribution

weiplot functions

weiplot takes data and plots it on Weibull paper. Weilines adds lines to a Weibull plot.

Add Lines onto a Weibull Plot

Description

Weibull distributed data plots as a straight line on log-log plot using wlines(). It is best used after function wplot() has been called.

Usage

weilines(x, y, lty = NULL, lwd = NULL, col = "black", type = "l",
  pch = 0)

Arguments

x

vector of values
y

vector of values the same length as x
lty

line type
lwd

line width
col

line color
type

type of plotting
pch

symbol type for type = "b"

Details

A Weibull plot uses log paper and has log(1/(1-F(x)) versus x, where the data values x have an empirical cdf of F(x). The plot margins may need to be adjusted so that the right hand axis is visible.

See Also

wplot() creates the Weibull plot

Examples

dummy <- c(0,0)
weiplot(dummy, xlim = c(1e-3, 10), type = "n")
x <- 10^seq(-3, 2, length = 100)
weilines(x, pexp(x), col = "red")
weilines(x, pweibull(x, 2), col = "blue")
weilines(x, pweibull(x, 3), col = "green")

Create Weibull Plot.

Description

A special type of plot where Weibull distributed data plots as a straight line. This was also originally called Rayleigh paper. Both Rayleigh and exponential distributions also plot as straight lines.

Usage

weiplot(data, n = 70, type = "p", xlim = NULL, ylim = c(0.01, 10),
  main = "Weibull Plot", sub = NULL, ylab = "log(1/1-F(x))",
  ylab2 = "F(x)", xlab = "x", percent = "False")

Arguments

data

data values from which a cumulative density function will be estimated using ecdf(data)
n

number of points required in plot (default n = 70).
type

plot type
xlim

the minimum and maximum to be used for the x-axis
ylim

the minimum and maximum to be used for the y-axis
main

the title of the plot
sub

the sub-title of the plot
ylab

the title of the left y-axis
ylab2

the title of the right y-axis
xlab

the title of the x-axis
percent

logical; display right hand axis as percentages

Details

A Weibull plot uses log paper and has log(1/(1-F(x)) versus x, where the data values x have an empirical cdf of F(x). The plot margins may need to be adjusted so that the right hand axis is visible.

See Also

weilines() adds lines to a Weibull plot

Examples

graphics::par(mar = c(5, 5, 5, 5))
r <- rexp(100000)
weiplot(r, xlim = c(1e-3, 10))
x <- 10^seq(-3, 2, length = 100)
weilines(x, pexp(x))