Cayley: The symmetric Cayley distribution

In rotations: Working with Rotation Data

The symmetric Cayley distribution

Description

Density, distribution function and random generation for the Cayley distribution with concentration kappa κ.

Usage

dcayley(r, kappa = 1, nu = NULL, Haar = TRUE)

pcayley(q, kappa = 1, nu = NULL, lower.tail = TRUE)

rcayley(n, kappa = 1, nu = NULL)

Arguments

r, q	vector of quantiles.
kappa	concentration parameter.
nu	circular variance, can be used in place of kappa.
Haar	logical; if TRUE density is evaluated with respect to the Haar measure.
lower.tail	logical; if TRUE (default) probabilities are P(X≤ x) otherwise, P(X>x).
n	number of observations. If length(n)>1, the length is taken to be the number required.

Details

The symmetric Cayley distribution with concentration κ has density

C(r |κ)= Γ(κ+2)(1+cos r)^κ(1-cos r)/[Γ(κ+1/2)2^(κ+1)√π].

The Cayley distribution is equivalent to the de la Vallee Poussin distribution of Schaeben1997.

Schaeben1997 leon2006

Value

dcayley	gives the density
pcayley	gives the distribution function
rcayley	generates a vector of random deviates

See Also

Angular-distributions for other distributions in the rotations package.

Examples

r <- seq(-pi, pi, length = 500)

#Visualize the Cayley density fucntion with respect to the Haar measure
plot(r, dcayley(r, kappa = 10), type = "l", ylab = "f(r)")

#Visualize the Cayley density fucntion with respect to the Lebesgue measure
plot(r, dcayley(r, kappa = 10, Haar = FALSE), type = "l", ylab = "f(r)")

#Plot the Cayley CDF
plot(r,pcayley(r,kappa = 10), type = "l", ylab = "F(r)")

#Generate random observations from Cayley distribution
rs <- rcayley(20, kappa = 1)
hist(rs, breaks = 10)

rotations documentation built on June 25, 2022, 1:06 a.m.