Haar: Uniform distribution

Description

Density, distribution function and random generation for the uniform distribution.

Usage

dhaar(r)
phaar(q, lower.tail = TRUE)
rhaar(n)

Value

dhaar: gives the density
phaar: gives the distribution function
rhaar: generates random deviates

Arguments

r, q: vector of quantiles.
lower.tail: logical; if TRUE (default), probabilities are $P(X \le x)$ otherwise, $P(X > x)$.
n: number of observations. If length(n)>1, the length is taken to be the number required.

Details

The uniform distribution has density $$C_U(r)=\frac{[1-cos(r)]}{2\pi}$$ with respect to the Lebesgue measure. The Haar measure is the volume invariant measure for SO(3) that plays the role of the uniform measure on SO(3) and $C_U(r)$ is the angular distribution that corresponds to the uniform distribution on SO(3), see UARS. The uniform distribution with respect to the Haar measure is given by $$C_U(r)=\frac{1}{2\pi}.$$ Because the uniform distribution with respect to the Haar measure gives a horizontal line at 1 with respect to the Lebesgue measure, we called this distribution 'Haar.'

Examples

Run this code

r <- seq(-pi, pi, length = 1000)

#Visualize the uniform distribution with respect to Lebesgue measure
plot(r, dhaar(r), type = "l", ylab = "f(r)")

#Visualize the uniform distribution with respect to Haar measure, which is
#a horizontal line at 1
plot(r, 2*pi*dhaar(r)/(1-cos(r)), type = "l", ylab = "f(r)")

#Plot the uniform CDF
plot(r,phaar(r), type = "l", ylab = "F(r)")

#Generate random observations from uniform distribution
rs <- rhaar(50)

#Visualize on the real line
hist(rs, breaks = 10)