Haar {rotations}

Uniform distribution

Description

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

Usage

dhaar(r)

phaar(q, lower.tail = TRUE)

rhaar(n)

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.'

Value

dhaar

gives the density

phaar

gives the distribution function

rhaar

generates random deviates

See Also

Angular-distributions for other distributions in the rotations package.

Examples

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)