unif: Uniform distribution on the hypersphere

Description

Density and simulation of the uniform distribution on $S^{p-1}:=\{\mathbf{x}\in R^p:||\mathbf{x}||=1\}$, $p\ge 1$. The density is just the inverse of the surface area of $S^{p-1}$, given by $$\omega_p:=2\pi^{p/2}/\Gamma(p/2).$$

Usage

d_unif_sphere(x, log = FALSE)
r_unif_sphere(n, p)
w_p(p, log = FALSE)

Arguments

locations in $S^{p-1}$ to evaluate the density. Either a matrix of size c(nx, p) or a vector of length p. Normalized internally if required (with a warning message).

log

flag to indicate if the logarithm of the density (or the normalizing constant) is to be computed.

sample size, a positive integer.

dimension of the ambient space $R^p$ that contains $S^{p-1}$. A positive integer.

Value

Depending on the function:

d_unif_sphere: a vector of length nx or 1 with the evaluated density at x.
r_unif_sphere: a matrix of size c(n, p) with the random sample.
w_p: the surface area of $S^{p-1}$.

Details

If $p = 1$, then $S^{0} = \{-1, 1\}$ and the "surface area" is $2$. The function w_p is vectorized on p.

Examples

Run this code

# NOT RUN {
## Area of S^{p - 1}

# Areas of S^0, S^1, and S^2
w_p(p = 1:3)

# Area as a function of p
p <- 1:20
plot(p, w_p(p = p), type = "o", pch = 16, xlab = "p", ylab = "Area",
     main = expression("Surface area of " * S^{p - 1}), axes = FALSE)
box()
axis(1, at = p)
axis(2, at = seq(0, 34, by = 2))

## Simulation and density evaluation for p = 1, 2, 3

# p = 1
n <- 500
x <- r_unif_sphere(n = n, p = 1)
barplot(table(x) / n)
head(d_unif_sphere(x))

# p = 2
x <- r_unif_sphere(n = n, p = 3)
plot(x)
head(d_unif_sphere(x))

# p = 3
x <- r_unif_sphere(n = n, p = 3)
rgl::plot3d(x)
head(d_unif_sphere(x))
# }

Run the code above in your browser using DataCamp Workspace