Description Usage Arguments Details Value Author(s) Examples
Density and simulation of the uniform distribution on S^{p1} := {x \in R^p : x = 1}, p≥ 1. The density is just the inverse of the surface area of S^{p1}, given by
ω_p := 2π^{p/2} / Γ(p/2).
1 2 3 4 5  d_unif_sphere(x, log = FALSE)
r_unif_sphere(n, p)
w_p(p, log = FALSE)

x 
locations in S^{p1} to evaluate the density. Either a
matrix of size 
log 
flag to indicate if the logarithm of the density (or the normalizing constant) is to be computed. 
n 
sample size, a positive integer. 
p 
dimension of the ambient space R^p that contains S^{p1}. A positive integer. 
If p = 1, then S^{0} = \{1, 1\} and the "surface area" is
2. The function w_p
is vectorized on p
.
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^{p1}.
Eduardo GarcíaPortugués, Davy Paindaveine, and Thomas Verdebout.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32  ## 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)
if (requireNamespace("rgl")) {
rgl::plot3d(x)
}
head(d_unif_sphere(x))

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