proj_unif function

Projection of the spherical uniform distribution

Density, distribution, and quantile functions of the projection of the spherical uniform random variable on an arbitrary direction, that is, the random variable $\gamma'X$, where $X$

is uniformly distributed on the (hyper)sphere c("$S^{p-1}:=\n$", "${x\\in R^p:||x||=1}$"), $p\ge 2$, and $\gamma\in S^{p-1}$ is an arbitrary projection direction. Note that the distribution is invariant to the choice of $\gamma$. Also, efficient simulation of $\gamma'X$.

d_proj_unif(x, p, log = FALSE)

p_proj_unif(x, p, log = FALSE)

q_proj_unif(u, p)

r_proj_unif(n, p)

Arguments

• x: a vector of size nx or a matrix of size c(nx, 1).
• p: integer giving the dimension of the ambient space $R^p$ that contains $S^{p-1}$.
• log: compute the logarithm of the density or distribution?
• u: vector of probabilities.
• n: sample size.

Returns

A matrix of size c(nx, 1) with the evaluation of the density, distribution, or quantile function at x or u. For r_proj_unif, a random vector of size n.

Examples

# Density function
curve(d_proj_unif(x, p = 2), from = -2, to = 2, n = 2e2, ylim = c(0, 2))
curve(d_proj_unif(x, p = 3), n = 2e2, col = 2, add = TRUE)
curve(d_proj_unif(x, p = 4), n = 2e2, col = 3, add = TRUE)
curve(d_proj_unif(x, p = 5), n = 2e2, col = 4, add = TRUE)
curve(d_proj_unif(x, p = 6), n = 2e2, col = 5, add = TRUE)

# Distribution function
curve(p_proj_unif(x, p = 2), from = -2, to = 2, n = 2e2, ylim = c(0, 1))
curve(p_proj_unif(x, p = 3), n = 2e2, col = 2, add = TRUE)
curve(p_proj_unif(x, p = 4), n = 2e2, col = 3, add = TRUE)
curve(p_proj_unif(x, p = 5), n = 2e2, col = 4, add = TRUE)
curve(p_proj_unif(x, p = 6), n = 2e2, col = 5, add = TRUE)

# Quantile function
curve(q_proj_unif(u = x, p = 2), from = 0, to = 1, n = 2e2, ylim = c(-1, 1))
curve(q_proj_unif(u = x, p = 3), n = 2e2, col = 2, add = TRUE)
curve(q_proj_unif(u = x, p = 4), n = 2e2, col = 3, add = TRUE)
curve(q_proj_unif(u = x, p = 5), n = 2e2, col = 4, add = TRUE)
curve(q_proj_unif(u = x, p = 6), n = 2e2, col = 5, add = TRUE)

# Sampling
hist(r_proj_unif(n = 1e4, p = 4), freq = FALSE, breaks = 50)
curve(d_proj_unif(x, p = 4), n = 2e2, col = 3, add = TRUE)

Author(s)

Eduardo García-Portugués and Paula Navarro-Esteban.