unif_cap function

Uniform spherical cap distribution

Density, simulation, and associated functions for a uniform distribution within a spherical cap of angle $0\leq \alpha\leq \pi$ about a direction $\mu$ on $S^{p-1}:={x \in R^p:||x||=1}$, $p \geq 2$. The density at $x \in S^{p-1}$ is given by [REMOVE_ME]$c_{p,r} 1_{[1 - r, 1]}(\mathbf{x}' \boldsymbol{\mu}) \quad\mathrm{with}\quad c_{p,r} := \omega_{p}\left[1 - F_p(1 - r)\right],c_{p,r} 1_{[1 - r, 1]}(x' \mu) with \quadc_{p,r}:=\omega_p \left[1 - F_p(1 - r)\right], [REMOVE_ME_2]$ where $r=\cos(\alpha)$ is the projected radius of the spherical cap about $\mu$, $\omega_p$ is the surface area of $S^{p-1}$, and $F_p$ is the projected uniform distribution (see p_proj_unif).

The angular function of the uniform cap distribution is $g(t):=1_{[1 - r, 1]}(t)$. The associated projected density is c("$\\tilde{g}(t):=\\omega_{p-1}\n$", "$c_{p,r} (1 - t^2)^{(p - 3) / 2} 1_{[1 - r, 1]}(t)$").

d_unif_cap(x, mu, angle = pi/10)

c_unif_cap(p, angle = pi/10)

r_unif_cap(n, mu, angle = pi/10)

p_proj_unif_cap(x, p, angle = pi/10)

q_proj_unif_cap(u, p, angle = pi/10)

d_proj_unif_cap(x, p, angle = pi/10, scaled = TRUE)

r_proj_unif_cap(n, p, angle = pi/10)

Arguments

• x: locations to evaluate the density or distribution. For d_unif_cap, positions on $S^{p - 1}$ given either as a matrix of size c(nx, p) or a vector of length p. Normalized internally if required (with a warning message). For d_unif_proj_cap and p_unif_proj_cap, a vector with values in $[-1, 1]$.
• mu: the directional mean $\mu$ of the distribution. A unit-norm vector of length p.
• angle: angle $\alpha$ defining the spherical cap about $\mu$. A scalar in $[0, \pi]$. Defaults to $\pi / 10$.
• p: integer giving the dimension of the ambient space $R^p$ that contains $S^{p-1}$.
• n: sample size, a positive integer.
• u: vector of probabilities.
• scaled: whether to scale the angular function by the normalizing constant. Defaults to TRUE.

Returns

Depending on the function:

• d_unif_cap: a vector of length nx or 1 with the evaluated density at x.
• r_unif_cap: a matrix of size c(n, p) with the random sample.
• c_unif_cap: the normalizing constant.
• p_proj_unif_cap: a vector of length x with the evaluated distribution function at x.
• q_proj_unif_cap: a vector of length u with the evaluated quantile function at u.
• d_proj_unif_cap: a vector of size nx with the evaluated angular function.
• r_proj_unif_cap: a vector of length n containing simulated values from the cosines density associated to the angular function.

Description

Density, simulation, and associated functions for a uniform distribution within a spherical cap of angle $0\leq \alpha\leq \pi$ about a direction $\mu$ on $S^{p-1}:={x \in R^p:||x||=1}$, $p \geq 2$. The density at $x \in S^{p-1}$ is given by

where $r=\cos(\alpha)$ is the projected radius of the spherical cap about $\mu$, $\omega_p$ is the surface area of $S^{p-1}$, and $F_p$ is the projected uniform distribution (see p_proj_unif).

The angular function of the uniform cap distribution is $g(t):=1_{[1 - r, 1]}(t)$. The associated projected density is c("$\\tilde{g}(t):=\\omega_{p-1}\n$", "$c_{p,r} (1 - t^2)^{(p - 3) / 2} 1_{[1 - r, 1]}(t)$").

Examples

# Simulation and density evaluation for p = 2
mu <- c(0, 1)
angle <- pi / 5
n <- 1e2
x <- r_unif_cap(n = n, mu = mu, angle = angle)
col <- viridisLite::viridis(n)
r_noise <- runif(n, 0.95, 1.05) # Perturbation to improve visualization
color <- col[rank(d_unif_cap(x = x, mu = mu, angle = angle))]
plot(r_noise * x, pch = 16, col = color, xlim = c(-1, 1), ylim = c(-1, 1))

# Simulation and density evaluation for p = 3
mu <- c(0, 0, 1)
angle <- pi / 5
x <- r_unif_cap(n = n, mu = mu, angle = angle)
color <- col[rank(d_unif_cap(x = x, mu = mu, angle = angle))]
scatterplot3d::scatterplot3d(x, size = 5, xlim = c(-1, 1), ylim = c(-1, 1),
                             zlim = c(-1, 1), color = color)

# Simulated data from the cosines density
n <- 1e3
p <- 3
angle <- pi / 3
hist(r_proj_unif_cap(n = n, p = p, angle = angle),
     breaks = seq(cos(angle), 1, l = 10), probability = TRUE,
     main = "Simulated data from proj_unif_cap", xlab = "t", xlim = c(-1, 1))
t <- seq(-1, 1, by = 0.01)
lines(t, d_proj_unif_cap(x = t, p = p, angle = angle), col = "red")

Author(s)

Alberto Fernández-de-Marcos and Eduardo García-Portugués.