ACG: Angular central Gaussian distribution

In rotasym: Tests for Rotational Symmetry on the Hypersphere

Angular central Gaussian distribution

Description

Density and simulation of the Angular Central Gaussian (ACG) distribution on \mathcal{S}^{p-1}:=\{\boldsymbol{x}\in \mathbb{R}^p:\|\boldsymbol{x}\|=1\}, p\ge 1. The density at \boldsymbol{x} \in \mathcal{S}^{p-1}, p\ge 2, is given by

c^{\mathrm{ACG}}_{p,\boldsymbol{\Lambda}} (\boldsymbol{x}' \boldsymbol{\Lambda}^{-1} \boldsymbol{x})^{-p/2} \quad\mathrm{with}\quad c^{\mathrm{ACG}}_{p,\boldsymbol{\Lambda}}:= 1 / (\omega_p |\boldsymbol{\Lambda}|^{1/2})

where \boldsymbol{\Lambda} is the shape matrix, a p\times p symmetric and positive definite matrix, and \omega_p is the surface area of \mathcal{S}^{p-1}.

Usage

d_ACG(x, Lambda, log = FALSE)

c_ACG(p, Lambda, log = FALSE)

r_ACG(n, Lambda)

Arguments

x	locations in \mathcal{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).
Lambda	the shape matrix \boldsymbol{\Lambda} of the ACG. A symmetric and positive definite matrix of size c(p, p).
log	flag to indicate if the logarithm of the density (or the normalizing constant) is to be computed.
p	dimension of the ambient space \mathbb{R}^p that contains \mathcal{S}^{p-1}. A positive integer.
n	sample size, a positive integer.

Details

Due to the projection of the ACG, the shape matrix \boldsymbol{\Lambda} is only identified up to a constant, that is, \boldsymbol{\Lambda} and c\boldsymbol{\Lambda} give the same ACG distribution. Usually, \boldsymbol{\Lambda} is normalized to have trace equal to p.

c_ACG is vectorized on p. If p = 1, then the ACG is the uniform distribution in the set \{-1, 1\}.

Value

Depending on the function:

• d_ACG: a vector of length nx or 1 with the evaluated density at x.

• r_ACG: a matrix of size c(n, p) with the random sample.

• c_ACG: the normalizing constant.

References

Tyler, D. E. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika, 74(3):579–589. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/74.3.579")}

See Also

tangent-elliptical, unif.

Examples

# Simulation and density evaluation for p = 2
Lambda <- diag(c(5, 1))
n <- 1e3
x <- r_ACG(n = n, Lambda = Lambda)
col <- viridisLite::viridis(n)
r <- runif(n, 0.95, 1.05) # Radius perturbation to improve visualization
dens <- d_ACG(x = x, Lambda = Lambda)
plot(r * x, pch = 16, col = col[rank(dens)])

# Simulation and density evaluation for p = 3
Lambda <- rbind(c(5, 1, 0.5),
                c(1, 2, 1),
                c(0.5, 1, 1))
x <- r_ACG(n = n, Lambda = Lambda)
dens <- d_ACG(x = x, Lambda = Lambda)
scatterplot3d::scatterplot3d(x, xlim = c(-1, 1), ylim = c(-1, 1),
                             zlim = c(-1, 1), color = col[rank(dens)],
                             pch = 16, xlab = "", ylab = "", zlab = "",
                             angle = 20)

rotasym documentation built on July 26, 2026, 9:06 a.m.