rotasym (version 1.0-6)

tangent-elliptical: Tangent elliptical distribution

Description

Density and simulation of the Tangent Elliptical (TE) distribution on \(S^{p-1}:=\{\mathbf{x}\in R^p:||\mathbf{x}||=1\}\), \(p\ge 2\). The distribution arises by considering the tangent-normal decomposition with multivariate signs distributed as an Angular Central Gaussian distribution.

Usage

d_TE(x, theta, g_scaled, d_V, Lambda, log = FALSE)
r_TE(n, theta, r_V, Lambda)

Arguments

x

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).

theta

a unit norm vector of size p giving the axis of rotational symmetry.

g_scaled

the scaled angular density \(c_g g\). In the form g_scaled <- function(t, log = TRUE) {...}. See examples.

d_V

the density \(f_V\). In the form d_V <- function(v, log = TRUE) {...}. See examples.

Lambda

the shape matrix \(\boldsymbol{\Lambda}\) of the ACG used in the multivariate signs. A symmetric and positive definite matrix of size c(p - 1, p - 1).

log

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

n

sample size, a positive integer.

r_V

a function for simulating \(V\). Its first argument must be the sample size. See examples.

Value

Depending on the function:

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

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

Details

The functions are wrappers for d_tang_norm and r_tang_norm with d_U = d_ACG and r_U = r_ACG.

References

Garc<U+00ED>a-Portugu<U+00E9>s, E., Paindaveine, D., Verdebout, T. (2019) On optimal tests for rotational symmetry against new classes of hyperspherical distributions. arXiv:1706.05030. https://arxiv.org/abs/1706.05030

See Also

tang-norm-decomp, tangent-vMF, ACG.

Examples

# NOT RUN {
## Simulation and density evaluation for p = 2

# Parameters
p <- 2
n <- 1e3
theta <- c(rep(0, p - 1), 1)
Lambda <- matrix(0.5, nrow = p - 1, ncol = p - 1)
diag(Lambda) <- 1
kappa_V <- 2

# Required functions
r_V <- function(n) r_g_vMF(n = n, p = p, kappa = kappa_V)
g_scaled <- function(t, log) {
  g_vMF(t, p = p - 1, kappa = kappa_V, scaled = TRUE, log = log)
}

# Sample and color according to density
x <- r_TE(n = n, theta = theta, r_V = r_V, Lambda = Lambda)
col <- viridisLite::viridis(n)
r <- runif(n, 0.95, 1.05) # Radius perturbation to improve visualization
dens <- d_TE(x = x, theta = theta, g_scaled = g_scaled, Lambda = Lambda)
plot(r * x, pch = 16, col = col[rank(dens)])

## Simulation and density evaluation for p = 3

# Parameters
p <- 3
n <- 5e3
theta <- c(rep(0, p - 1), 1)
Lambda <- matrix(0.5, nrow = p - 1, ncol = p - 1)
diag(Lambda) <- 1
kappa_V <- 2

# Sample and color according to density
x <- r_TE(n = n, theta = theta, r_V = r_V, Lambda = Lambda)
col <- viridisLite::viridis(n)
dens <- d_TE(x = x, theta = theta, g_scaled = g_scaled, Lambda = Lambda)
rgl::plot3d(x, col = col[rank(dens)], size = 5)

## A non-vMF angular function: g(t) = 1 - t^2. It is sssociated to the
## Beta(1/2, (p + 1)/2) distribution.

# Scaled angular function
g_scaled <- function(t, log) {
  log_c_g <- lgamma(0.5 * p) + log(0.5 * p / (p - 1)) - 0.5 * p * log(pi)
  log_g <- log_c_g + log(1 - t^2)
  switch(log + 1, exp(log_g), log_g)
}

# Simulation
r_V <- function(n) {
  sample(x = c(-1, 1), size = n, replace = TRUE) *
    sqrt(rbeta(n = n, shape1 = 0.5, shape2 = 0.5 * (p + 1)))
}

# Sample and color according to density
kappa_V <- 1
Lambda <- matrix(0.75, nrow = p - 1, ncol = p - 1)
diag(Lambda) <- 1
x <- r_TE(n = n, theta = theta, r_V = r_V, Lambda = Lambda)
col <- viridisLite::viridis(n)
dens <- d_TE(x = x, theta = theta, g_scaled = g_scaled, Lambda = Lambda)
rgl::plot3d(x, col = col[rank(dens)], size = 5)
# }