tangent-vMF: Tangent von Mises-Fisher distribution
In rotasym: Tests for Rotational Symmetry on the Hypersphere

Description Usage Arguments Details Value Author(s) References See Also Examples

Density and simulation of the Tangent von Mises–Fisher (TM) distribution on S^{p-1} := {x \in R^p : ||x|| = 1}, p≥ 2. The distribution arises by considering the tangent-normal decomposition with multivariate signs distributed as a von Mises–Fisher distribution.

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d_TM(x, theta, g_scaled, d_V, mu, kappa, log = FALSE)

r_TM(n, theta, r_V, mu, kappa)

`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.
`mu`	the directional mean μ of the vMF used in the multivariate signs. A unit-norm vector of length `p - 1`.
`kappa`	concentration parameter κ of the vMF used in the multivariate signs. A nonnegative scalar.
`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.

The functions are wrappers for d_tang_norm and r_tang_norm with d_U = d_vMF and r_U = r_vMF.

Depending on the function:

d_TM: a vector of length nx or 1 with the evaluated density at x.
r_TM: a matrix of size c(n, p) with the random sample.

Eduardo García-Portugués, Davy Paindaveine, and Thomas Verdebout.

García-Portugués, E., Paindaveine, D., Verdebout, T. (2020) On optimal tests for rotational symmetry against new classes of hyperspherical distributions. Journal of the American Statistical Association, 115(532):1873–1887. doi: 10.1080/01621459.2019.1665527

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

## Simulation and density evaluation for p = 2

# Parameters
p <- 2
n <- 1e3
theta <- c(rep(0, p - 1), 1)
mu <- c(rep(0, p - 2), 1)
kappa <- 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_TM(n = n, theta = theta, r_V = r_V, mu = 1, kappa = kappa)
col <- viridisLite::viridis(n)
r <- runif(n, 0.95, 1.05) # Radius perturbation to improve visualization
dens <- d_TM(x = x, theta = theta, g_scaled = g_scaled, mu = mu,
             kappa = kappa)
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)
mu <- c(rep(0, p - 2), 1)
kappa <- 1
kappa_V <- 2

# Sample and color according to density
x <- r_TM(n = n, theta = theta, r_V = r_V, mu = mu, kappa = kappa)
col <- viridisLite::viridis(n)
dens <- d_TM(x = x, theta = theta, g_scaled = g_scaled, mu = mu,
             kappa = kappa)
if (requireNamespace("rgl")) {
  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 <- 0.5
x <- r_TM(n = n, theta = theta, r_V = r_V, mu = mu, kappa = kappa)
col <- viridisLite::viridis(n)
dens <- d_TM(x = x, theta = theta, g_scaled = g_scaled,
             mu = mu, kappa = kappa)
if (requireNamespace("rgl")) {
  rgl::plot3d(x, col = col[rank(dens)], size = 5)
}