tangent-vMF {rotasym}

Tangent von Mises–Fisher distribution

Description

Density and simulation of the Tangent von Mises–Fisher (TM) 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 a von Mises–Fisher distribution.

Usage

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

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

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.

mu

the directional mean \boldsymbol{\mu} of the vMF used in the multivariate signs. A unit-norm vector of length p - 1.

kappa

concentration parameter \kappa 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.

Details

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

Value

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.

Author(s)

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

References

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

See Also

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

Examples

## 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)
}