tangent-vMF | R Documentation |
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.
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 |
theta |
a unit norm vector of size |
g_scaled |
the scaled angular density c_g g. In the
form |
d_V |
the density f_V. In the form
|
mu |
the directional mean μ of the vMF
used in the multivariate signs. A unit-norm vector of length |
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) }
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