estimators {rotasym} | R Documentation |
Estimators for the axis of rotational symmetry
Description
Estimation of the axis of rotational symmetry
of a rotational symmetric unit-norm
random vector
in
,
, from a
hyperspherical sample
.
Usage
spherical_mean(data)
spherical_loc_PCA(data)
Arguments
data |
hyperspherical data, a matrix of size |
Details
The spherical_mean
estimator computes the sample mean of
and normalizes it
by its norm (if the norm is different from zero). It estimates consistently
for rotational symmetric models based on
angular functions
that are
monotone increasing.
The estimator in spherical_loc_PCA
is based on the fact that, under
rotational symmetry, the expectation of
is
for certain constants
. Therefore,
is the eigenvector with unique
multiplicity of the expectation of
. Its
use is recommended if the rotationally symmetric data is not unimodal.
Value
A vector of length p
with an estimate for
.
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
Examples
# Sample from a vMF
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
set.seed(123456789)
data <- r_vMF(n = n, mu = theta, kappa = 3)
theta_mean <- spherical_mean(data)
theta_PCA <- spherical_loc_PCA(data)
sqrt(sum((theta - theta_mean)^2)) # More efficient
sqrt(sum((theta - theta_PCA)^2))
# Sample from a mixture of antipodal vMF's
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
set.seed(123456789)
data <- rbind(r_vMF(n = n, mu = theta, kappa = 3),
r_vMF(n = n, mu = -theta, kappa = 3))
theta_mean <- spherical_mean(data)
theta_PCA <- spherical_loc_PCA(data)
sqrt(sum((theta - theta_mean)^2))
sqrt(sum((theta - theta_PCA)^2)) # Better suited in this case