test_rotasym: Tests of rotational symmetry for hyperspherical data
In rotasym: Tests for Rotational Symmetry on the Hypersphere
test_rotasym R Documentation
Tests of rotational symmetry for hyperspherical data
Description
Tests for assessing the rotational symmetry of a unit-norm
random vector \mathbf{X} in
S^{p-1}:=\{\mathbf{x}\in R^p:||\mathbf{x}||=1\}, p \ge 2, about a location
\boldsymbol{\theta}\in S^{p-1}, from a
hyperspherical sample \mathbf{X}_1,\ldots,\mathbf{X}_n\in S^{p-1}.
The vector \mathbf{X} is said to be rotational symmetric about
\boldsymbol{\theta} if the distributions of
\mathbf{OX} and \mathbf{X} coincide, where
\mathbf{O} is any p\times p rotation matrix
that fixes \boldsymbol{\theta}, i.e.,
\mathbf{O}\boldsymbol{\theta}=\boldsymbol{\theta}.
Usage
test_rotasym(data, theta = spherical_mean, type = c("sc", "loc", "loc_vMF",
"hyb", "hyb_vMF")[5], Fisher = FALSE, U = NULL, V = NULL)
Arguments
data
hyperspherical data, a matrix of size c(n, p) with unit
norm rows. Normalized internally if any row does not have unit norm
(with a warning message). NAs are ignored.
theta
either a unit norm vector of size p giving the axis of
rotational symmetry (for the specified-\boldsymbol{\theta}
case) or a function that implements an estimator
\hat{\boldsymbol{\theta}} of
\boldsymbol{\theta} (for the
unspecified-\boldsymbol{\theta} case). The default calls
the spherical_mean function. See examples.
type
a character string (case insensitive) indicating the type of
test to conduct:
-
"sc": "scatter" test based on the statistic
Q_{\boldsymbol{\theta}}^{\mathrm{sc}}. Evaluates if the
covariance matrix of the multivariate signs is isotropic.
-
"loc": "location" test based on the statistic
Q_{\boldsymbol{\theta}}^{\mathrm{loc}}. Evaluates if
the expectation of the multivariate signs is zero.
-
"loc_vMF": adapted "location" test, based on the statistic
Q_{\mathrm{vMF}}^{\mathrm{loc}}.
-
"hyb": "hybrid" test based on the statistics
Q_{\boldsymbol{\theta}}^{\mathrm{sc}} and
Q_{\boldsymbol{\theta}}^{\mathrm{loc}}.
-
"hyb_vMF" (default): adapted "hybrid" test based on the
statistics Q_{\boldsymbol{\theta}}^{\mathrm{sc}} and
Q_{\mathrm{vMF}}^{\mathrm{loc}}.
See the details below for further explanations of the tests.
Fisher
if TRUE, then Fisher's method is employed to aggregate
the scatter and location tests in the hybrid test, see details below.
Otherwise, the hybrid statistic is the sum of the scatter and location
statistics. Defaults to FALSE.
U
multivariate signs of data, a matrix of size
c(n, p - 1). Computed if NULL (the default).
V
cosines of data, a vector of size n. Computed
if NULL (the default).
Details
Descriptions of the tests:
The "scatter" test is locally and asymptotically optimal against
tangent elliptical alternatives to rotational
symmetry. However, it is not consistent against
tangent von Mises–Fisher (vMF) alternatives.
The asymptotic null distribution of
Q_{\boldsymbol{\theta}}^{\mathrm{sc}}
is unaffected if \boldsymbol{\theta} is estimated, that is,
the asymptotic null distributions of
Q_{\boldsymbol{\theta}}^{\mathrm{sc}} and
Q_{\hat{\boldsymbol{\theta}}}^{\mathrm{sc}} are
the same.
The "location" test is locally and asymptotically most powerful
against vMF alternatives to rotational symmetry. However, it is not
consistent against tangent elliptical alternatives. The asymptotic
null distribution of
Q_{\boldsymbol{\theta}}^{\mathrm{loc}}
for known \boldsymbol{\theta} (the one implemented in
test_rotasym) does change if \boldsymbol{\theta}
is estimated by \hat{\boldsymbol{\theta}}. Therefore, if
the test is performed with an estimated \boldsymbol{\theta}
(if theta is a function)
Q_{\hat{\boldsymbol{\theta}}}^{\mathrm{loc}}
will not be properly calibrated. test_rotasym will give a warning in
such case.
The "vMF location" test is a modification of the "location" test
designed to make its null asymptotic distribution invariant from the
estimation of \boldsymbol{\theta} (as the "scatter" test is).
The test is optimal against tangent vMF alternatives with a specific,
vMF-based, angular function g_vMF. Despite not
being optimal against all tangent vMF alternatives, it is
consistent for all of them. As the location test,
it is not consistent against tangent elliptical alternatives.
The "hybrid" test combines (see below how) the "scatter" and
"location" tests. The test is neither optimal against tangent elliptical nor
tangent vMF alternatives, but it is consistent against both. Since it is
based on the "location" test, if computed with an estimator
\hat{\boldsymbol{\theta}}, the test statistic will not
be properly calibrated. test_rotasym will give a warning in such case.
The "vMF hybrid" test is the analogous of the "hybrid" test but
replaces the "location" test by the "vMF location" test.
The combination of the scatter and location tests in the hybrid tests is
done in two different ways:
If Fisher = FALSE, then the scatter and location tests
statistics give the hybrid test statistic
Q^{\mathrm{hyb}}:=Q_{\boldsymbol{\theta}}^{\mathrm{sc}}+
Q_{\boldsymbol{\theta}}^{\mathrm{loc}}.
If Fisher = TRUE, then Fisher's method for aggregating
independent tests (the two test statistics are independent under rotational
symmetry) is considered, resulting the hybrid test statistic:
Q_{\boldsymbol{\theta}}^{\mathrm{hyb}}
:=-2(\log(p_{\mathrm{sc}})+\log(p_{\mathrm{loc}}))
where p_{\mathrm{sc}} and p_{\mathrm{loc}} are
the p-values of the scatter and location tests, respectively.
The hybrid test statistic Q_{\mathrm{vMF}}^{\mathrm{hyb}} follows analogously to
Q_{\boldsymbol{\theta}}^{\mathrm{hyb}} by replacing
Q_{\boldsymbol{\theta}}^{\mathrm{loc}} with
Q_{\mathrm{vMF}}^{\mathrm{loc}}.
Finally, recall that the tests are designed to test implications of
rotational symmetry. Therefore, the tests are not consistent against
all types of alternatives to rotational symmetry.
Value
An object of the htest class with the following elements:
-
statistic: test statistic.
-
parameter: degrees of freedom of the chi-square distribution
appearing in all the null asymptotic distributions.
-
p.value: p-value of the test.
-
method: information on the type of test performed.
-
data.name: name of the value of data.
-
U: multivariate signs of data.
-
V: cosines of data.
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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2019.1665527")}
See Also
tangent-elliptical, tangent-vMF,
spherical_mean.
Examples
## Rotational symmetry holds
# Sample data from a vMF (rotational symmetric distribution about mu)
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
set.seed(123456789)
data_0 <- r_vMF(n = n, mu = theta, kappa = 1)
# theta known
test_rotasym(data = data_0, theta = theta, type = "sc")
test_rotasym(data = data_0, theta = theta, type = "loc")
test_rotasym(data = data_0, theta = theta, type = "loc_vMF")
test_rotasym(data = data_0, theta = theta, type = "hyb")
test_rotasym(data = data_0, theta = theta, type = "hyb", Fisher = TRUE)
test_rotasym(data = data_0, theta = theta, type = "hyb_vMF")
test_rotasym(data = data_0, theta = theta, type = "hyb_vMF", Fisher = TRUE)
# theta unknown (employs the spherical mean as estimator)
test_rotasym(data = data_0, type = "sc")
test_rotasym(data = data_0, type = "loc") # Warning
test_rotasym(data = data_0, type = "loc_vMF")
test_rotasym(data = data_0, type = "hyb") # Warning
test_rotasym(data = data_0, type = "hyb", Fisher = TRUE) # Warning
test_rotasym(data = data_0, type = "hyb_vMF")
test_rotasym(data = data_0, type = "hyb_vMF", Fisher = TRUE)
## Rotational symmetry does not hold
# Sample non-rotational symmetric data from a tangent-vMF distribution
# The scatter test is blind to these deviations, while the location tests
# are optimal
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
mu <- c(rep(0, p - 2), 1)
kappa <- 2
set.seed(123456789)
r_V <- function(n) {
r_g_vMF(n = n, p = p, kappa = 1)
}
data_1 <- r_TM(n = n, r_V = r_V, theta = theta, mu = mu, kappa = kappa)
# theta known
test_rotasym(data = data_1, theta = theta, type = "sc")
test_rotasym(data = data_1, theta = theta, type = "loc")
test_rotasym(data = data_1, theta = theta, type = "loc_vMF")
test_rotasym(data = data_1, theta = theta, type = "hyb")
test_rotasym(data = data_1, theta = theta, type = "hyb", Fisher = TRUE)
test_rotasym(data = data_1, theta = theta, type = "hyb_vMF")
test_rotasym(data = data_1, theta = theta, type = "hyb_vMF", Fisher = TRUE)
# theta unknown (employs the spherical mean as estimator)
test_rotasym(data = data_1, type = "sc")
test_rotasym(data = data_1, type = "loc") # Warning
test_rotasym(data = data_1, type = "loc_vMF")
test_rotasym(data = data_1, type = "hyb") # Warning
test_rotasym(data = data_1, type = "hyb", Fisher = TRUE) # Warning
test_rotasym(data = data_1, type = "hyb_vMF")
test_rotasym(data = data_1, type = "hyb_vMF", Fisher = TRUE)
# Sample non-rotational symmetric data from a tangent-elliptical distribution
# The location tests are blind to these deviations, while the
# scatter test is optimal
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
Lambda <- matrix(0.5, nrow = p - 1, ncol = p - 1)
diag(Lambda) <- 1
set.seed(123456789)
r_V <- function(n) {
r_g_vMF(n = n, p = p, kappa = 1)
}
data_2 <- r_TE(n = n, r_V = r_V, theta = theta, Lambda = Lambda)
# theta known
test_rotasym(data = data_2, theta = theta, type = "sc")
test_rotasym(data = data_2, theta = theta, type = "loc")
test_rotasym(data = data_2, theta = theta, type = "loc_vMF")
test_rotasym(data = data_2, theta = theta, type = "hyb")
test_rotasym(data = data_2, theta = theta, type = "hyb", Fisher = TRUE)
test_rotasym(data = data_2, theta = theta, type = "hyb_vMF")
test_rotasym(data = data_2, theta = theta, type = "hyb_vMF", Fisher = TRUE)
# theta unknown (employs the spherical mean as estimator)
test_rotasym(data = data_2, type = "sc")
test_rotasym(data = data_2, type = "loc") # Warning
test_rotasym(data = data_2, type = "loc_vMF")
test_rotasym(data = data_2, type = "hyb") # Warning
test_rotasym(data = data_2, type = "hyb", Fisher = TRUE) # Warning
test_rotasym(data = data_2, type = "hyb_vMF")
test_rotasym(data = data_2, type = "hyb_vMF", Fisher = TRUE)
## Sunspots births data
# Load data
data("sunspots_births")
sunspots_births$X <-
cbind(cos(sunspots_births$phi) * cos(sunspots_births$theta),
cos(sunspots_births$phi) * sin(sunspots_births$theta),
sin(sunspots_births$phi))
# Test rotational symmetry for the 23rd cycle, specified theta
sunspots_23 <- subset(sunspots_births, cycle == 23)
test_rotasym(data = sunspots_23$X, type = "sc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_23$X, type = "loc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_23$X, type = "hyb", theta = c(0, 0, 1))
# Test rotational symmetry for the 23rd cycle, unspecified theta
spherical_loc_PCA(sunspots_23$X)
test_rotasym(data = sunspots_23$X, type = "sc", theta = spherical_loc_PCA)
test_rotasym(data = sunspots_23$X, type = "loc_vMF",
theta = spherical_loc_PCA)
test_rotasym(data = sunspots_23$X, type = "hyb_vMF",
theta = spherical_loc_PCA)
# Test rotational symmetry for the 22nd cycle, specified theta
sunspots_22 <- subset(sunspots_births, cycle == 22)
test_rotasym(data = sunspots_22$X, type = "sc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_22$X, type = "loc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_22$X, type = "hyb", theta = c(0, 0, 1))
# Test rotational symmetry for the 22nd cycle, unspecified theta
spherical_loc_PCA(sunspots_22$X)
test_rotasym(data = sunspots_22$X, type = "sc", theta = spherical_loc_PCA)
test_rotasym(data = sunspots_22$X, type = "loc_vMF",
theta = spherical_loc_PCA)
test_rotasym(data = sunspots_22$X, type = "hyb_vMF",
theta = spherical_loc_PCA)
rotasym documentation built on Aug. 19, 2023, 9:06 a.m.
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| test_rotasym | R Documentation |
Tests of rotational symmetry for hyperspherical data
Description
Tests for assessing the rotational symmetry of a unit-norm
random vector \mathbf{X} in
S^{p-1}:=\{\mathbf{x}\in R^p:||\mathbf{x}||=1\}, p \ge 2, about a location
\boldsymbol{\theta}\in S^{p-1}, from a
hyperspherical sample \mathbf{X}_1,\ldots,\mathbf{X}_n\in S^{p-1}.
The vector \mathbf{X} is said to be rotational symmetric about
\boldsymbol{\theta} if the distributions of
\mathbf{OX} and \mathbf{X} coincide, where
\mathbf{O} is any p\times p rotation matrix
that fixes \boldsymbol{\theta}, i.e.,
\mathbf{O}\boldsymbol{\theta}=\boldsymbol{\theta}.
Usage
test_rotasym(data, theta = spherical_mean, type = c("sc", "loc", "loc_vMF",
"hyb", "hyb_vMF")[5], Fisher = FALSE, U = NULL, V = NULL)
Arguments
data |
hyperspherical data, a matrix of size |
theta |
either a unit norm vector of size |
type |
a character string (case insensitive) indicating the type of test to conduct:
See the details below for further explanations of the tests. |
Fisher |
if |
U |
multivariate signs of |
V |
cosines of |
Details
Descriptions of the tests:
The "scatter" test is locally and asymptotically optimal against tangent elliptical alternatives to rotational symmetry. However, it is not consistent against tangent von Mises–Fisher (vMF) alternatives. The asymptotic null distribution of
Q_{\boldsymbol{\theta}}^{\mathrm{sc}}is unaffected if\boldsymbol{\theta}is estimated, that is, the asymptotic null distributions ofQ_{\boldsymbol{\theta}}^{\mathrm{sc}}andQ_{\hat{\boldsymbol{\theta}}}^{\mathrm{sc}}are the same.The "location" test is locally and asymptotically most powerful against vMF alternatives to rotational symmetry. However, it is not consistent against tangent elliptical alternatives. The asymptotic null distribution of
Q_{\boldsymbol{\theta}}^{\mathrm{loc}}for known\boldsymbol{\theta}(the one implemented intest_rotasym) does change if\boldsymbol{\theta}is estimated by\hat{\boldsymbol{\theta}}. Therefore, if the test is performed with an estimated\boldsymbol{\theta}(ifthetais a function)Q_{\hat{\boldsymbol{\theta}}}^{\mathrm{loc}}will not be properly calibrated.test_rotasymwill give a warning in such case.The "vMF location" test is a modification of the "location" test designed to make its null asymptotic distribution invariant from the estimation of
\boldsymbol{\theta}(as the "scatter" test is). The test is optimal against tangent vMF alternatives with a specific, vMF-based, angular functiong_vMF. Despite not being optimal against all tangent vMF alternatives, it is consistent for all of them. As the location test, it is not consistent against tangent elliptical alternatives.The "hybrid" test combines (see below how) the "scatter" and "location" tests. The test is neither optimal against tangent elliptical nor tangent vMF alternatives, but it is consistent against both. Since it is based on the "location" test, if computed with an estimator
\hat{\boldsymbol{\theta}}, the test statistic will not be properly calibrated.test_rotasymwill give a warning in such case.The "vMF hybrid" test is the analogous of the "hybrid" test but replaces the "location" test by the "vMF location" test.
The combination of the scatter and location tests in the hybrid tests is done in two different ways:
If
Fisher = FALSE, then the scatter and location tests statistics give the hybrid test statisticQ^{\mathrm{hyb}}:=Q_{\boldsymbol{\theta}}^{\mathrm{sc}}+ Q_{\boldsymbol{\theta}}^{\mathrm{loc}}.If
Fisher = TRUE, then Fisher's method for aggregating independent tests (the two test statistics are independent under rotational symmetry) is considered, resulting the hybrid test statistic:Q_{\boldsymbol{\theta}}^{\mathrm{hyb}} :=-2(\log(p_{\mathrm{sc}})+\log(p_{\mathrm{loc}}))where
p_{\mathrm{sc}}andp_{\mathrm{loc}}are thep-values of the scatter and location tests, respectively.
The hybrid test statistic Q_{\mathrm{vMF}}^{\mathrm{hyb}} follows analogously to
Q_{\boldsymbol{\theta}}^{\mathrm{hyb}} by replacing
Q_{\boldsymbol{\theta}}^{\mathrm{loc}} with
Q_{\mathrm{vMF}}^{\mathrm{loc}}.
Finally, recall that the tests are designed to test implications of rotational symmetry. Therefore, the tests are not consistent against all types of alternatives to rotational symmetry.
Value
An object of the htest class with the following elements:
-
statistic: test statistic. -
parameter: degrees of freedom of the chi-square distribution appearing in all the null asymptotic distributions. -
p.value:p-value of the test. -
method: information on the type of test performed. -
data.name: name of the value ofdata. -
U: multivariate signs ofdata. -
V: cosines ofdata.
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2019.1665527")}
See Also
tangent-elliptical, tangent-vMF,
spherical_mean.
Examples
## Rotational symmetry holds
# Sample data from a vMF (rotational symmetric distribution about mu)
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
set.seed(123456789)
data_0 <- r_vMF(n = n, mu = theta, kappa = 1)
# theta known
test_rotasym(data = data_0, theta = theta, type = "sc")
test_rotasym(data = data_0, theta = theta, type = "loc")
test_rotasym(data = data_0, theta = theta, type = "loc_vMF")
test_rotasym(data = data_0, theta = theta, type = "hyb")
test_rotasym(data = data_0, theta = theta, type = "hyb", Fisher = TRUE)
test_rotasym(data = data_0, theta = theta, type = "hyb_vMF")
test_rotasym(data = data_0, theta = theta, type = "hyb_vMF", Fisher = TRUE)
# theta unknown (employs the spherical mean as estimator)
test_rotasym(data = data_0, type = "sc")
test_rotasym(data = data_0, type = "loc") # Warning
test_rotasym(data = data_0, type = "loc_vMF")
test_rotasym(data = data_0, type = "hyb") # Warning
test_rotasym(data = data_0, type = "hyb", Fisher = TRUE) # Warning
test_rotasym(data = data_0, type = "hyb_vMF")
test_rotasym(data = data_0, type = "hyb_vMF", Fisher = TRUE)
## Rotational symmetry does not hold
# Sample non-rotational symmetric data from a tangent-vMF distribution
# The scatter test is blind to these deviations, while the location tests
# are optimal
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
mu <- c(rep(0, p - 2), 1)
kappa <- 2
set.seed(123456789)
r_V <- function(n) {
r_g_vMF(n = n, p = p, kappa = 1)
}
data_1 <- r_TM(n = n, r_V = r_V, theta = theta, mu = mu, kappa = kappa)
# theta known
test_rotasym(data = data_1, theta = theta, type = "sc")
test_rotasym(data = data_1, theta = theta, type = "loc")
test_rotasym(data = data_1, theta = theta, type = "loc_vMF")
test_rotasym(data = data_1, theta = theta, type = "hyb")
test_rotasym(data = data_1, theta = theta, type = "hyb", Fisher = TRUE)
test_rotasym(data = data_1, theta = theta, type = "hyb_vMF")
test_rotasym(data = data_1, theta = theta, type = "hyb_vMF", Fisher = TRUE)
# theta unknown (employs the spherical mean as estimator)
test_rotasym(data = data_1, type = "sc")
test_rotasym(data = data_1, type = "loc") # Warning
test_rotasym(data = data_1, type = "loc_vMF")
test_rotasym(data = data_1, type = "hyb") # Warning
test_rotasym(data = data_1, type = "hyb", Fisher = TRUE) # Warning
test_rotasym(data = data_1, type = "hyb_vMF")
test_rotasym(data = data_1, type = "hyb_vMF", Fisher = TRUE)
# Sample non-rotational symmetric data from a tangent-elliptical distribution
# The location tests are blind to these deviations, while the
# scatter test is optimal
n <- 200
p <- 10
theta <- c(1, rep(0, p - 1))
Lambda <- matrix(0.5, nrow = p - 1, ncol = p - 1)
diag(Lambda) <- 1
set.seed(123456789)
r_V <- function(n) {
r_g_vMF(n = n, p = p, kappa = 1)
}
data_2 <- r_TE(n = n, r_V = r_V, theta = theta, Lambda = Lambda)
# theta known
test_rotasym(data = data_2, theta = theta, type = "sc")
test_rotasym(data = data_2, theta = theta, type = "loc")
test_rotasym(data = data_2, theta = theta, type = "loc_vMF")
test_rotasym(data = data_2, theta = theta, type = "hyb")
test_rotasym(data = data_2, theta = theta, type = "hyb", Fisher = TRUE)
test_rotasym(data = data_2, theta = theta, type = "hyb_vMF")
test_rotasym(data = data_2, theta = theta, type = "hyb_vMF", Fisher = TRUE)
# theta unknown (employs the spherical mean as estimator)
test_rotasym(data = data_2, type = "sc")
test_rotasym(data = data_2, type = "loc") # Warning
test_rotasym(data = data_2, type = "loc_vMF")
test_rotasym(data = data_2, type = "hyb") # Warning
test_rotasym(data = data_2, type = "hyb", Fisher = TRUE) # Warning
test_rotasym(data = data_2, type = "hyb_vMF")
test_rotasym(data = data_2, type = "hyb_vMF", Fisher = TRUE)
## Sunspots births data
# Load data
data("sunspots_births")
sunspots_births$X <-
cbind(cos(sunspots_births$phi) * cos(sunspots_births$theta),
cos(sunspots_births$phi) * sin(sunspots_births$theta),
sin(sunspots_births$phi))
# Test rotational symmetry for the 23rd cycle, specified theta
sunspots_23 <- subset(sunspots_births, cycle == 23)
test_rotasym(data = sunspots_23$X, type = "sc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_23$X, type = "loc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_23$X, type = "hyb", theta = c(0, 0, 1))
# Test rotational symmetry for the 23rd cycle, unspecified theta
spherical_loc_PCA(sunspots_23$X)
test_rotasym(data = sunspots_23$X, type = "sc", theta = spherical_loc_PCA)
test_rotasym(data = sunspots_23$X, type = "loc_vMF",
theta = spherical_loc_PCA)
test_rotasym(data = sunspots_23$X, type = "hyb_vMF",
theta = spherical_loc_PCA)
# Test rotational symmetry for the 22nd cycle, specified theta
sunspots_22 <- subset(sunspots_births, cycle == 22)
test_rotasym(data = sunspots_22$X, type = "sc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_22$X, type = "loc", theta = c(0, 0, 1))
test_rotasym(data = sunspots_22$X, type = "hyb", theta = c(0, 0, 1))
# Test rotational symmetry for the 22nd cycle, unspecified theta
spherical_loc_PCA(sunspots_22$X)
test_rotasym(data = sunspots_22$X, type = "sc", theta = spherical_loc_PCA)
test_rotasym(data = sunspots_22$X, type = "loc_vMF",
theta = spherical_loc_PCA)
test_rotasym(data = sunspots_22$X, type = "hyb_vMF",
theta = spherical_loc_PCA)
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