cosines-signs: Cosines and multivariate signs of a hyperspherical sample...

In rotasym: Tests for Rotational Symmetry on the Hypersphere

Cosines and multivariate signs of a hyperspherical sample about a given location

Description

Computation of the cosines and multivariate signs of the hyperspherical sample \boldsymbol{X}_1,\ldots,\boldsymbol{X}_n\in \mathcal{S}^{p-1} about a location \boldsymbol{\theta}\in \mathcal{S}^{p-1}, for \mathcal{S}^{p-1}:=\{\boldsymbol{x}\in \mathbb{R}^p:\|\boldsymbol{x}\|=1\} with p\ge 2. The cosines are defined as

V_i:=\boldsymbol{X}_i'\boldsymbol{\theta},\quad i=1,\ldots,n,

whereas the multivariate signs are the vectors \boldsymbol{U}_1,\ldots,\boldsymbol{U}_n\in \mathcal{S}^{p-2} defined as

\boldsymbol{U}_i := \boldsymbol{\Gamma}_{\boldsymbol{\theta}}\boldsymbol{X}_i/ \|\boldsymbol{\Gamma}_{\boldsymbol{\theta}}\boldsymbol{X}_i\|,\quad i=1,\ldots,n.

The projection matrix \boldsymbol{\Gamma}_{\boldsymbol{\theta}} is a p\times (p-1) semi-orthogonal matrix that satisfies

\boldsymbol{\Gamma}_{\boldsymbol{\theta}}' \boldsymbol{\Gamma}_{\boldsymbol{\theta}}=\boldsymbol{I}_{p-1} \quad\mathrm{and}\quad\boldsymbol{\Gamma}_{\boldsymbol{\theta}} \boldsymbol{\Gamma}_{\boldsymbol{\theta}}'= \boldsymbol{I}_p-\boldsymbol{\theta}\boldsymbol{\theta}'.

where \boldsymbol{I}_p is the identity matrix of dimension p.

Usage

signs(X, theta, Gamma = NULL, check_X = FALSE)

cosines(X, theta, check_X = FALSE)

Gamma_theta(theta, eig = FALSE)

Arguments

X	hyperspherical data, a matrix of size c(n, p) with unit-norm rows. NAs are allowed.
theta	a unit-norm vector of length p. Normalized internally if it does not have unit norm (with a warning message).
Gamma	output from Gamma_theta(theta = theta). If NULL (default), it is computed internally.
check_X	whether to check the unit norms on the rows of X. Defaults to FALSE for performance reasons.
eig	whether \boldsymbol{\Gamma}_{\boldsymbol{\theta}} is to be found using an eigendecomposition of \boldsymbol{I}_p-\boldsymbol{\theta}\boldsymbol{\theta}' (inefficient). Defaults to FALSE.

Details

Note that the projection matrix \boldsymbol{\Gamma}_{\boldsymbol{\theta}} is not unique. In particular, any completion of \boldsymbol{\theta} to an orthonormal basis \{\boldsymbol{\theta},\boldsymbol{v}_1,\ldots,\boldsymbol{v}_{p-1}\} gives a set of p-1 orthonormal p-vectors \{\boldsymbol{v}_1,\ldots,\boldsymbol{v}_{p-1}\} that conform the columns of \boldsymbol{\Gamma}_{\boldsymbol{\theta}}. If eig = FALSE, this approach is employed by rotating the canonical completion of \boldsymbol{e}_1=(1,0,\ldots,0), \{\boldsymbol{e}_2,\ldots,\boldsymbol{e}_p\}, by the rotation matrix that rotates \boldsymbol{e}_1 to \boldsymbol{\theta}:

\boldsymbol{H}_{\boldsymbol{\theta}}= (\boldsymbol{\theta}+\boldsymbol{e}_1) (\boldsymbol{\theta}+\boldsymbol{e}_1)'/(1+\theta_1)-\boldsymbol{I}_p.

If eig = TRUE, then a much more expensive eigendecomposition of \boldsymbol{\Gamma}_{\boldsymbol{\theta}} \boldsymbol{\Gamma}_{\boldsymbol{\theta}}'= \boldsymbol{I}_p-\boldsymbol{\theta}\boldsymbol{\theta}' is performed for determining \{\boldsymbol{v}_1,\ldots,\boldsymbol{v}_{p-1}\}.

If signs and cosines are called with X without unit norms in the rows, then the results will be spurious. Setting check_X = TRUE prevents this from happening.

Value

Depending on the function:

• cosines: a vector of length n with the cosines of X.

• signs: a matrix of size c(n, p - 1) with the multivariate signs of X.

• Gamma_theta: a projection matrix \boldsymbol{\Gamma}_{\boldsymbol{\theta}} of size c(p, p - 1).

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

tang-norm-decomp, test_rotasym.

Examples

# Gamma_theta
theta <- c(0, 1)
Gamma_theta(theta = theta)

# Signs and cosines for p = 2
L <- rbind(c(1, 0.5),
           c(0.5, 1))
X <- r_ACG(n = 1e3, Lambda = L)
old_par <- par(mfrow = c(1, 2))
plot(signs(X = X, theta = theta), main = "Signs", xlab = expression(x[1]),
     ylab = expression(x[2]))
hist(cosines(X = X, theta = theta), prob = TRUE, main = "Cosines",
     xlab = expression(x * "'" * theta))
par(old_par)

# Signs and cosines for p = 3
L <- rbind(c(2, 0.25, 0.25),
           c(0.25, 0.5, 0.25),
           c(0.25, 0.25, 0.5))
X <- r_ACG(n = 1e3, Lambda = L)
old_par <- par(mfrow = c(1, 2))
theta <- c(0, 1, 0)
plot(signs(X = X, theta = theta), main = "Signs", xlab = expression(x[1]),
     ylab = expression(x[2]))
hist(cosines(X = X, theta = theta), prob = TRUE, main = "Cosines",
     xlab = expression(x * "'" * theta))
par(old_par)

rotasym documentation built on July 26, 2026, 9:06 a.m.