rotasym (version 1.0-5)

cosines-signs: Cosines and multivariate signs of a hyperspherical sample about a given location

Description

Computation of the cosines and multivariate signs of the hyperspherical sample \(\mathbf{X}_1,\ldots,\mathbf{X}_n\in S^{p-1}\) about a location \(\boldsymbol{\theta}\in S^{p-1}\), for \(S^{p-1}:=\{\mathbf{x}\in R^p:||\mathbf{x}||=1\}\) with \(p\ge 2\). The cosines are defined as $$V_i:=\mathbf{X}_i'\boldsymbol{\theta},\quad i=1,\ldots,n,$$ whereas the multivariate signs are the vectors \(\mathbf{U}_1,\ldots,\mathbf{U}_n\in S^{p-2}\) defined as $$\mathbf{U}_i := \boldsymbol{\Gamma}_{\boldsymbol{\theta}}\mathbf{X}_i/ ||\boldsymbol{\Gamma}_{\boldsymbol{\theta}}\mathbf{X}_i||,\quad i=1,\ldots,n.$$ The projection matrix \(\boldsymbol{\Gamma}_{\boldsymbol{\theta}}\) is a \(p\times (p-1)\) semi-orthogonal matrix that satisfies $$\boldsymbol{\Gamma}_{\boldsymbol{\theta}}' \boldsymbol{\Gamma}_{\boldsymbol{\theta}}=\mathbf{I}_{p-1} \quad\mathrm{and}\quad\boldsymbol{\Gamma}_{\boldsymbol{\theta}} \boldsymbol{\Gamma}_{\boldsymbol{\theta}}'= \mathbf{I}_p-\boldsymbol{\theta}\boldsymbol{\theta}'.$$ where \(\mathbf{I}_p\) 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 \(\mathbf{I}_p-\boldsymbol{\theta}\boldsymbol{\theta}'\) (inefficient). Defaults to FALSE.

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).

Details

Note that the projection matrix \(\boldsymbol{\Gamma}_{\boldsymbol{\theta}}\) is not unique. In particular, any completion of \(\boldsymbol{\theta}\) to an orthonormal basis \(\{\boldsymbol{\theta},\mathbf{v}_1,\ldots,\mathbf{v}_{p-1}\}\) gives a set of \(p-1\) orthonormal \(p\)-vectors \(\{\mathbf{v}_1,\ldots,\mathbf{v}_{p-1}\}\) that conform the columns of \(\boldsymbol{\Gamma}_{\boldsymbol{\theta}}\). If eig = FALSE, this approach is employed by rotating the canonical completion of \(\mathbf{e}_1=(1,0,\ldots,0)\), \(\{\mathbf{e}_2,\ldots,\mathbf{e}_p\}\), by the rotation matrix that rotates \(\mathbf{e}_1\) to \(\boldsymbol{\theta}\): $$\mathbf{H}_{\boldsymbol{\theta}}= (\boldsymbol{\theta}+\mathbf{e}_1)(\boldsymbol{\theta}+\mathbf{e}_1)'/ (1+\theta_1)-\mathbf{I}_p.$$ If eig = TRUE, then a much more expensive eigendecomposition of \(\boldsymbol{\Gamma}_{\boldsymbol{\theta}} \boldsymbol{\Gamma}_{\boldsymbol{\theta}}'= \mathbf{I}_p-\boldsymbol{\theta}\boldsymbol{\theta}'\) is performed for determining \(\{\mathbf{v}_1,\ldots,\mathbf{v}_{p-1}\}\).

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.

References

Garc<U+00ED>a-Portugu<U+00E9>s, E., Paindaveine, D., Verdebout, T. (2019) On optimal tests for rotational symmetry against new classes of hyperspherical distributions. arXiv:1706.05030. https://arxiv.org/abs/1706.05030

Examples

# NOT RUN {
# 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)
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))

# 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)
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))
# }