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 $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_n\in \mathcal{S}^{p-1}$ 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)

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).

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.

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. tools:::Rd_expr_doi("10.1080/01621459.2019.1665527")

Examples

Run this code

# 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)

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