Description Usage Arguments Details Value Author(s) References Examples
Computation of the cosines and multivariate signs of the hyperspherical sample X_1, …, X_n \in S^{p1} about a location θ \in S^{p1}, for S^{p1} := {x\in R^p : x = 1} with p≥ 2. The cosines are defined as
V_i := X_i'θ, i = 1, …, n,
whereas the multivariate signs are the vectors U_1, …, U_n \in S^{p2} defined as
U_i := Γ_θ X_i / Γ_θ X_i, i = 1, …, n.
The projection matrix Γ_θ is a p x (p1) semiorthogonal matrix that satisfies
Γ_θ'Γ_θ = I_{p1} and Γ_θΓ_θ' = I_p  θθ',
where I_p is the identity matrix of dimension p.
1 2 3 4 5 
X 
hyperspherical data, a matrix of size 
theta 
a unitnorm vector of length 
Gamma 
output from 
check_X 
whether to check the unit norms on the rows of 
eig 
whether
Γ_θ is to be found using an eigendecomposition of
I_p  θθ' (inefficient). Defaults to 
Note that the projection matrix
Γ_θ is not
unique. In particular, any completion of θ
to an orthonormal basis
{θ, v_1, …, v_{p1}} gives a set of p1 orthonormal
pvectors
{v_1, …, v_{p1}} that conform the columns of
Γ_θ. If
eig = FALSE
, this approach is employed by rotating the canonical
completion of e_1=(1, 0, …, 0),
{e_1, …, e_p}, by the rotation matrix that rotates
e_1 to θ:
H_θ = (θ + e_1)(θ + e_1)' / (1 + θ_1)  I_p.
If eig = TRUE
, then a much more expensive
eigendecomposition of
Γ_θΓ_θ' = I_p  θθ' is performed for
determining
{v_1, …, v_{p1}}.
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.
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
Γ_θ of size
c(p, p  1)
.
Eduardo GarcíaPortugués, Davy Paindaveine, and Thomas Verdebout.
GarcíaPortugués, 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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  # 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))

Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.