sph_stat_Sobolev {sphunif}

Finite Sobolev statistics for testing (hyper)spherical uniformity

Description

Computes the finite Sobolev statistic

S_{n, p}(\{b_{k, p}\}_{k=1}^K) = \sum_{i, j = 1}^n \sum_{k = 1}^K b_{k, p}C_k^(p / 2 - 1)(\cos^{-1}({\bf X}_i'{\bf X}_j)),

for a sequence \{b_{k, p}\}_{k = 1}^K of non-negative weights. For p = 2, the Gegenbauer polynomials are replaced by Chebyshev ones.

Usage

sph_stat_Sobolev(X, Psi_in_X = FALSE, p = 0, vk2 = c(0, 0, 1))

cir_stat_Sobolev(Theta, Psi_in_Theta = FALSE, vk2 = c(0, 0, 1))

Arguments

X

an array of size c(n, p, M) containing the Cartesian coordinates of M samples of size n of directions on S^{p-1}. Must not contain NA's.

Psi_in_X

does X contain the shortest angles matrix \boldsymbol\Psi that is obtained with Psi_mat(X)? If FALSE (default), \boldsymbol\Psi is computed internally.

p

integer giving the dimension of the ambient space R^p that contains S^{p-1}.

vk2

weights for the finite Sobolev test. A non-negative vector or matrix. Defaults to c(0, 0, 1).

Theta

a matrix of size c(n, M) with M samples of size n of circular data on [0, 2\pi). Must not contain NA's.

Psi_in_Theta

does Theta contain the shortest angles matrix \boldsymbol\Psi that is obtained with
Psi_mat(array(Theta, dim = c(n, 1, M)))? If FALSE (default), \boldsymbol\Psi is computed internally.

Value

A matrix of size c(M, ncol(vk2)) containing the statistics for each of the M samples.