sph_stat_Sobolev function

Finite Sobolev statistics for testing (hyper)spherical uniformity

Computes the finite Sobolev statistic [REMOVE_ME]$S_{n, p}(\{b_{k, p}\}_{k=1}^K) = \sum_{i, j = 1}^n\sum_{k = 1}^K b_{k, p}C_k^(p / 2 - 1)(\cos^{-1}({\bf X}_i'{\bf X}_j)), [REMOVE_ME_2]$

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

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 $\Psi$ that is obtained with Psi_mat(X)? If FALSE (default), $\Psi$ 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 $\Psi$ that is obtained with

  Psi_mat(array(Theta, dim = c(n, 1, M)))? If FALSE

  (default), $\Psi$ is computed internally.

Returns

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

Description

Computes the finite Sobolev statistic

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