sphunif (version 1.4.3)

sph_stat_Rayleigh: Statistics for testing (hyper)spherical uniformity

Description

Low-level implementation of several statistics for assessing uniformity on the (hyper)sphere \(S^{p-1}:=\{{\bf x}\in R^p:||{\bf x}||=1\}\), \(p\ge 2\).

Usage

sph_stat_Rayleigh(X)
sph_stat_Bingham(X)
sph_stat_Ajne(X, Psi_in_X = FALSE)
sph_stat_Gine_Gn(X, Psi_in_X = FALSE, p = 0L)
sph_stat_Gine_Fn(X, Psi_in_X = FALSE, p = 0L)
sph_stat_Pycke(X, Psi_in_X = FALSE, p = 0L)
sph_stat_Bakshaev(X, Psi_in_X = FALSE, p = 0L)
sph_stat_Riesz(X, Psi_in_X = FALSE, p = 0L, s = 1)
sph_stat_PCvM(X, Psi_in_X = FALSE, p = 0L, N = 160L, L = 1000L)
sph_stat_PRt(X, Psi_in_X = FALSE, p = 0L, t = 1/3, N = 160L,
  L = 1000L)
sph_stat_PAD(X, Psi_in_X = FALSE, p = 0L, N = 160L, L = 1000L)
sph_stat_Poisson(X, Psi_in_X = FALSE, p = 0L, rho = 0.5)
sph_stat_Softmax(X, Psi_in_X = FALSE, p = 0L, kappa = 1)
sph_stat_Stereo(X, Psi_in_X = FALSE, p = 0L, a = 0)
sph_stat_CCF09(X, dirs, K_CCF09 = 25L, original = FALSE)
sph_stat_Rayleigh_HD(X)
sph_stat_CJ12(X, Psi_in_X = FALSE, p = 0L, regime = 3L)

Value

A matrix of size c(M, 1) containing the statistics for each of the M samples.

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 \(S^{p-1}\).

s

\(s\) parameter for the \(s\)-Riesz test, a real in \((0, 2)\). Defaults to 1.

N

number of points used in the Gauss-Legendre quadrature. Defaults to 160.

L

number of discretization points to interpolate angular functions that require evaluating an integral. Defaults to 1e3.

t

\(t\) parameter for the Rothman and Cressie tests, a real in \((0, 1)\). Defaults to 1 / 3.

rho

\(\rho\) parameter for the Poisson test, a real in \([0, 1)\). Defaults to 0.5.

kappa

\(\kappa\) parameter for the Softmax test, a non-negative real. Defaults to 1.

a

either:

• \(a_n = a / n\) parameter used in the length of the arcs of the coverage-based tests. Must be positive. Defaults to 2 * pi.

• \(a\) parameter for the Stereo test, a real in \([-1, 1]\). Defaults to 0.

dirs

a matrix of size c(n_proj, p) containing n_proj random directions (in Cartesian coordinates) on \(S^{p-1}\) to perform the CCF09 test.

K_CCF09

integer giving the truncation of the series present in the asymptotic distribution of the Kolmogorov-Smirnov statistic. Defaults to 25.

original

return the CCF09 statistic as originally defined? If FALSE (default), a faster and equivalent statistic is computed, and rejection happens for large values of the statistic, which is consistent with the rest of tests. Otherwise, rejection happens for low values.

regime

type of asymptotic regime for the CJ12 test, either 1 (sub-exponential regime), 2 (exponential), or 3 (super-exponential; default).

Warning

Be careful on avoiding the next bad usages of the functions, which will produce spurious results:

• The directions in X do not have unit norm.

• X does not contain Psi_mat(X) when X_in_Theta = TRUE.

• The parameter p does not match with the dimension of \(R^p\).

• Not passing the scalar products matrix to sph_stat_CJ12 when Psi_in_X = TRUE.

• The directions in dirs do not have unit norm.

Details

Detailed descriptions and references of the statistics are available in García-Portugués and Verdebout (2018).

The Pycke and CJ12 statistics employ the scalar products matrix, rather than the shortest angles matrix, when Psi_in_X = TRUE. This matrix is obtained by setting scalar_prod = TRUE in Psi_mat.

References

García-Portugués, E. and Verdebout, T. (2018) An overview of uniformity tests on the hypersphere. arXiv:1804.00286. tools:::Rd_expr_doi("10.48550/arXiv.1804.00286")

Examples

## Sample uniform spherical data

M <- 2
n <- 100
p <- 3
set.seed(123456789)
X <- r_unif_sph(n = n, p = p, M = M)

## Sobolev tests

# Rayleigh
sph_stat_Rayleigh(X)

# Bingham
sph_stat_Bingham(X)

# Ajne
Psi <- Psi_mat(X)
dim(Psi) <- c(dim(Psi), 1)
sph_stat_Ajne(X)
sph_stat_Ajne(Psi, Psi_in_X = TRUE)

# Gine Gn
sph_stat_Gine_Gn(X)
sph_stat_Gine_Gn(Psi, Psi_in_X = TRUE, p = p)

# Gine Fn
sph_stat_Gine_Fn(X)
sph_stat_Gine_Fn(Psi, Psi_in_X = TRUE, p = p)

# Pycke
sph_stat_Pycke(X)
sph_stat_Pycke(Psi, Psi_in_X = TRUE, p = p)

# Bakshaev
sph_stat_Bakshaev(X)
sph_stat_Bakshaev(Psi, Psi_in_X = TRUE, p = p)

# Riesz
sph_stat_Riesz(X, s = 1)
sph_stat_Riesz(Psi, Psi_in_X = TRUE, p = p, s = 1)

# Projected Cramér-von Mises
sph_stat_PCvM(X)
sph_stat_PCvM(Psi, Psi_in_X = TRUE, p = p)

# Projected Rothman
sph_stat_PRt(X)
sph_stat_PRt(Psi, Psi_in_X = TRUE, p = p)

# Projected Anderson-Darling
sph_stat_PAD(X)
sph_stat_PAD(Psi, Psi_in_X = TRUE, p = p)

## Other tests

# CCF09
dirs <- r_unif_sph(n = 3, p = p, M = 1)[, , 1]
sph_stat_CCF09(X, dirs = dirs)

## High-dimensional tests

# Rayleigh HD-Standardized
sph_stat_Rayleigh_HD(X)

# CJ12
sph_stat_CJ12(X, regime = 1)
sph_stat_CJ12(Psi, regime = 1, Psi_in_X = TRUE, p = p)
sph_stat_CJ12(X, regime = 2)
sph_stat_CJ12(Psi, regime = 2, Psi_in_X = TRUE, p = p)
sph_stat_CJ12(X, regime = 3)
sph_stat_CJ12(Psi, regime = 3, Psi_in_X = TRUE, p = p)