unif_stat: Circular and (hyper)spherical uniformity statistics

In sphunif: Uniformity Tests on the Circle, Sphere, and Hypersphere

Circular and (hyper)spherical uniformity statistics

Description

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, for a sample {\bf X}_1,\ldots,{\bf X}_n\in S^{p-1}.

unif_stat receives a (several) sample(s) of directions in Cartesian coordinates, except for the circular case (p=2) in which the sample(s) can be angles \Theta_1,\ldots,\Theta_n\in [0, 2\pi).

unif_stat allows to compute several statistics to several samples within a single call, facilitating thus Monte Carlo experiments.

Usage

unif_stat(data, type = "all", data_sorted = FALSE, CCF09_dirs = NULL,
  CJ12_reg = 3, cov_a = 2 * pi, Cressie_t = 1/3, K_CCF09 = 25,
  Poisson_rho = 0.5, Pycke_q = 0.5, Rayleigh_m = 1, Riesz_s = 1,
  Rothman_t = 1/3, Sobolev_vk2 = c(0, 0, 1), Softmax_kappa = 1,
  Stereo_a = 0)

Arguments

data	sample to compute the test statistic. An array of size c(n, p, M) containing M samples of size n of directions (in Cartesian coordinates) on S^{p-1}. Alternatively, a matrix of size c(n, M) with the angles on [0, 2\pi) of the M circular samples of size n on S^{1}. Other objects accepted are an array of size c(n, 1, M) or a vector of size n with angular data. Must not contain NA's.
type	type of test to be applied. A character vector containing any of the following types of tests, depending on the dimension p: Circular data: any of the names available at object avail_cir_tests. (Hyper)spherical data: any of the names available at object avail_sph_tests. If type = "all" (default), then type is set as avail_cir_tests or avail_sph_tests, depending on the value of p.
data_sorted	is the circular data sorted? If TRUE, certain statistics are faster to compute. Defaults to FALSE.
CCF09_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. If NULL (default), a sample of size n_proj = 50 directions is computed internally.
CJ12_reg	type of asymptotic regime for CJ12 test, either 1 (sub-exponential regime), 2 (exponential), or 3 (super-exponential; default).
cov_a	a_n = a / n parameter used in the length of the arcs of the coverage-based tests. Must be positive. Defaults to 2 * pi.
Cressie_t	t parameter for the Cressie test, a real in (0, 1). Defaults to 1 / 3.
K_CCF09	integer giving the truncation of the series present in the asymptotic distribution of the Kolmogorov-Smirnov statistic. Defaults to 25.
Poisson_rho	\rho parameter for the Poisson test, a real in [0, 1). Defaults to 0.5.
Pycke_q	q parameter for the Pycke "q-test", a real in (0, 1). Defaults to 1 / 2.
Rayleigh_m	integer m for the m-modal Rayleigh test. Defaults to m = 1 (the standard Rayleigh test).
Riesz_s	s parameter for the s-Riesz test, a real in (0, 2). Defaults to 1.
Rothman_t	t parameter for the Rothman test, a real in (0, 1). Defaults to 1 / 3.
Sobolev_vk2	weights for the finite Sobolev test. A non-negative vector or matrix. Defaults to c(0, 0, 1).
Softmax_kappa	\kappa parameter for the Softmax test, a non-negative real. Defaults to 1.
Stereo_a	a parameter for the Stereo test, a real in [-1, 1]. Defaults to 0.

Details

Except for CCF09_dirs, K_CCF09, and CJ12_reg, all the test-specific parameters are vectorized.

Descriptions and references for most of the statistics are available in García-Portugués and Verdebout (2018).

Value

A data frame of size c(M, length(type)), with column names given by type, that contains the values of the test statistics.

References

García-Portugués, E. and Verdebout, T. (2018) An overview of uniformity tests on the hypersphere. arXiv:1804.00286. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.1804.00286")}.

Examples

## Circular data

# Sample
n <- 10
M <- 2
Theta <- r_unif_cir(n = n, M = M)

# Matrix
unif_stat(data = Theta, type = "all")

# Array
unif_stat(data = array(Theta, dim = c(n, 1, M)), type = "all")

# Vector
unif_stat(data = Theta[, 1], type = "all")

## Spherical data

# Circular sample in Cartesian coordinates
n <- 10
M <- 2
X <- array(dim = c(n, 2, M))
for (i in 1:M) X[, , i] <- cbind(cos(Theta[, i]), sin(Theta[, i]))

# Array
unif_stat(data = X, type = "all")

# High-dimensional data
X <- r_unif_sph(n = n, p = 3, M = M)
unif_stat(data = X, type = "all")

## Specific arguments

# Rothman
unif_stat(data = Theta, type = "Rothman", Rothman_t = 0.5)

# CCF09
unif_stat(data = X, type = "CCF09", CCF09_dirs = X[, , 1])
unif_stat(data = X, type = "CCF09", CCF09_dirs = X[, , 1], K_CCF09 = 1)

# CJ12
unif_stat(data = X, type = "CJ12", CJ12_reg = 3)
unif_stat(data = X, type = "CJ12", CJ12_reg = 1)

sphunif documentation built on May 29, 2024, 4:19 a.m.