unif_stat function

Circular and (hyper)spherical uniformity statistics

Implementation of several statistics for assessing uniformity on the (hyper)sphere c("$\n$", "$S^{p-1} := {x \\in R^p : ||x|| = 1}$"), $p\ge 2$, for a sample $X_1,\ldots,X_n\in S^{p-1}$.

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.

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.

Returns

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

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).

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)

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") .