Sobolev {sphunif}

Asymptotic distributions of Sobolev statistics of spherical uniformity

Description

Approximated density, distribution, and quantile functions for the asymptotic null distributions of Sobolev statistics of uniformity on S^{p-1}:=\{{\bf x}\in R^p:||{\bf x}||=1\}. These asymptotic distributions are infinite weighted sums of (central) chi squared random variables:

\sum_{k = 1}^\infty v_k^2 \chi^2_{d_{p, k}},

where

d_{p, k} := {{p + k - 3}\choose{p - 2}} + {{p + k - 2}\choose{p - 2}}

is the dimension of the space of eigenfunctions of the Laplacian on S^{p-1}, p\ge 2, associated to the k-th eigenvalue, k\ge 1.

Usage

d_p_k(p, k, log = FALSE)

weights_dfs_Sobolev(p, K_max = 1000, thre = 0.001, type, Rothman_t = 1/3,
  Pycke_q = 0.5, Riesz_s = 1, Poisson_rho = 0.5, Softmax_kappa = 1,
  Stereo_a = 0, Sobolev_vk2 = c(0, 0, 1), log = FALSE, verbose = TRUE,
  Gauss = TRUE, N = 320, tol = 1e-06, force_positive = TRUE,
  x_tail = NULL)

d_Sobolev(x, p, type, method = c("I", "SW", "HBE")[1], K_max = 1000,
  thre = 0.001, Rothman_t = 1/3, Pycke_q = 0.5, Riesz_s = 1,
  Poisson_rho = 0.5, Softmax_kappa = 1, Stereo_a = 0,
  Sobolev_vk2 = c(0, 0, 1), ncps = 0, verbose = TRUE, N = 320,
  x_tail = NULL, ...)

p_Sobolev(x, p, type, method = c("I", "SW", "HBE", "MC")[1], K_max = 1000,
  thre = 0.001, Rothman_t = 1/3, Pycke_q = 0.5, Riesz_s = 1,
  Poisson_rho = 0.5, Softmax_kappa = 1, Stereo_a = 0,
  Sobolev_vk2 = c(0, 0, 1), ncps = 0, verbose = TRUE, N = 320,
  x_tail = NULL, ...)

q_Sobolev(u, p, type, method = c("I", "SW", "HBE", "MC")[1], K_max = 1000,
  thre = 0.001, Rothman_t = 1/3, Pycke_q = 0.5, Riesz_s = 1,
  Poisson_rho = 0.5, Softmax_kappa = 1, Stereo_a = 0,
  Sobolev_vk2 = c(0, 0, 1), ncps = 0, verbose = TRUE, N = 320,
  x_tail = NULL, ...)

Arguments

p

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

k

sequence of integer indexes.

log

compute the logarithm of d_{p,k}? Defaults to FALSE.

K_max

integer giving the truncation of the series that compute the asymptotic p-value of a Sobolev test. Defaults to 1e3.

thre

error threshold for the tail probability given by the the first terms of the truncated series of a Sobolev test. Defaults to 1e-3.

type

name of the Sobolev statistic, using the naming from avail_cir_tests and avail_sph_tests.

Rothman_t

t parameter for the Rothman test, a real in (0, 1). Defaults to 1 / 3.

Pycke_q

q parameter for the Pycke "q-test", a real in (0, 1). Defaults to 1 / 2.

Riesz_s

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

Poisson_rho

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

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.

Sobolev_vk2

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

verbose

output information about the truncation? Defaults to TRUE.

Gauss

use a Gauss–Legendre quadrature rule of N nodes in the computation of the Gegenbauer coefficients? Otherwise, call integrate. Defaults to TRUE.

N

number of points used in the Gauss–Legendre quadrature for computing the Gegenbauer coefficients. Defaults to 320.

tol

tolerance passed to integrate's rel.tol and abs.tol if Gauss = FALSE. Defaults to 1e-6.

force_positive

set negative weights to zero? Defaults to TRUE.

x_tail

scalar evaluation point for determining the upper tail probability. If NULL, set to the 0.90 quantile of the whole series, computed by the "HBE" approximation.

x

vector of quantiles.

method

method for approximating the density, distribution, or quantile function of the weighted sum of chi squared random variables. Must be "I" (Imhof), "SW" (Satterthwaite–Welch), "HBE" (Hall–Buckley–Eagleson), or "MC" (Monte Carlo; only for distribution or quantile functions). Defaults to "I".

ncps

non-centrality parameters. Either 0 (default) or a vector with the same length as weights.

...

further parameters passed to *_wschisq.

u

vector of probabilities.

Details

The truncation of \sum_{k = 1}^\infty v_k^2 \chi^2_{d_{p, k}} is done to the first K_max terms and then up to the index such that the first terms explain the tail probability at the x_tail with an absolute error smaller than thre (see details in cutoff_wschisq). This automatic truncation takes place when calling *_Sobolev. Setting thre = 0 truncates to K_max terms exactly. If the series only contains odd or even non-zero terms, then only K_max / 2 addends are effectively taken into account in the first truncation.

Value

• d_p_k: a vector of size length(k) with the evaluation of d_{p,k}.

• weights_dfs_Sobolev: a list with entries weights and dfs, automatically truncated according to K_max and thre (see details).

• d_Sobolev: density function evaluated at x, a vector.

• p_Sobolev: distribution function evaluated at x, a vector.

• q_Sobolev: quantile function evaluated at u, a vector.

Author(s)

Eduardo García-Portugués and Paula Navarro-Esteban.

Examples

# Circular-specific statistics
curve(p_Sobolev(x = x, p = 2, type = "Watson", method = "HBE"),
      n = 2e2, ylab = "Distribution", main = "Watson")
curve(p_Sobolev(x = x, p = 2, type = "Rothman", method = "HBE"),
      n = 2e2, ylab = "Distribution", main = "Rothman")
curve(p_Sobolev(x = x, p = 2, type = "Pycke_q", method = "HBE"), to = 10,
      n = 2e2, ylab = "Distribution", main = "Pycke_q")
curve(p_Sobolev(x = x, p = 2, type = "Hermans_Rasson", method = "HBE"),
      to = 10, n = 2e2, ylab = "Distribution", main = "Hermans_Rasson")

# Statistics for arbitrary dimensions
test_statistic <- function(type, to = 1, pmax = 5, M = 1e3, ...) {

  col <- viridisLite::viridis(pmax - 1)
  curve(p_Sobolev(x = x, p = 2, type = type, method = "MC", M = M,
                  ...), to = to, n = 2e2, col = col[pmax - 1],
                  ylab = "Distribution", main = type, ylim = c(0, 1))
  for (p in 3:pmax) {
    curve(p_Sobolev(x = x, p = p, type = type, method = "MC", M = M,
                    ...), add = TRUE, n = 2e2, col = col[pmax - p + 1])
  }
  legend("bottomright", legend = paste("p =", 2:pmax), col = rev(col),
         lwd = 2)

}

# Ajne
test_statistic(type = "Ajne")

# Gine_Gn
test_statistic(type = "Gine_Gn", to = 1.5)

# Gine_Fn
test_statistic(type = "Gine_Fn", to = 2)

# Bakshaev
test_statistic(type = "Bakshaev", to = 3)

# Riesz
test_statistic(type = "Riesz", Riesz_s = 0.5, to = 3)

# PCvM
test_statistic(type = "PCvM", to = 0.6)

# PAD
test_statistic(type = "PAD", to = 3)

# PRt
test_statistic(type = "PRt", Rothman_t = 0.5)

# Quantiles
p <- c(2, 3, 4, 11)
t(sapply(p, function(p) q_Sobolev(u = c(0.10, 0.05, 0.01), p = p,
                                  type = "PCvM")))
t(sapply(p, function(p) q_Sobolev(u = c(0.10, 0.05, 0.01), p = p,
                                  type = "PAD")))
t(sapply(p, function(p) q_Sobolev(u = c(0.10, 0.05, 0.01), p = p,
                                  type = "PRt")))

# Series truncation for thre = 1e-5
sapply(p, function(p) length(weights_dfs_Sobolev(p = p, type = "PCvM")$dfs))
sapply(p, function(p) length(weights_dfs_Sobolev(p = p, type = "PRt")$dfs))
sapply(p, function(p) length(weights_dfs_Sobolev(p = p, type = "PAD")$dfs))