R: Asymptotic distributions of Sobolev statistics of spherical...

Sobolev {sphunif}

R Documentation

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`	Sobolev statistic. For `p = 2`, either `"Watson"`, `"Rothman"`, `"Pycke_q"`, or `"Hermans_Rasson"`. For `p \ge 2`, `"Ajne"`, `"Gine_Gn"`, `"Gine_Fn"`, `"Bakshaev"`, `"Riesz"`, `"PCvM"`, `"PAD"`, or `"PRt"`.
`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
`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. 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))

[Package sphunif version 1.3.0 Index]