sph_stat_distr: Asymptotic distributions for spherical uniformity statistics

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

Asymptotic distributions for spherical uniformity statistics

Description

Computation of the asymptotic null distributions of spherical uniformity statistics.

Usage

p_sph_stat_Bingham(x, p)

d_sph_stat_Bingham(x, p)

p_sph_stat_CJ12(x, regime = 1L, beta = 0)

d_sph_stat_CJ12(x, regime = 3L, beta = 0)

p_sph_stat_Rayleigh(x, p)

d_sph_stat_Rayleigh(x, p)

p_sph_stat_Rayleigh_HD(x, p)

d_sph_stat_Rayleigh_HD(x, p)

p_sph_stat_Ajne(x, p, K_max = 1000, thre = 0, method = "I", ...)

d_sph_stat_Ajne(x, p, K_max = 1000, thre = 0, method = "I", ...)

p_sph_stat_Bakshaev(x, p, K_max = 1000, thre = 0, method = "I", ...)

d_sph_stat_Bakshaev(x, p, K_max = 1000, thre = 0, method = "I", ...)

p_sph_stat_Gine_Fn(x, p, K_max = 1000, thre = 0, method = "I", ...)

d_sph_stat_Gine_Fn(x, p, K_max = 1000, thre = 0, method = "I", ...)

p_sph_stat_Gine_Gn(x, p, K_max = 1000, thre = 0, method = "I", ...)

d_sph_stat_Gine_Gn(x, p, K_max = 1000, thre = 0, method = "I", ...)

p_sph_stat_PAD(x, p, K_max = 1000, thre = 0, method = "I", ...)

d_sph_stat_PAD(x, p, K_max = 1000, thre = 0, method = "I", ...)

p_sph_stat_PCvM(x, p, K_max = 1000, thre = 0, method = "I", ...)

d_sph_stat_PCvM(x, p, K_max = 1000, thre = 0, method = "I", ...)

p_sph_stat_Poisson(x, p, rho = 0.5, K_max = 1000, thre = 0,
  method = "I", ...)

d_sph_stat_Poisson(x, p, rho = 0.5, K_max = 1000, thre = 0,
  method = "I", ...)

p_sph_stat_PRt(x, p, t = 1/3, K_max = 1000, thre = 0, method = "I",
  ...)

d_sph_stat_PRt(x, p, t = 1/3, K_max = 1000, thre = 0, method = "I",
  ...)

p_sph_stat_Riesz(x, p, s = 1, K_max = 1000, thre = 0, method = "I",
  ...)

d_sph_stat_Riesz(x, p, s = 1, K_max = 1000, thre = 0, method = "I",
  ...)

p_sph_stat_Sobolev(x, p, vk2 = c(0, 0, 1), method = "I", ...)

d_sph_stat_Sobolev(x, p, vk2 = c(0, 0, 1), method = "I", ...)

p_sph_stat_Softmax(x, p, kappa = 1, K_max = 1000, thre = 0,
  method = "I", ...)

d_sph_stat_Softmax(x, p, kappa = 1, K_max = 1000, thre = 0,
  method = "I", ...)

p_sph_stat_Stereo(x, p, a = 0, K_max = 1000, method = "I", ...)

d_sph_stat_Stereo(x, p, a = 0, K_max = 1000, method = "I", ...)

Arguments

x	a vector of size nx or a matrix of size c(nx, 1).
p	integer giving the dimension of the ambient space R^p that contains S^{p-1}.
regime	type of asymptotic regime for the CJ12 test, either 1 (sub-exponential regime), 2 (exponential), or 3 (super-exponential; default).
beta	\beta parameter in the exponential regime of the CJ12 test, a non-negative real. Defaults to 0.
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 0 (no further truncation).
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".
...	further parameters passed to p_Sobolev or d_Sobolev (such as x_tail).
rho	\rho parameter for the Poisson test, a real in [0, 1). Defaults to 0.5.
t	t parameter for the Rothman and Cressie tests, a real in (0, 1). Defaults to 1 / 3.
s	s parameter for the s-Riesz test, a real in (0, 2). Defaults to 1.
vk2	weights for the finite Sobolev test. A non-negative vector or matrix. Defaults to c(0, 0, 1).
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.

Details

Descriptions and references on most of the asymptotic distributions are available in García-Portugués and Verdebout (2018).

Value

• r_sph_stat_*: a matrix of size c(n, 1) containing the sample.

• p_sph_stat_*, d_sph_stat_*: a matrix of size c(nx, 1) with the evaluation of the distribution or density functions at x.

Examples

# Ajne
curve(d_sph_stat_Ajne(x, p = 3, method = "HBE"), n = 2e2, ylim = c(0, 4))
curve(p_sph_stat_Ajne(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# Bakshaev
curve(d_sph_stat_Bakshaev(x, p = 3, method = "HBE"), to = 5, n = 2e2,
      ylim = c(0, 2))
curve(p_sph_stat_Bakshaev(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# Bingham
curve(d_sph_stat_Bingham(x, p = 3), to = 20, n = 2e2, ylim = c(0, 1))
curve(p_sph_stat_Bingham(x, p = 3), n = 2e2, col = 2, add = TRUE)

# CJ12
curve(d_sph_stat_CJ12(x, regime = 1), from = -10, to = 10, n = 2e2,
      ylim = c(0, 1))
curve(d_sph_stat_CJ12(x, regime = 2, beta = 0.1), n = 2e2, col = 2,
      add = TRUE)
curve(d_sph_stat_CJ12(x, regime = 3), n = 2e2, col = 3, add = TRUE)
curve(p_sph_stat_CJ12(x, regime = 1), n = 2e2, col = 1, add = TRUE)
curve(p_sph_stat_CJ12(x, regime = 2, beta = 0.1), n = 2e2, col = 2,
      add = TRUE)
curve(p_sph_stat_CJ12(x, regime = 3), col = 3, add = TRUE)

# Gine Fn
curve(d_sph_stat_Gine_Fn(x, p = 3, method = "HBE"), to = 2, n = 2e2,
      ylim = c(0, 2))
curve(p_sph_stat_Gine_Fn(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# Gine Gn
curve(d_sph_stat_Gine_Gn(x, p = 3, method = "HBE"), to = 1.5, n = 2e2,
      ylim = c(0, 2.5))
curve(p_sph_stat_Gine_Gn(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# PAD
curve(d_sph_stat_PAD(x, p = 3, method = "HBE"), to = 3, n = 2e2,
      ylim = c(0, 1.5))
curve(p_sph_stat_PAD(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# PCvM
curve(d_sph_stat_PCvM(x, p = 3, method = "HBE"), to = 0.6, n = 2e2,
      ylim = c(0, 7))
curve(p_sph_stat_PCvM(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# Poisson
curve(d_sph_stat_Poisson(x, p = 3, method = "HBE"), to = 2, n = 2e2,
      ylim = c(0, 2))
curve(p_sph_stat_Poisson(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# PRt
curve(d_sph_stat_PRt(x, p = 3, method = "HBE"), n = 2e2, ylim = c(0, 5))
curve(p_sph_stat_PRt(x, p = 3, method = "HBE"), n = 2e2, col = 2, add = TRUE)

# Rayleigh
curve(d_sph_stat_Rayleigh(x, p = 3), to = 15, n = 2e2, ylim = c(0, 1))
curve(p_sph_stat_Rayleigh(x, p = 3), n = 2e2, col = 2, add = TRUE)

# HD-standardized Rayleigh
curve(d_sph_stat_Rayleigh_HD(x, p = 3), from = -4, to = 4, n = 2e2,
      ylim = c(0, 1))
curve(p_sph_stat_Rayleigh_HD(x, p = 3), n = 2e2, col = 2, add = TRUE)

# Riesz
curve(d_sph_stat_Riesz(x, p = 3, method = "HBE"), n = 2e2, from = 0, to = 5,
      ylim = c(0, 2))
curve(p_sph_stat_Riesz(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# Sobolev
x <- seq(-1, 5, by = 0.05)
vk2 <- diag(rep(0.3, 2))
matplot(x, d_sph_stat_Sobolev(x = x, vk2 = vk2, p = 3), type = "l",
        ylim = c(0, 1), lty = 1)
matlines(x, p_sph_stat_Sobolev(x = x, vk2 = vk2, p = 3), lty = 1)
matlines(x, d_sph_stat_Sobolev(x = x, vk2 = vk2 + 0.01, p = 3), lty = 2)
matlines(x, p_sph_stat_Sobolev(x = x, vk2 = vk2 + 0.01, p = 3), lty = 2)

# Softmax
curve(d_sph_stat_Softmax(x, p = 3, method = "HBE"), to = 2, n = 2e2,
      ylim = c(0, 2))
curve(p_sph_stat_Softmax(x, p = 3, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

# Stereo
curve(d_sph_stat_Stereo(x, p = 4, method = "HBE"), from=-5,to = 10, n = 2e2,
      ylim = c(0, 2))
curve(p_sph_stat_Stereo(x, p = 4, method = "HBE"), n = 2e2, col = 2,
      add = TRUE)

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