sph_stat_distr function

Asymptotic distributions for spherical uniformity statistics

Computation of the asymptotic null distributions of spherical uniformity statistics.

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.

Returns

• 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.

Details

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

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)