sphunif (version 1.4.3)

p_Kolmogorov: Asymptotic distributions for circular uniformity statistics

Description

Computation of the asymptotic null distributions of circular uniformity statistics.

Usage

p_Kolmogorov(x, K_Kolmogorov = 25L, alternating = TRUE)
d_Kolmogorov(x, K_Kolmogorov = 25L, alternating = TRUE)
p_cir_stat_Ajne(x, K_Ajne = 15L)
d_cir_stat_Ajne(x, K_Ajne = 15L)
p_cir_stat_Bingham(x)
d_cir_stat_Bingham(x)
p_cir_stat_Greenwood(x)
d_cir_stat_Greenwood(x)
p_cir_stat_Gini(x)
d_cir_stat_Gini(x)
p_cir_stat_Gini_squared(x)
d_cir_stat_Gini_squared(x)
p_cir_stat_Hodges_Ajne2(x, n, asymp_std = FALSE)
p_cir_stat_Hodges_Ajne(x, n, exact = TRUE, asymp_std = FALSE)
d_cir_stat_Hodges_Ajne(x, n, exact = TRUE, asymp_std = FALSE)
p_cir_stat_Kuiper(x, n, K_Kuiper = 12L, second_term = TRUE,
  Stephens = FALSE)
d_cir_stat_Kuiper(x, n, K_Kuiper = 12L, second_term = TRUE,
  Stephens = FALSE)
p_cir_stat_Log_gaps(x, abs_val = TRUE)
d_cir_stat_Log_gaps(x, abs_val = TRUE)
p_cir_stat_Max_uncover(x)
d_cir_stat_Max_uncover(x)
p_cir_stat_Num_uncover(x)
d_cir_stat_Num_uncover(x)
p_cir_stat_Pycke(x)
d_cir_stat_Pycke(x)
p_cir_stat_Vacancy(x)
d_cir_stat_Vacancy(x)
p_cir_stat_Watson(x, n = 0L, K_Watson = 25L, Stephens = FALSE)
d_cir_stat_Watson(x, n = 0L, K_Watson = 25L, Stephens = FALSE)
p_cir_stat_Watson_1976(x, K_Watson_1976 = 8L, N = 40L)
d_cir_stat_Watson_1976(x, K_Watson_1976 = 8L)
p_cir_stat_Range(x, n, max_gap = TRUE)
d_cir_stat_Range(x, n, max_gap = TRUE)
p_cir_stat_Rao(x)
d_cir_stat_Rao(x)
p_cir_stat_Rayleigh(x)
d_cir_stat_Rayleigh(x)
p_cir_stat_Bakshaev(x, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_Bakshaev(x, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_Gine_Fn(x, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_Gine_Fn(x, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_Gine_Gn(x, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_Gine_Gn(x, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_Hermans_Rasson(x, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_Hermans_Rasson(x, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_PAD(x, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_PAD(x, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_PCvM(x, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_PCvM(x, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_PRt(x, t = 1/3, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_PRt(x, t = 1/3, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_Poisson(x, rho = 0.5, K_max = 1000, thre = 0, method = "I",
  ...)
d_cir_stat_Poisson(x, rho = 0.5, K_max = 1000, thre = 0, method = "I",
  ...)
p_cir_stat_Pycke_q(x, q = 0.5, K_max = 1000, thre = 0, method = "I",
  ...)
d_cir_stat_Pycke_q(x, q = 0.5, K_max = 1000, thre = 0, method = "I",
  ...)
p_cir_stat_Rothman(x, t = 1/3, K_max = 1000, thre = 0, method = "I",
  ...)
d_cir_stat_Rothman(x, t = 1/3, K_max = 1000, thre = 0, method = "I",
  ...)
p_cir_stat_Riesz(x, s = 1, K_max = 1000, thre = 0, method = "I", ...)
d_cir_stat_Riesz(x, s = 1, K_max = 1000, thre = 0, method = "I", ...)
p_cir_stat_Sobolev(x, vk2 = c(0, 0, 1), method = "I", ...)
d_cir_stat_Sobolev(x, vk2 = c(0, 0, 1), method = "I", ...)
p_cir_stat_Softmax(x, kappa = 1, K_max = 1000, thre = 0, method = "I",
  ...)
d_cir_stat_Softmax(x, kappa = 1, K_max = 1000, thre = 0, method = "I",
  ...)
p_cir_stat_Stein(x, Stein_K = 10, Stein_cf = FALSE, method = "I", ...)
d_cir_stat_Stein(x, Stein_K = 10, Stein_cf = FALSE, method = "I", ...)

Value

A matrix of size c(nx, 1) with the evaluation of the distribution or density function at x.

Arguments

x

a vector of size nx or a matrix of size c(nx, 1).

K_Kolmogorov, K_Kuiper, K_Watson, K_Watson_1976, K_Ajne

integer giving the truncation of the series present in the null asymptotic distributions. For the Kolmogorov-Smirnov-related series defaults to 25; for the others series defaults to a smaller number.

alternating

use the alternating series expansion for the distribution of the Kolmogorov-Smirnov statistic? Defaults to TRUE.

n

sample size employed for computing the statistic.

asymp_std

compute the distribution associated to the normalized Hodges-Ajne statistic? Defaults to FALSE.

exact

use the exact distribution for the Hodges-Ajne statistic? Defaults to TRUE.

second_term

use the second-order series expansion for the distribution of the Kuiper statistic? Defaults to TRUE.

Stephens

compute Stephens (1970) modification so that the null distribution of the is less dependent on the sample size?

abs_val

compute the distribution associated to the absolute value of the Darling's log gaps statistic? Defaults to TRUE.

N

number of points used in the Gauss-Legendre quadrature. Defaults to 40.

max_gap

compute the distribution associated to the maximum gap for the range statistic? Defaults to TRUE.

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).

t

\(t\) parameter for the Rothman and Cressie tests, a real in \((0, 1)\). Defaults to 1 / 3.

rho

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

q

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

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.

Stein_K

truncation \(K\) parameter for the Stein test, a positive integer. Defaults to 10.

Stein_cf

logical indicating whether to use the characteristic function in the Stein test. Defaults to FALSE (moment generating function).

Details

Descriptions and references for most of the tests are available in García-Portugués and Verdebout (2018).

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")

Examples

# Ajne
curve(d_cir_stat_Ajne(x), to = 1.5, n = 2e2, ylim = c(0, 4))
curve(p_cir_stat_Ajne(x), n = 2e2, col = 2, add = TRUE)

# Bakshaev
curve(d_cir_stat_Bakshaev(x, method = "HBE"), to = 6, n = 2e2,
      ylim = c(0, 1))
curve(p_cir_stat_Bakshaev(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Bingham
curve(d_cir_stat_Bingham(x), to = 12, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Bingham(x), n = 2e2, col = 2, add = TRUE)
# Greenwood
curve(d_cir_stat_Greenwood(x), from = -6, to = 6, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Greenwood(x), n = 2e2, col = 2, add = TRUE)

# Hermans-Rasson
curve(p_cir_stat_Hermans_Rasson(x, method = "HBE"), to = 10, n = 2e2,
      ylim = c(0, 1))
curve(d_cir_stat_Hermans_Rasson(x, method = "HBE"), n = 2e2, add = TRUE,
      col = 2)

# Hodges-Ajne
plot(25:45, d_cir_stat_Hodges_Ajne(cbind(25:45), n = 50), type = "h",
     lwd = 2, ylim = c(0, 1))
lines(25:45, p_cir_stat_Hodges_Ajne(cbind(25:45), n = 50), type = "s",
      col = 2)

# Kolmogorov-Smirnov
curve(d_Kolmogorov(x), to = 3, n = 2e2, ylim = c(0, 2))
curve(p_Kolmogorov(x), n = 2e2, col = 2, add = TRUE)

# Kuiper
curve(d_cir_stat_Kuiper(x, n = 50), to = 3, n = 2e2, ylim = c(0, 2))
curve(p_cir_stat_Kuiper(x, n = 50), n = 2e2, col = 2, add = TRUE)

# Kuiper and Watson with Stephens modification
curve(d_cir_stat_Kuiper(x, n = 8, Stephens = TRUE), to = 2.5, n = 2e2,
      ylim = c(0, 10))
curve(d_cir_stat_Watson(x, n = 8, Stephens = TRUE), n = 2e2, lty = 2,
      add = TRUE)
n <- c(10, 20, 30, 40, 50, 100, 500)
col <- rainbow(length(n))
for (i in seq_along(n)) {
  curve(d_cir_stat_Kuiper(x, n = n[i], Stephens = TRUE), n = 2e2,
        col = col[i], add = TRUE)
  curve(d_cir_stat_Watson(x, n = n[i], Stephens = TRUE), n = 2e2,
        col = col[i], lty = 2, add = TRUE)
}

# Maximum uncovered spacing
curve(d_cir_stat_Max_uncover(x), from = -3, to = 6, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Max_uncover(x), n = 2e2, col = 2, add = TRUE)

# Number of uncovered spacing
curve(d_cir_stat_Num_uncover(x), from = -4, to = 4, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Num_uncover(x), n = 2e2, col = 2, add = TRUE)

# Log gaps
curve(d_cir_stat_Log_gaps(x), from = -1, to = 4, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Log_gaps(x), n = 2e2, col = 2, add = TRUE)

# Gine Fn
curve(d_cir_stat_Gine_Fn(x, method = "HBE"), to = 2.5, n = 2e2,
      ylim = c(0, 2))
curve(p_cir_stat_Gine_Fn(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Gine Gn
curve(d_cir_stat_Gine_Gn(x, method = "HBE"), to = 2.5, n = 2e2,
      ylim = c(0, 2))
curve(p_cir_stat_Gine_Gn(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Gini mean difference
curve(d_cir_stat_Gini(x), from = -4, to = 4, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Gini(x), n = 2e2, col = 2, add = TRUE)

# Gini mean squared difference
curve(d_cir_stat_Gini_squared(x), from = -10, to = 10, n = 2e2,
      ylim = c(0, 1))
curve(p_cir_stat_Gini_squared(x), n = 2e2, col = 2, add = TRUE)

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

# PCvM
curve(d_cir_stat_PCvM(x, method = "HBE"), to = 4, n = 2e2, ylim = c(0, 2))
curve(p_cir_stat_PCvM(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# PRt
curve(d_cir_stat_PRt(x, method = "HBE"), n = 2e2, ylim = c(0, 5))
curve(p_cir_stat_PRt(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Poisson
curve(d_cir_stat_Poisson(x, method = "HBE"), from = -1, to = 5, n = 2e2,
      ylim = c(0, 2))
curve(p_cir_stat_Poisson(x, method = "HBE"), n = 2e2, col = 2, add = TRUE)

# Pycke
curve(d_cir_stat_Pycke(x), from = -5, to = 10, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Pycke(x), n = 2e2, col = 2, add = TRUE)

# Pycke q
curve(d_cir_stat_Pycke_q(x, method = "HBE"), to = 15, n = 2e2,
      ylim = c(0, 1))
curve(p_cir_stat_Pycke_q(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Range
curve(d_cir_stat_Range(x, n = 50), to = 2, n = 2e2, ylim = c(0, 4))
curve(p_cir_stat_Range(x, n = 50), n = 2e2, col = 2, add = TRUE)

# Rao
curve(d_cir_stat_Rao(x), from = -6, to = 6, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Rao(x), n = 2e2, col = 2, add = TRUE)

# Rayleigh
curve(d_cir_stat_Rayleigh(x), to = 12, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Rayleigh(x), n = 2e2, col = 2, add = TRUE)

# Riesz
curve(d_cir_stat_Riesz(x, method = "HBE"), to = 6, n = 2e2,
      ylim = c(0, 1))
curve(p_cir_stat_Riesz(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Rothman
curve(d_cir_stat_Rothman(x, method = "HBE"), n = 2e2, ylim = c(0, 5))
curve(p_cir_stat_Rothman(x, method = "HBE"), n = 2e2, add = TRUE, col = 2)

# Vacancy
curve(d_cir_stat_Vacancy(x), from = -4, to = 4, n = 2e2, ylim = c(0, 1))
curve(p_cir_stat_Vacancy(x), n = 2e2, col = 2, add = TRUE)

# Watson
curve(d_cir_stat_Watson(x), to = 0.5, n = 2e2, ylim = c(0, 15))
curve(p_cir_stat_Watson(x), n = 2e2, col = 2, add = TRUE)

# Watson (1976)
curve(d_cir_stat_Watson_1976(x), to = 1.5, n = 2e2, ylim = c(0, 3))
curve(p_cir_stat_Watson_1976(x), n = 2e2, col = 2, add = TRUE)

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

# Sobolev
vk2 <- c(0.5, 0)
curve(d_cir_stat_Sobolev(x = x, vk2 = vk2), to = 3, n = 2e2, ylim = c(0, 2))
curve(p_cir_stat_Sobolev(x = x, vk2 = vk2), n = 2e2, col = 2, add = TRUE)