cir_stat_distr() R function from [sphunif]

Asymptotic distributions for circular uniformity statistics

Computation of the asymptotic null distributions of circular uniformity statistics.


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",
  ...)

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? The modification does not alter the test decision.
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.

Returns

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

Details

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

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)

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") .

sphunif package Read PDF manual

Maintainer: Eduardo García-Portugués
License: GPL-3
Last published: 2024-05-24

Useful links

cir_stat_distr function