Implementation of more than 35 tests of uniformity on the circle, sphere, and hypersphere. Software companion for the (evolving) review An overview of uniformity tests on the hypersphere (García-Portugués and Verdebout, 2018) and the paper On a projection-based class of uniformity tests on the hypersphere (García-Portugués, Navarro-Esteban and Cuesta-Albertos, 2023).
Get the latest version from GitHub:
# Install the package and the vignettes
library(devtools)
install_github("egarpor/sphunif", build_vignettes = TRUE)
# Load package
library(sphunif)
# See main vignette
vignette("sphunif")
You want to test if a sample of circular data is uniformly distributed. For example, the following circular uniform sample in radians:
set.seed(987202226) cir_data <- runif(n = 10, min = 0, max = 2 * pi)
Call the main function in the sphunif package, unif_test, specifying the type of test to be performed. For example, the "Watson" test:
library(sphunif) unif_test(data = cir_data, type = "Watson") # An htest object #> #> Watson test of circular uniformity #> #> data: cir_data #> statistic = 0.036003, p-value = 0.8694 #> alternative hypothesis: any alternative to circular uniformity
By default, the calibration of the test statistic is done by Monte Carlo. This can be changed with p_value = "asymp" to employ asymptotic distributions (faster, but not available for all tests):
unif_test(data = cir_data, type = "Watson", p_value = "MC") # Monte Carlo #> #> Watson test of circular uniformity #> #> data: cir_data #> statistic = 0.036003, p-value = 0.8831 #> alternative hypothesis: any alternative to circular uniformity unif_test(data = cir_data, type = "Watson", p_value = "asymp") # Asymp. distr. #> #> Watson test of circular uniformity #> #> data: cir_data #> statistic = 0.036003, p-value = 0.8694 #> alternative hypothesis: any alternative to circular uniformity
You can perform several tests with a single call to unif_test. Choose the available circular uniformity tests from
avail_cir_tests #> [1] "Ajne" "Bakshaev" "Bingham" "Cressie" #> [5] "CCF09" "FG01" "Gine_Fn" "Gine_Gn" #> [9] "Gini" "Gini_squared" "Greenwood" "Hermans_Rasson" #> [13] "Hodges_Ajne" "Kuiper" "Log_gaps" "Max_uncover" #> [17] "Num_uncover" "PAD" "PCvM" "Poisson" #> [21] "PRt" "Pycke" "Pycke_q" "Range" #> [25] "Rao" "Rayleigh" "Riesz" "Rothman" #> [29] "Sobolev" "Softmax" "Vacancy" "Watson" #> [33] "Watson_1976"
For example:
# A *list* of htest objects
unif_test(data = cir_data, type = c("Watson", "PAD", "Ajne"))
#> $Watson
#>
#> Watson test of circular uniformity
#>
#> data: cir_data
#> statistic = 0.036003, p-value = 0.8694
#> alternative hypothesis: any alternative to circular uniformity
#>
#>
#> $PAD
#>
#> Projected Anderson-Darling test of circular uniformity
#>
#> data: cir_data
#> statistic = 0.47247, p-value = 0.9051
#> alternative hypothesis: any alternative to circular uniformity
#>
#>
#> $Ajne
#>
#> Ajne test of circular uniformity
#>
#> data: cir_data
#> statistic = 0.11016, p-value = 0.7361
#> alternative hypothesis: any non-axial alternative to circular uniformity
You want to test if a sample of spherical data is uniformly distributed. Consider a non-uniformly-generated sample of directions in Cartesian coordinates:
# Sample data on S^2 set.seed(987202226) theta <- runif(n = 50, min = 0, max = 2 * pi) phi <- runif(n = 50, min = 0, max = pi) sph_data <- cbind(cos(theta) * sin(phi), sin(theta) * sin(phi), cos(phi))
The available spherical uniformity tests:
avail_sph_tests #> [1] "Ajne" "Bakshaev" "Bingham" "CCF09" "CJ12" #> [6] "Gine_Fn" "Gine_Gn" "PAD" "PCvM" "Poisson" #> [11] "PRt" "Pycke" "Sobolev" "Softmax" "Stereo" #> [16] "Rayleigh" "Rayleigh_HD" "Riesz"
The default type = "all" equals type = avail_sph_tests:
head(unif_test(data = sph_data, type = "all", p_value = "MC")) #> $Ajne #> #> Ajne test of spherical uniformity #> #> data: sph_data #> statistic = 0.079876, p-value = 0.9578 #> alternative hypothesis: any non-axial alternative to spherical uniformity #> #> #> $Bakshaev #> #> Bakshaev (2010) test of spherical uniformity #> #> data: sph_data #> statistic = 1.2727, p-value = 0.4418 #> alternative hypothesis: any alternative to spherical uniformity #> #> #> $Bingham #> #> Bingham test of spherical uniformity #> #> data: sph_data #> statistic = 22.455, p-value < 2.2e-16 #> alternative hypothesis: scatter matrix different from constant #> #> #> $CCF09 #> #> Cuesta-Albertos et al. (2009) test of spherical uniformity with k = 50 #> #> data: sph_data #> statistic = 1.4619, p-value = 0.2775 #> alternative hypothesis: any alternative to spherical uniformity #> #> #> $CJ12 #> #> Cai and Jiang (2012) test of spherical uniformity #> #> data: sph_data #> statistic = 27.401, p-value = 0.262 #> alternative hypothesis: unclear consistency #> #> #> $Gine_Fn #> #> Gine's Fn test of spherical uniformity #> #> data: sph_data #> statistic = 1.8889, p-value = 0.2216 #> alternative hypothesis: any alternative to spherical uniformity unif_test(data = sph_data, type = "Rayleigh", p_value = "asymp") #> #> Rayleigh test of spherical uniformity #> #> data: sph_data #> statistic = 0.52692, p-value = 0.9129 #> alternative hypothesis: mean direction different from zero
The hyperspherical setting is treated analogously to the spherical setting, and the available tests are exactly the same (avail_sph_tests). An example of testing uniformity with a sample of size 100 on the $9$ -sphere:
# Sample data on S^9 set.seed(987202226) hyp_data <- r_unif_sph(n = 50, p = 10) # Test unif_test(data = hyp_data, type = "Rayleigh", p_value = "asymp") #> #> Rayleigh test of spherical uniformity #> #> data: hyp_data #> statistic = 11.784, p-value = 0.2997 #> alternative hypothesis: mean direction different from zeroOn a projection-based class of uniformity tests on the hypersphere
The data application in García-Portugués, Navarro-Esteban, and Cuesta-Albertos (2023) regarding the testing of uniformity of locations for craters in Rhea can be reproduced through the script data-application-ecdf.R. The code snippet below is a simplified version.
# Load data
data(rhea)
# Add Cartesian coordinates
rhea$X <- cbind(cos(rhea$theta) * cos(rhea$phi),
sin(rhea$theta) * cos(rhea$phi),
sin(rhea$phi))
# Distribution of diameter
quantile(rhea$diameter)
#> 0% 25% 50% 75% 100%
#> 10.0000 13.2475 17.0500 24.5600 449.8200
# Subsets of craters, according to diameter
ind_15_20 <- rhea$diameter > 15 & rhea$diameter < 20
ind_20 <- rhea$diameter > 20
# Sample sizes
nrow(rhea)
#> [1] 3596
sum(ind_15_20)
#> [1] 867
sum(ind_20)
#> [1] 1373
# Tests to be performed
type_tests <- c("PCvM", "PAD", "PRt")
# Tests
tests_rhea_15_20 <- unif_test(data = rhea$X[ind_15_20, ], type = type_tests,
p_value = "asymp", K_max = 5e4)
tests_rhea_20 <- unif_test(data = rhea$X[ind_20, ], type = type_tests,
p_value = "asymp", K_max = 5e4)
tests_rhea_15_20
#> $PCvM
#>
#> Projected Cramer-von Mises test of spherical uniformity
#>
#> data: rhea$X[ind_15_20, ]
#> statistic = 0.26452, p-value = 0.1176
#> alternative hypothesis: any alternative to spherical uniformity
#>
#>
#> $PAD
#>
#> Projected Anderson-Darling test of spherical uniformity
#>
#> data: rhea$X[ind_15_20, ]
#> statistic = 1.6854, p-value = 0.07209
#> alternative hypothesis: any alternative to spherical uniformity
#>
#>
#> $PRt
#>
#> Projected Rothman test of spherical uniformity with t = 0.333
#>
#> data: rhea$X[ind_15_20, ]
#> statistic = 0.31352, p-value = 0.1856
#> alternative hypothesis: any alternative to spherical uniformity if t is irrational
tests_rhea_20
#> $PCvM
#>
#> Projected Cramer-von Mises test of spherical uniformity
#>
#> data: rhea$X[ind_20, ]
#> statistic = 3.6494, p-value = 2.202e-09
#> alternative hypothesis: any alternative to spherical uniformity
#>
#>
#> $PAD
#>
#> Projected Anderson-Darling test of spherical uniformity
#>
#> data: rhea$X[ind_20, ]
#> statistic = 18.482, p-value < 2.2e-16
#> alternative hypothesis: any alternative to spherical uniformity
#>
#>
#> $PRt
#>
#> Projected Rothman test of spherical uniformity with t = 0.333
#>
#> data: rhea$X[ind_20, ]
#> statistic = 5.3485, p-value = 3.481e-09
#> alternative hypothesis: any alternative to spherical uniformity if t is irrationalA Cramér–von Mises test of uniformity on the hypersphere
The data application in García-Portugués, Navarro-Esteban, and Cuesta-Albertos (2021) regarding the testing of the uniformity of locations for craters in Venus can be reproduced through the script data-application-cvm.R.
On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powersThe data application in García-Portugués, Paindaveine, and Verdebout (2024) regarding the symmetry of comet orbits can be reproduced through the script data-application-sobolev.R.
A stereographic test of spherical uniformityThe script simulations-stereo.R contains some of the numerical experiments in Fernández-de-Marcos and García-Portugués (2024).
Fernández-de-Marcos, A. and García-Portugués, E. (2024). A stereographic test of spherical uniformity. arXiv:2405.13531. doi:10.48550/arXiv.2405.13531.
García-Portugués, E., Navarro-Esteban, P., and Cuesta-Albertos, J. A. (2023). On a projection-based class of uniformity tests on the hypersphere. Bernoulli, 29(1):181–204. doi:10.1007/978-3-030-69944-4_12.
García-Portugués, E., Navarro-Esteban, P., and Cuesta-Albertos, J. A. (2021). A Cramér–von Mises test of uniformity on the hypersphere. In Balzano, S., Porzio, G. C., Salvatore, R., Vistocco, D., and Vichi, M. (Eds.), Statistical Learning and Modeling in Data Analysis, Studies in Classification, Data Analysis and Knowledge Organization, pp. 107–116. Springer, Cham. doi:10.1007/978-3-030-69944-4_12.
García-Portugués, E., Paindaveine, D., and Verdebout, T. (2024). On a class of Sobolev tests for symmetry of directions, their detection thresholds, and asymptotic powers. arXiv:2108.09874v2. doi:10.48550/arXiv.2108.09874.
García-Portugués, E. and Verdebout, T. (2018). An overview of uniformity tests on the hypersphere. arXiv:1804.00286. doi:10.48550/arXiv.1804.00286.
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