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R/unif_test.R
In sphunif: Uniformity Tests on the Circle, Sphere, and Hypersphere
Defines functions
unif_test
Documented in
unif_test
#' @title Circular and (hyper)spherical uniformity tests
#'
#' @description Implementation of several uniformity tests on the (hyper)sphere
#' \eqn{S^{p-1}:=\{{\bf x}\in R^p:||{\bf x}||=1\}}{
#' S^{p-1}:=\{x\in R^p:||x||=1\}}, \eqn{p\ge 2}, with calibration either in
#' terms of their asymptotic/exact distributions, if available, or Monte Carlo.
#'
#' \code{unif_test} receives a sample of directions
#' \eqn{{\bf X}_1,\ldots,{\bf X}_n\in S^{p-1}}{X_1,\ldots,X_n\in S^{p-1}} in
#' \emph{Cartesian coordinates}, except for the circular case (\eqn{p=2}) in
#' which the sample can be represented in terms of \emph{angles}
#' \eqn{\Theta_1,\ldots,\Theta_n\in [0, 2\pi)}.
#'
#' \code{unif_test} allows to perform several tests within a single call,
#' facilitating thus the exploration of a dataset by applying several tests.
#'
#' @param data sample to perform the test. A matrix of size \code{c(n, p)}
#' containing a sample of size \code{n} of directions (in Cartesian
#' coordinates) on \eqn{S^{p-1}}. Alternatively if \code{p = 2}, a matrix of
#' size \code{c(n, 1)} containing the \code{n} angles on \eqn{[0, 2\pi)} of the
#' circular sample on \eqn{S^{1}}. Other objects accepted are an array of size
#' \code{c(n, p, 1)} with directions (in Cartesian coordinates), or a vector of
#' size \code{n} or an array of size \code{c(n, 1, 1)} with angular data.
#' Must not contain \code{NA}'s.
#' @param type type of test to be applied. A character vector containing any of
#' the following types of tests, depending on the dimension \eqn{p}:
#' \itemize{
#' \item Circular data: any of the names available at object
#' \code{\link{avail_cir_tests}}.
#' \item (Hyper)spherical data: any of the names available at object
#' \code{\link{avail_sph_tests}}.
#' }
#' If \code{type = "all"} (default), then \code{type} is set as
#' \code{avail_cir_tests} or \code{avail_sph_tests}, depending on the value of
#' \eqn{p}.
#' @param p_value type of \eqn{p}-value computation. Either \code{"MC"} for
#' employing the approximation by Monte Carlo of the exact null distribution,
#' \code{"asymp"} (default) for the use of the asymptotic/exact null
#' distribution (if available), or \code{"crit_val"} for approximation by means
#' of the table of critical values \code{crit_val}.
#' @param alpha vector with significance levels. Defaults to
#' \code{c(0.10, 0.05, 0.01)}.
#' @inheritParams unif_stat_distr
#' @inheritParams unif_stat_MC
#' @param crit_val table with critical values for the tests, to be used if
#' \code{p_value = "crit_val"}. A data frame, with column names containing the
#' character vector \code{type} and rows corresponding to the significance
#' levels \code{alpha}, that results from extracting \code{$crit_val_MC} from
#' a call to \code{\link{unif_stat_MC}}. Internally computed if
#' \code{NULL} (default).
#' @inheritParams unif_stat
#' @param ... If \code{p_value = "MC"} or \code{p_value = "crit_val"}, optional
#' performance parameters to be passed to \code{\link{unif_stat_MC}}:
#' \code{chunks}, \code{cores}, and \code{seed}.
#' @return If only a \bold{single test} is performed, a list with class
#' \code{htest} containing the following components:
#' \itemize{
#' \item \code{statistic}: the value of the test statistic.
#' \item \code{p.value}: the p-value of the test. If
#' \code{p_value = "crit_val"}, an \code{NA}.
#' \item \code{alternative}: a character string describing the alternative
#' hypothesis.
#' \item \code{method}: a character string indicating what type of test was
#' performed.
#' \item \code{data.name}: a character string giving the name of the data.
#' \item \code{reject}: the rejection decision for the levels of significance
#' \code{alpha}.
#' \item \code{crit_val}: a vector with the critical values for the
#' significance levels \code{alpha} used with \code{p_value = "MC"} or
#' \code{p_value = "asymp"}.
#' }
#' If \bold{several tests} are performed, a \code{type}-named list with
#' entries for each test given by the above list.
#' @details
#' All the tests reject for large values of the test statistic, so the critical
#' values for the significance levels \code{alpha} correspond to the
#' \code{alpha}-upper quantiles of the null distribution of the test statistic.
#'
#' When \code{p_value = "asymp"}, tests that do not have an implemented or
#' known asymptotic are omitted, and a warning is generated.
#'
#' When \code{p_value = "MC"}, it is possible to have a progress bar indicating
#' the Monte Carlo simulation progress if \code{unif_test} is wrapped with
#' \code{\link[progressr:with_progress]{progressr::with_progress}} or if
#' \code{progressr::handlers(global = TRUE)} is invoked (once) by the user.
#' See the examples below. The progress bar is updated with the number of
#' finished chunks.
#'
#' All the statistics are continuous random variables except the
#' Hodges--Ajne statistic (\code{"Hodges_Ajne"}), the Cressie statistic
#' (\code{"Cressie"}), and the number of (different) uncovered spacings
#' (\code{"Num_uncover"}). These three statistics are discrete random variables.
#'
#' The Monte Carlo calibration for the CCF09 test is made conditionally
#' on the choice of \cr\code{CCF09_dirs}. That is, all the Monte
#' Carlo statistics share the same random directions.
#'
#' 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. \emph{arXiv:1804.00286}.
#' \url{https://arxiv.org/abs/1804.00286}.
#' @examples
#' ## Asymptotic distribution
#'
#' # Circular data
#' n <- 10
#' samp_cir <- r_unif_cir(n = n)
#'
#' # Matrix
#' unif_test(data = samp_cir, type = "Ajne", p_value = "asymp")
#'
#' # Vector
#' unif_test(data = samp_cir[, 1], type = "Ajne", p_value = "asymp")
#'
#' # Array
#' unif_test(data = array(samp_cir, dim = c(n, 1, 1)), type = "Ajne",
#' p_value = "asymp")
#' \donttest{
#' # Several tests
#' unif_test(data = samp_cir, type = avail_cir_tests, p_value = "asymp")
#' }
#' # Spherical data
#' n <- 10
#' samp_sph <- r_unif_sph(n = n, p = 3)
#'
#' # Array
#' unif_test(data = samp_sph, type = "Bingham", p_value = "asymp")
#'
#' # Matrix
#' unif_test(data = samp_sph[, , 1], type = "Bingham", p_value = "asymp")
#' \donttest{
#' # Several tests
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "asymp")
#'
#' ## Monte Carlo
#'
#' # Circular data
#' unif_test(data = samp_cir, type = "Ajne", p_value = "MC")
#' unif_test(data = samp_cir, type = avail_cir_tests, p_value = "MC")
#'
#' # Spherical data
#' unif_test(data = samp_sph, type = "Bingham", p_value = "MC")
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC")
#'
#' # Caching stats_MC
#' stats_MC_cir <- unif_stat_MC(n = nrow(samp_cir), p = 2)$stats_MC
#' stats_MC_sph <- unif_stat_MC(n = nrow(samp_sph), p = 3)$stats_MC
#' unif_test(data = samp_cir, type = avail_cir_tests,
#' p_value = "MC", stats_MC = stats_MC_cir)
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC",
#' stats_MC = stats_MC_sph)
#'
#' ## Critical values
#'
#' # Circular data
#' unif_test(data = samp_cir, type = avail_cir_tests, p_value = "crit_val")
#'
#' # Spherical data
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "crit_val")
#'
#' # Caching crit_val
#' crit_val_cir <- unif_stat_MC(n = n, p = 2)$crit_val_MC
#' crit_val_sph <- unif_stat_MC(n = n, p = 3)$crit_val_MC
#' unif_test(data = samp_cir, type = avail_cir_tests,
#' p_value = "crit_val", crit_val = crit_val_cir)
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "crit_val",
#' crit_val = crit_val_sph)
#'
#' ## Specific arguments
#'
#' # Rothman
#' unif_test(data = samp_cir, type = "Rothman", Rothman_t = 0.5)
#'
#' # CCF09
#' unif_test(data = samp_sph, type = "CCF09", p_value = "MC",
#' CCF09_dirs = samp_sph[1:2, , 1])
#' unif_test(data = samp_sph, type = "CCF09", p_value = "MC",
#' CCF09_dirs = samp_sph[3:4, , 1])
#'
#' ## Using a progress bar when p_value = "MC"
#'
#' # Define a progress bar
#' require(progress)
#' require(progressr)
#' handlers(handler_progress(
#' format = ":spin [:bar] :percent Total: :elapsedfull End \u2248 :eta",
#' clear = FALSE))
#'
#' # Call unif_test() within with_progress()
#' with_progress(
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC",
#' chunks = 10, M = 1e3)
#' )
#'
#' # With several cores
#' with_progress(
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC",
#' cores = 2, chunks = 10, M = 1e3)
#' )
#'
#' # Instead of using with_progress() each time, it is more practical to run
#' # handlers(global = TRUE)
#' # once to activate progress bars in your R session
#' }
#' @export
unif_test <- function(data, type = "all", p_value = "asymp",
alpha = c(0.10, 0.05, 0.01), M = 1e4, stats_MC = NULL,
crit_val = NULL, data_sorted = FALSE, Rayleigh_m = 1,
cov_a = 2 * pi, Rothman_t = 1 / 3, Cressie_t = 1 / 3,
Pycke_q = 0.5, Riesz_s = 1, CCF09_dirs = NULL,
K_CCF09 = 25, CJ12_reg = 3, CJ12_beta = 0, K_max = 1e4,
...) {
# Read data's name
data_name <- deparse(substitute(data))
# Stop if NA's
if (anyNA(data)) {
stop("NAs present in data, please remove them.")
}
# If data is a vector, transform it to matrix
if (is.vector(data)) {
data <- matrix(data, ncol = 1)
}
# If data is an array, transform it to matrix
d <- dim(data)
l <- length(d)
if (l == 3) {
# As matrix
data <- matrix(data[, , 1], nrow = d[1], ncol = d[2])
# First slice only
if (d[3] != 1) {
message(paste("data is an array with more than one slice,",
"only the first one is employed."))
}
} else if (l > 3) {
stop("data must be a vector, matrix, or a 3-dimensional array.")
}
# Sample size and dimension
n <- nrow(data)
d <- ncol(data)
# Circular or spherical data?
if (d == 1 | d == 2) {
avail_stats <- avail_cir_tests
prefix_stat <- "cir_stat_"
p <- 2
# As polar coordinates
if (d == 2) {
dim(data) <- c(n, d, 1)
data <- X_to_Theta(X = data)
}
} else {
avail_stats <- avail_sph_tests
prefix_stat <- "sph_stat_"
p <- d
dim(data) <- c(n, d, 1) # As an array
}
# Get the type of statistics
if (is.character(type)) {
type <- gsub("-", "_", type)
type <- unique(type)
if ("all" %in% type) {
stats_type <- avail_stats
} else {
stats_type <- match.arg(arg = type, choices = avail_stats,
several.ok = TRUE)
}
} else if (is.numeric(type)) {
type <- unique(type)
stats_type <- avail_stats[type]
} else {
stop("type must be a character or a numeric vector.")
}
# Omit statistics that do not have asymptotic distribution
if (p_value == "asymp") {
ind_asymp <- sapply(paste0("p_", prefix_stat, stats_type),
exists, where = asNamespace("sphunif"))
if (!all(ind_asymp)) {
warning(paste0(paste("Omitting the following statistics with not",
"implemented or unknown asymptotic distributions: "),
paste(stats_type[!ind_asymp], collapse = ", "), "."))
stats_type <- stats_type[ind_asymp]
if (length(stats_type) == 0) {
stop("No remaining statistics to use.")
}
}
}
# Number of statistics
n_stats <- length(stats_type)
## Statistics
# Sample random directions for the CCF09 test
if ("CCF09" %in% stats_type & is.null(CCF09_dirs)) {
CCF09_dirs <- r_unif_sph(n = 50, p = p, M = 1)[, , 1]
}
# Compute statistics
stat <- unif_stat(data = data, type = stats_type, data_sorted = data_sorted,
Rayleigh_m = Rayleigh_m, cov_a = cov_a,
Rothman_t = Rothman_t, Cressie_t = Cressie_t,
Pycke_q = Pycke_q, Riesz_s = Riesz_s,
CCF09_dirs = CCF09_dirs, K_CCF09 = K_CCF09,
CJ12_reg = CJ12_reg)
## Calibration
if (p_value == "crit_val") {
# Get the critical values
if (is.null(crit_val)) {
crit_val <- unif_stat_MC(n = n, type = stats_type, p = p, M = M,
r_H1 = NULL, crit_val = NULL, alpha = alpha,
return_stats = FALSE, stats_sorted = FALSE,
Rayleigh_m = Rayleigh_m, cov_a = cov_a,
Rothman_t = Rothman_t, Cressie_t = Cressie_t,
Pycke_q = Pycke_q, Riesz_s = Riesz_s,
CCF09_dirs = CCF09_dirs, K_CCF09 = K_CCF09,
CJ12_reg = CJ12_reg, ...)$crit_val_MC
} else {
# Check if there is any missing statistic in crit_val
names_crit_val <- names(crit_val)
missing_crit_val <- !(stats_type %in% names_crit_val)
if (any(missing_crit_val)) {
stop(paste0("Missing statistics in crit_val: \"",
paste(stats_type[missing_crit_val],
collapse = "\", \""), "\"."))
}
# Match columns of crit_val with stats_type
crit_val <- crit_val[, pmatch(x = stats_type, table = names_crit_val),
drop = FALSE]
}
# Rejection?
reject <- rbind(apply(crit_val, 1, function(x) x > stat))
# p-values
p_val <- matrix(NA, nrow = 1, ncol = n_stats)
} else if (p_value == "MC") {
# Get the stats_MC
if (is.null(stats_MC)) {
stats_MC <- unif_stat_MC(n = n, type = stats_type, p = p, M = M,
r_H1 = NULL, crit_val = NULL, alpha = alpha,
return_stats = TRUE, stats_sorted = TRUE,
Rothman_t = Rothman_t, Pycke_q = Pycke_q,
Riesz_s = Riesz_s, CCF09_dirs = CCF09_dirs,
CJ12_reg = CJ12_reg, K_CCF09 = K_CCF09,
...)$stats_MC
}
# p-values
p_val <- 1 - as.data.frame(sapply(stats_type, function(distr) {
ecdf_bin(data = stats_MC[[distr]], sorted_x = stat[[distr]],
data_sorted = TRUE, efic = TRUE, divide_n = TRUE)
}, simplify = FALSE))
# Critical values
crit_val <- apply(stats_MC, 2, quantile, probs = 1 - alpha, na.rm = TRUE)
rownames(crit_val) <- alpha
# Rejection?
reject <- rbind(apply(crit_val, 1, function(x) stat >= x))
} else if (p_value == "asymp") {
# p-values
p_val <- 1 - unif_stat_distr(x = stat, type = stats_type, p = p, n = n,
approx = "asymp", stats_MC = NULL, M = M,
cov_a = cov_a, Rothman_t = Rothman_t,
Cressie_t = Cressie_t, Pycke_q = Pycke_q,
Riesz_s = Riesz_s, CCF09_dirs = CCF09_dirs,
K_CCF09 = K_CCF09, CJ12_reg = CJ12_reg,
CJ12_beta = CJ12_beta, K_max = K_max,
thre = 0, ...)
# Critical values
crit_val <- as.data.frame(matrix(NA, nrow = length(alpha), ncol = n_stats))
rownames(crit_val) <- alpha
# Rejection?
reject <- rbind(sapply(alpha, function(a) p_val < a))
colnames(reject) <- alpha
} else {
stop(paste("Wrong choice for calibration, must be \"MC\", \"asymp\",",
"or \"crit_val\"."))
}
## Prepare return object
# Create list of htest objects
test <- vector(mode = "list", length = n_stats)
names(test) <- stats_type
for (i in seq_along(stats_type)) {
# Type of test
if (p == 2) {
method <- switch(stats_type[i],
"Ajne" = "Ajne test of circular uniformity",
"Bakshaev" = "Bakshaev (2010) test of circular uniformity",
"Bingham" = "Bingham test of circular uniformity",
"CCF09" = paste("Cuesta-Albertos et al. (2009) test of circular",
"uniformity with k =", nrow(CCF09_dirs)),
"Cressie" = paste("Cressie test of circular uniformity with t =",
round(Cressie_t, 3)),
"FG01" = paste("Cramer-von Mises 4-point test of Feltz and",
"Goldin (2001)"),
"Gine_Fn" = "Gine's Fn test of circular uniformity",
"Gine_Gn" = "Gine's Gn test of circular uniformity",
"Gini" = "Gini mean difference test of circular uniformity",
"Gini_squared" = paste("Gini squared mean difference test of",
"circular uniformity"),
"Greenwood" = "Greenwood spacings test of circular uniformity",
"Hermans_Rasson" = "Hermans-Rasson test of circular uniformity",
"Hodges_Ajne" = "Hodges-Ajne test of circular uniformity",
"Kuiper" = "Kuiper test of circular uniformity",
"Log_gaps" = paste("Darling\'s log-gaps spacings test of",
"circular uniformity"),
"Max_uncover" = paste("Maximum uncovered spacing test of",
"circular uniformity with a =",
round(cov_a, 3)),
"Num_uncover" = paste("Number of uncovered spacings test of",
"circular uniformity with a =",
round(cov_a, 3)),
"PAD" = paste("Projected Anderson-Darling test of",
"circular uniformity"),
"PCvM" = paste("Projected Cramer-von Mises test of",
"circular uniformity"),
"PRt" = paste("Projected Rothman test of circular uniformity",
"with t =", round(Rothman_t, 3)),
"Pycke" = "Pycke test of circular uniformity",
"Pycke_q" = paste("Pycke \"q-test\" of spherical uniformity with q =",
round(Pycke_q, 3)),
"Range" = "Range test of circular uniformity",
"Rao" = "Rao\'s spacings test of circular uniformity",
"Rayleigh" = paste0("Rayleigh test of circular uniformity",
ifelse(Rayleigh_m > 1, paste0(" with m = ",
Rayleigh_m), "")),
"Riesz" = "Warning! This is an experimental test not meant to be used",
"Rothman" = paste("Rothman test of circular uniformity with t =",
round(Rothman_t, 3)),
"Vacancy" = paste("Vacancy test of circular uniformity with a =",
round(cov_a, 3)),
"Watson" = "Watson test of circular uniformity",
"Watson_1976" = "Watson (1976) test of circular uniformity"
)
alternative <- switch(stats_type[i],
"Ajne" = "any non-axial alternative to circular uniformity",
"Bakshaev" = "any alternative to circular uniformity",
"Bingham" = "scatter matrix different from constant",
"Cressie" = paste("any alternative to circular uniformity",
"if t is irrational (conjectured)"),
"CCF09" = "any alternative to circular uniformity",
"FG01" = "any alternative to circular uniformity",
"Gine_Fn" = "any alternative to circular uniformity",
"Gine_Gn" = "any axial alternative to circular uniformity",
"Gini" = "any alternative to circular uniformity",
"Gini_squared" = "any alternative to circular uniformity",
"Greenwood" = "any alternative to circular uniformity",
"Hermans_Rasson" = "any alternative to circular uniformity",
"Hodges_Ajne" = "any non-axial alternative to circular uniformity",
"Kuiper" = "any alternative to circular uniformity",
"Log_gaps" = "any alternative to circular uniformity",
"Max_uncover" = "any alternative to circular uniformity",
"Num_uncover" = paste("any alternative to circular uniformity",
"if a <= 2 * pi"),
"PAD" = "any alternative to circular uniformity",
"PCvM" = "any alternative to circular uniformity",
"PRt" = paste("any alternative to circular uniformity",
"if t is irrational"),
"Pycke" = "any alternative to circular uniformity",
"Pycke_q" = "any alternative to circular uniformity",
"Range" = "any alternative to circular uniformity",
"Rao" = "any alternative to circular uniformity",
"Rayleigh" = "mean direction different from zero",
"Rothman" = paste("any alternative to circular uniformity",
"if t is irrational"),
"Riesz" = "unclear, experimental test",
"Vacancy" = "any alternative to circular uniformity",
"Watson" = "any alternative to circular uniformity",
"Watson_1976" = "unclear consistency"
)
} else {
method <- switch(stats_type[i],
"Ajne" = "Ajne test of spherical uniformity",
"Bakshaev" = "Bakshaev (2010) test of spherical uniformity",
"Bingham" = "Bingham test of spherical uniformity",
"CJ12" = "Cai and Jiang (2012) test of spherical uniformity",
"CCF09" = paste("Cuesta-Albertos et al. (2009) test of spherical",
"uniformity with k =", nrow(CCF09_dirs)),
"Gine_Fn" = "Gine's Fn test of spherical uniformity",
"Gine_Gn" = "Gine's Gn test of spherical uniformity",
"PAD" = paste("Projected Anderson-Darling test of",
"spherical uniformity"),
"PCvM" = paste("Projected Cramer-von Mises test of",
"spherical uniformity"),
"PRt" = paste("Projected Rothman test of spherical uniformity",
"with t =", round(Rothman_t, 3)),
"Pycke" = "Pycke test of spherical uniformity",
"Rayleigh" = "Rayleigh test of spherical uniformity",
"Rayleigh_HD" = paste("HD-standardized Rayleigh test of",
"spherical uniformity"),
"Riesz" = "Warning! This is an experimental test not meant to be used"
)
alternative <- switch(stats_type[i],
"Ajne" = "any non-axial alternative to spherical uniformity",
"Bakshaev" = "any alternative to spherical uniformity",
"Bingham" = "scatter matrix different from constant",
"CJ12" = "unclear consistency",
"CCF09" = "any alternative to spherical uniformity",
"Gine_Fn" = "any alternative to spherical uniformity",
"Gine_Gn" = "any axial alternative to spherical uniformity",
"PCvM" = "any alternative to spherical uniformity",
"PAD" = "any alternative to spherical uniformity",
"PRt" = paste("any alternative to spherical uniformity",
"if t is irrational"),
"Pycke" = "any alternative to spherical uniformity",
"Rayleigh" = "mean direction different from zero",
"Rayleigh_HD" = "mean direction different from zero",
"Riesz" = "unclear, experimental test"
)
}
# htest object
test[[i]] <- list(statistic = c("statistic" = stat[1, i]),
p.value = p_val[1, i], alternative = alternative,
method = method, data.name = data_name,
reject = reject[i, ], crit_val = crit_val[, i])
class(test[[i]]) <- "htest"
}
# If there is only one test, return htest directly
if (n_stats == 1) {
test <- test[[1]]
}
return(test)
}
#' @title Available circular and (hyper)spherical uniformity tests
#'
#' @description Listing of the tests implemented in the \code{\link{sphunif}}
#' package.
#'
#' @return A character vector whose elements are valid inputs for the
#' \code{type} argument in \code{\link{unif_test}}, \code{\link{unif_stat}},
#' \code{\link{unif_stat_distr}}, and \code{\link{unif_stat_MC}}.
#' \code{avail_cir_tests} provides the available circular tests and
#' \code{avail_sph_tests} the (hyper)spherical tests.
#' @examples
#' # Circular tests
#' avail_cir_tests
#'
#' # Spherical tests
#' avail_sph_tests
#' @name avail_tests
#' @rdname avail_tests
#' @export
avail_cir_tests <- c("Ajne",
"Bakshaev",
"Bingham",
"Cressie",
"CCF09",
"FG01",
"Gine_Fn",
"Gine_Gn",
"Gini",
"Gini_squared",
"Greenwood",
"Hermans_Rasson",
"Hodges_Ajne",
"Kuiper",
"Log_gaps",
"Max_uncover",
"Num_uncover",
"PAD",
"PCvM",
"PRt",
"Pycke",
"Pycke_q",
"Range",
"Rao",
"Rayleigh",
"Riesz",
"Rothman",
"Vacancy",
"Watson",
"Watson_1976")
#' @rdname avail_tests
#' @export
avail_sph_tests <- c("Ajne",
"Bakshaev",
"Bingham",
"CJ12",
"CCF09",
"Gine_Fn",
"Gine_Gn",
"PAD",
"PCvM",
"PRt",
"Pycke",
"Rayleigh",
"Rayleigh_HD",
"Riesz")
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Defines functions unif_test
Documented in unif_test
#' @title Circular and (hyper)spherical uniformity tests
#'
#' @description Implementation of several uniformity tests on the (hyper)sphere
#' \eqn{S^{p-1}:=\{{\bf x}\in R^p:||{\bf x}||=1\}}{
#' S^{p-1}:=\{x\in R^p:||x||=1\}}, \eqn{p\ge 2}, with calibration either in
#' terms of their asymptotic/exact distributions, if available, or Monte Carlo.
#'
#' \code{unif_test} receives a sample of directions
#' \eqn{{\bf X}_1,\ldots,{\bf X}_n\in S^{p-1}}{X_1,\ldots,X_n\in S^{p-1}} in
#' \emph{Cartesian coordinates}, except for the circular case (\eqn{p=2}) in
#' which the sample can be represented in terms of \emph{angles}
#' \eqn{\Theta_1,\ldots,\Theta_n\in [0, 2\pi)}.
#'
#' \code{unif_test} allows to perform several tests within a single call,
#' facilitating thus the exploration of a dataset by applying several tests.
#'
#' @param data sample to perform the test. A matrix of size \code{c(n, p)}
#' containing a sample of size \code{n} of directions (in Cartesian
#' coordinates) on \eqn{S^{p-1}}. Alternatively if \code{p = 2}, a matrix of
#' size \code{c(n, 1)} containing the \code{n} angles on \eqn{[0, 2\pi)} of the
#' circular sample on \eqn{S^{1}}. Other objects accepted are an array of size
#' \code{c(n, p, 1)} with directions (in Cartesian coordinates), or a vector of
#' size \code{n} or an array of size \code{c(n, 1, 1)} with angular data.
#' Must not contain \code{NA}'s.
#' @param type type of test to be applied. A character vector containing any of
#' the following types of tests, depending on the dimension \eqn{p}:
#' \itemize{
#' \item Circular data: any of the names available at object
#' \code{\link{avail_cir_tests}}.
#' \item (Hyper)spherical data: any of the names available at object
#' \code{\link{avail_sph_tests}}.
#' }
#' If \code{type = "all"} (default), then \code{type} is set as
#' \code{avail_cir_tests} or \code{avail_sph_tests}, depending on the value of
#' \eqn{p}.
#' @param p_value type of \eqn{p}-value computation. Either \code{"MC"} for
#' employing the approximation by Monte Carlo of the exact null distribution,
#' \code{"asymp"} (default) for the use of the asymptotic/exact null
#' distribution (if available), or \code{"crit_val"} for approximation by means
#' of the table of critical values \code{crit_val}.
#' @param alpha vector with significance levels. Defaults to
#' \code{c(0.10, 0.05, 0.01)}.
#' @inheritParams unif_stat_distr
#' @inheritParams unif_stat_MC
#' @param crit_val table with critical values for the tests, to be used if
#' \code{p_value = "crit_val"}. A data frame, with column names containing the
#' character vector \code{type} and rows corresponding to the significance
#' levels \code{alpha}, that results from extracting \code{$crit_val_MC} from
#' a call to \code{\link{unif_stat_MC}}. Internally computed if
#' \code{NULL} (default).
#' @inheritParams unif_stat
#' @param ... If \code{p_value = "MC"} or \code{p_value = "crit_val"}, optional
#' performance parameters to be passed to \code{\link{unif_stat_MC}}:
#' \code{chunks}, \code{cores}, and \code{seed}.
#' @return If only a \bold{single test} is performed, a list with class
#' \code{htest} containing the following components:
#' \itemize{
#' \item \code{statistic}: the value of the test statistic.
#' \item \code{p.value}: the p-value of the test. If
#' \code{p_value = "crit_val"}, an \code{NA}.
#' \item \code{alternative}: a character string describing the alternative
#' hypothesis.
#' \item \code{method}: a character string indicating what type of test was
#' performed.
#' \item \code{data.name}: a character string giving the name of the data.
#' \item \code{reject}: the rejection decision for the levels of significance
#' \code{alpha}.
#' \item \code{crit_val}: a vector with the critical values for the
#' significance levels \code{alpha} used with \code{p_value = "MC"} or
#' \code{p_value = "asymp"}.
#' }
#' If \bold{several tests} are performed, a \code{type}-named list with
#' entries for each test given by the above list.
#' @details
#' All the tests reject for large values of the test statistic, so the critical
#' values for the significance levels \code{alpha} correspond to the
#' \code{alpha}-upper quantiles of the null distribution of the test statistic.
#'
#' When \code{p_value = "asymp"}, tests that do not have an implemented or
#' known asymptotic are omitted, and a warning is generated.
#'
#' When \code{p_value = "MC"}, it is possible to have a progress bar indicating
#' the Monte Carlo simulation progress if \code{unif_test} is wrapped with
#' \code{\link[progressr:with_progress]{progressr::with_progress}} or if
#' \code{progressr::handlers(global = TRUE)} is invoked (once) by the user.
#' See the examples below. The progress bar is updated with the number of
#' finished chunks.
#'
#' All the statistics are continuous random variables except the
#' Hodges--Ajne statistic (\code{"Hodges_Ajne"}), the Cressie statistic
#' (\code{"Cressie"}), and the number of (different) uncovered spacings
#' (\code{"Num_uncover"}). These three statistics are discrete random variables.
#'
#' The Monte Carlo calibration for the CCF09 test is made conditionally
#' on the choice of \cr\code{CCF09_dirs}. That is, all the Monte
#' Carlo statistics share the same random directions.
#'
#' 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. \emph{arXiv:1804.00286}.
#' \url{https://arxiv.org/abs/1804.00286}.
#' @examples
#' ## Asymptotic distribution
#'
#' # Circular data
#' n <- 10
#' samp_cir <- r_unif_cir(n = n)
#'
#' # Matrix
#' unif_test(data = samp_cir, type = "Ajne", p_value = "asymp")
#'
#' # Vector
#' unif_test(data = samp_cir[, 1], type = "Ajne", p_value = "asymp")
#'
#' # Array
#' unif_test(data = array(samp_cir, dim = c(n, 1, 1)), type = "Ajne",
#' p_value = "asymp")
#' \donttest{
#' # Several tests
#' unif_test(data = samp_cir, type = avail_cir_tests, p_value = "asymp")
#' }
#' # Spherical data
#' n <- 10
#' samp_sph <- r_unif_sph(n = n, p = 3)
#'
#' # Array
#' unif_test(data = samp_sph, type = "Bingham", p_value = "asymp")
#'
#' # Matrix
#' unif_test(data = samp_sph[, , 1], type = "Bingham", p_value = "asymp")
#' \donttest{
#' # Several tests
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "asymp")
#'
#' ## Monte Carlo
#'
#' # Circular data
#' unif_test(data = samp_cir, type = "Ajne", p_value = "MC")
#' unif_test(data = samp_cir, type = avail_cir_tests, p_value = "MC")
#'
#' # Spherical data
#' unif_test(data = samp_sph, type = "Bingham", p_value = "MC")
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC")
#'
#' # Caching stats_MC
#' stats_MC_cir <- unif_stat_MC(n = nrow(samp_cir), p = 2)$stats_MC
#' stats_MC_sph <- unif_stat_MC(n = nrow(samp_sph), p = 3)$stats_MC
#' unif_test(data = samp_cir, type = avail_cir_tests,
#' p_value = "MC", stats_MC = stats_MC_cir)
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC",
#' stats_MC = stats_MC_sph)
#'
#' ## Critical values
#'
#' # Circular data
#' unif_test(data = samp_cir, type = avail_cir_tests, p_value = "crit_val")
#'
#' # Spherical data
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "crit_val")
#'
#' # Caching crit_val
#' crit_val_cir <- unif_stat_MC(n = n, p = 2)$crit_val_MC
#' crit_val_sph <- unif_stat_MC(n = n, p = 3)$crit_val_MC
#' unif_test(data = samp_cir, type = avail_cir_tests,
#' p_value = "crit_val", crit_val = crit_val_cir)
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "crit_val",
#' crit_val = crit_val_sph)
#'
#' ## Specific arguments
#'
#' # Rothman
#' unif_test(data = samp_cir, type = "Rothman", Rothman_t = 0.5)
#'
#' # CCF09
#' unif_test(data = samp_sph, type = "CCF09", p_value = "MC",
#' CCF09_dirs = samp_sph[1:2, , 1])
#' unif_test(data = samp_sph, type = "CCF09", p_value = "MC",
#' CCF09_dirs = samp_sph[3:4, , 1])
#'
#' ## Using a progress bar when p_value = "MC"
#'
#' # Define a progress bar
#' require(progress)
#' require(progressr)
#' handlers(handler_progress(
#' format = ":spin [:bar] :percent Total: :elapsedfull End \u2248 :eta",
#' clear = FALSE))
#'
#' # Call unif_test() within with_progress()
#' with_progress(
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC",
#' chunks = 10, M = 1e3)
#' )
#'
#' # With several cores
#' with_progress(
#' unif_test(data = samp_sph, type = avail_sph_tests, p_value = "MC",
#' cores = 2, chunks = 10, M = 1e3)
#' )
#'
#' # Instead of using with_progress() each time, it is more practical to run
#' # handlers(global = TRUE)
#' # once to activate progress bars in your R session
#' }
#' @export
unif_test <- function(data, type = "all", p_value = "asymp",
alpha = c(0.10, 0.05, 0.01), M = 1e4, stats_MC = NULL,
crit_val = NULL, data_sorted = FALSE, Rayleigh_m = 1,
cov_a = 2 * pi, Rothman_t = 1 / 3, Cressie_t = 1 / 3,
Pycke_q = 0.5, Riesz_s = 1, CCF09_dirs = NULL,
K_CCF09 = 25, CJ12_reg = 3, CJ12_beta = 0, K_max = 1e4,
...) {
# Read data's name
data_name <- deparse(substitute(data))
# Stop if NA's
if (anyNA(data)) {
stop("NAs present in data, please remove them.")
}
# If data is a vector, transform it to matrix
if (is.vector(data)) {
data <- matrix(data, ncol = 1)
}
# If data is an array, transform it to matrix
d <- dim(data)
l <- length(d)
if (l == 3) {
# As matrix
data <- matrix(data[, , 1], nrow = d[1], ncol = d[2])
# First slice only
if (d[3] != 1) {
message(paste("data is an array with more than one slice,",
"only the first one is employed."))
}
} else if (l > 3) {
stop("data must be a vector, matrix, or a 3-dimensional array.")
}
# Sample size and dimension
n <- nrow(data)
d <- ncol(data)
# Circular or spherical data?
if (d == 1 | d == 2) {
avail_stats <- avail_cir_tests
prefix_stat <- "cir_stat_"
p <- 2
# As polar coordinates
if (d == 2) {
dim(data) <- c(n, d, 1)
data <- X_to_Theta(X = data)
}
} else {
avail_stats <- avail_sph_tests
prefix_stat <- "sph_stat_"
p <- d
dim(data) <- c(n, d, 1) # As an array
}
# Get the type of statistics
if (is.character(type)) {
type <- gsub("-", "_", type)
type <- unique(type)
if ("all" %in% type) {
stats_type <- avail_stats
} else {
stats_type <- match.arg(arg = type, choices = avail_stats,
several.ok = TRUE)
}
} else if (is.numeric(type)) {
type <- unique(type)
stats_type <- avail_stats[type]
} else {
stop("type must be a character or a numeric vector.")
}
# Omit statistics that do not have asymptotic distribution
if (p_value == "asymp") {
ind_asymp <- sapply(paste0("p_", prefix_stat, stats_type),
exists, where = asNamespace("sphunif"))
if (!all(ind_asymp)) {
warning(paste0(paste("Omitting the following statistics with not",
"implemented or unknown asymptotic distributions: "),
paste(stats_type[!ind_asymp], collapse = ", "), "."))
stats_type <- stats_type[ind_asymp]
if (length(stats_type) == 0) {
stop("No remaining statistics to use.")
}
}
}
# Number of statistics
n_stats <- length(stats_type)
## Statistics
# Sample random directions for the CCF09 test
if ("CCF09" %in% stats_type & is.null(CCF09_dirs)) {
CCF09_dirs <- r_unif_sph(n = 50, p = p, M = 1)[, , 1]
}
# Compute statistics
stat <- unif_stat(data = data, type = stats_type, data_sorted = data_sorted,
Rayleigh_m = Rayleigh_m, cov_a = cov_a,
Rothman_t = Rothman_t, Cressie_t = Cressie_t,
Pycke_q = Pycke_q, Riesz_s = Riesz_s,
CCF09_dirs = CCF09_dirs, K_CCF09 = K_CCF09,
CJ12_reg = CJ12_reg)
## Calibration
if (p_value == "crit_val") {
# Get the critical values
if (is.null(crit_val)) {
crit_val <- unif_stat_MC(n = n, type = stats_type, p = p, M = M,
r_H1 = NULL, crit_val = NULL, alpha = alpha,
return_stats = FALSE, stats_sorted = FALSE,
Rayleigh_m = Rayleigh_m, cov_a = cov_a,
Rothman_t = Rothman_t, Cressie_t = Cressie_t,
Pycke_q = Pycke_q, Riesz_s = Riesz_s,
CCF09_dirs = CCF09_dirs, K_CCF09 = K_CCF09,
CJ12_reg = CJ12_reg, ...)$crit_val_MC
} else {
# Check if there is any missing statistic in crit_val
names_crit_val <- names(crit_val)
missing_crit_val <- !(stats_type %in% names_crit_val)
if (any(missing_crit_val)) {
stop(paste0("Missing statistics in crit_val: \"",
paste(stats_type[missing_crit_val],
collapse = "\", \""), "\"."))
}
# Match columns of crit_val with stats_type
crit_val <- crit_val[, pmatch(x = stats_type, table = names_crit_val),
drop = FALSE]
}
# Rejection?
reject <- rbind(apply(crit_val, 1, function(x) x > stat))
# p-values
p_val <- matrix(NA, nrow = 1, ncol = n_stats)
} else if (p_value == "MC") {
# Get the stats_MC
if (is.null(stats_MC)) {
stats_MC <- unif_stat_MC(n = n, type = stats_type, p = p, M = M,
r_H1 = NULL, crit_val = NULL, alpha = alpha,
return_stats = TRUE, stats_sorted = TRUE,
Rothman_t = Rothman_t, Pycke_q = Pycke_q,
Riesz_s = Riesz_s, CCF09_dirs = CCF09_dirs,
CJ12_reg = CJ12_reg, K_CCF09 = K_CCF09,
...)$stats_MC
}
# p-values
p_val <- 1 - as.data.frame(sapply(stats_type, function(distr) {
ecdf_bin(data = stats_MC[[distr]], sorted_x = stat[[distr]],
data_sorted = TRUE, efic = TRUE, divide_n = TRUE)
}, simplify = FALSE))
# Critical values
crit_val <- apply(stats_MC, 2, quantile, probs = 1 - alpha, na.rm = TRUE)
rownames(crit_val) <- alpha
# Rejection?
reject <- rbind(apply(crit_val, 1, function(x) stat >= x))
} else if (p_value == "asymp") {
# p-values
p_val <- 1 - unif_stat_distr(x = stat, type = stats_type, p = p, n = n,
approx = "asymp", stats_MC = NULL, M = M,
cov_a = cov_a, Rothman_t = Rothman_t,
Cressie_t = Cressie_t, Pycke_q = Pycke_q,
Riesz_s = Riesz_s, CCF09_dirs = CCF09_dirs,
K_CCF09 = K_CCF09, CJ12_reg = CJ12_reg,
CJ12_beta = CJ12_beta, K_max = K_max,
thre = 0, ...)
# Critical values
crit_val <- as.data.frame(matrix(NA, nrow = length(alpha), ncol = n_stats))
rownames(crit_val) <- alpha
# Rejection?
reject <- rbind(sapply(alpha, function(a) p_val < a))
colnames(reject) <- alpha
} else {
stop(paste("Wrong choice for calibration, must be \"MC\", \"asymp\",",
"or \"crit_val\"."))
}
## Prepare return object
# Create list of htest objects
test <- vector(mode = "list", length = n_stats)
names(test) <- stats_type
for (i in seq_along(stats_type)) {
# Type of test
if (p == 2) {
method <- switch(stats_type[i],
"Ajne" = "Ajne test of circular uniformity",
"Bakshaev" = "Bakshaev (2010) test of circular uniformity",
"Bingham" = "Bingham test of circular uniformity",
"CCF09" = paste("Cuesta-Albertos et al. (2009) test of circular",
"uniformity with k =", nrow(CCF09_dirs)),
"Cressie" = paste("Cressie test of circular uniformity with t =",
round(Cressie_t, 3)),
"FG01" = paste("Cramer-von Mises 4-point test of Feltz and",
"Goldin (2001)"),
"Gine_Fn" = "Gine's Fn test of circular uniformity",
"Gine_Gn" = "Gine's Gn test of circular uniformity",
"Gini" = "Gini mean difference test of circular uniformity",
"Gini_squared" = paste("Gini squared mean difference test of",
"circular uniformity"),
"Greenwood" = "Greenwood spacings test of circular uniformity",
"Hermans_Rasson" = "Hermans-Rasson test of circular uniformity",
"Hodges_Ajne" = "Hodges-Ajne test of circular uniformity",
"Kuiper" = "Kuiper test of circular uniformity",
"Log_gaps" = paste("Darling\'s log-gaps spacings test of",
"circular uniformity"),
"Max_uncover" = paste("Maximum uncovered spacing test of",
"circular uniformity with a =",
round(cov_a, 3)),
"Num_uncover" = paste("Number of uncovered spacings test of",
"circular uniformity with a =",
round(cov_a, 3)),
"PAD" = paste("Projected Anderson-Darling test of",
"circular uniformity"),
"PCvM" = paste("Projected Cramer-von Mises test of",
"circular uniformity"),
"PRt" = paste("Projected Rothman test of circular uniformity",
"with t =", round(Rothman_t, 3)),
"Pycke" = "Pycke test of circular uniformity",
"Pycke_q" = paste("Pycke \"q-test\" of spherical uniformity with q =",
round(Pycke_q, 3)),
"Range" = "Range test of circular uniformity",
"Rao" = "Rao\'s spacings test of circular uniformity",
"Rayleigh" = paste0("Rayleigh test of circular uniformity",
ifelse(Rayleigh_m > 1, paste0(" with m = ",
Rayleigh_m), "")),
"Riesz" = "Warning! This is an experimental test not meant to be used",
"Rothman" = paste("Rothman test of circular uniformity with t =",
round(Rothman_t, 3)),
"Vacancy" = paste("Vacancy test of circular uniformity with a =",
round(cov_a, 3)),
"Watson" = "Watson test of circular uniformity",
"Watson_1976" = "Watson (1976) test of circular uniformity"
)
alternative <- switch(stats_type[i],
"Ajne" = "any non-axial alternative to circular uniformity",
"Bakshaev" = "any alternative to circular uniformity",
"Bingham" = "scatter matrix different from constant",
"Cressie" = paste("any alternative to circular uniformity",
"if t is irrational (conjectured)"),
"CCF09" = "any alternative to circular uniformity",
"FG01" = "any alternative to circular uniformity",
"Gine_Fn" = "any alternative to circular uniformity",
"Gine_Gn" = "any axial alternative to circular uniformity",
"Gini" = "any alternative to circular uniformity",
"Gini_squared" = "any alternative to circular uniformity",
"Greenwood" = "any alternative to circular uniformity",
"Hermans_Rasson" = "any alternative to circular uniformity",
"Hodges_Ajne" = "any non-axial alternative to circular uniformity",
"Kuiper" = "any alternative to circular uniformity",
"Log_gaps" = "any alternative to circular uniformity",
"Max_uncover" = "any alternative to circular uniformity",
"Num_uncover" = paste("any alternative to circular uniformity",
"if a <= 2 * pi"),
"PAD" = "any alternative to circular uniformity",
"PCvM" = "any alternative to circular uniformity",
"PRt" = paste("any alternative to circular uniformity",
"if t is irrational"),
"Pycke" = "any alternative to circular uniformity",
"Pycke_q" = "any alternative to circular uniformity",
"Range" = "any alternative to circular uniformity",
"Rao" = "any alternative to circular uniformity",
"Rayleigh" = "mean direction different from zero",
"Rothman" = paste("any alternative to circular uniformity",
"if t is irrational"),
"Riesz" = "unclear, experimental test",
"Vacancy" = "any alternative to circular uniformity",
"Watson" = "any alternative to circular uniformity",
"Watson_1976" = "unclear consistency"
)
} else {
method <- switch(stats_type[i],
"Ajne" = "Ajne test of spherical uniformity",
"Bakshaev" = "Bakshaev (2010) test of spherical uniformity",
"Bingham" = "Bingham test of spherical uniformity",
"CJ12" = "Cai and Jiang (2012) test of spherical uniformity",
"CCF09" = paste("Cuesta-Albertos et al. (2009) test of spherical",
"uniformity with k =", nrow(CCF09_dirs)),
"Gine_Fn" = "Gine's Fn test of spherical uniformity",
"Gine_Gn" = "Gine's Gn test of spherical uniformity",
"PAD" = paste("Projected Anderson-Darling test of",
"spherical uniformity"),
"PCvM" = paste("Projected Cramer-von Mises test of",
"spherical uniformity"),
"PRt" = paste("Projected Rothman test of spherical uniformity",
"with t =", round(Rothman_t, 3)),
"Pycke" = "Pycke test of spherical uniformity",
"Rayleigh" = "Rayleigh test of spherical uniformity",
"Rayleigh_HD" = paste("HD-standardized Rayleigh test of",
"spherical uniformity"),
"Riesz" = "Warning! This is an experimental test not meant to be used"
)
alternative <- switch(stats_type[i],
"Ajne" = "any non-axial alternative to spherical uniformity",
"Bakshaev" = "any alternative to spherical uniformity",
"Bingham" = "scatter matrix different from constant",
"CJ12" = "unclear consistency",
"CCF09" = "any alternative to spherical uniformity",
"Gine_Fn" = "any alternative to spherical uniformity",
"Gine_Gn" = "any axial alternative to spherical uniformity",
"PCvM" = "any alternative to spherical uniformity",
"PAD" = "any alternative to spherical uniformity",
"PRt" = paste("any alternative to spherical uniformity",
"if t is irrational"),
"Pycke" = "any alternative to spherical uniformity",
"Rayleigh" = "mean direction different from zero",
"Rayleigh_HD" = "mean direction different from zero",
"Riesz" = "unclear, experimental test"
)
}
# htest object
test[[i]] <- list(statistic = c("statistic" = stat[1, i]),
p.value = p_val[1, i], alternative = alternative,
method = method, data.name = data_name,
reject = reject[i, ], crit_val = crit_val[, i])
class(test[[i]]) <- "htest"
}
# If there is only one test, return htest directly
if (n_stats == 1) {
test <- test[[1]]
}
return(test)
}
#' @title Available circular and (hyper)spherical uniformity tests
#'
#' @description Listing of the tests implemented in the \code{\link{sphunif}}
#' package.
#'
#' @return A character vector whose elements are valid inputs for the
#' \code{type} argument in \code{\link{unif_test}}, \code{\link{unif_stat}},
#' \code{\link{unif_stat_distr}}, and \code{\link{unif_stat_MC}}.
#' \code{avail_cir_tests} provides the available circular tests and
#' \code{avail_sph_tests} the (hyper)spherical tests.
#' @examples
#' # Circular tests
#' avail_cir_tests
#'
#' # Spherical tests
#' avail_sph_tests
#' @name avail_tests
#' @rdname avail_tests
#' @export
avail_cir_tests <- c("Ajne",
"Bakshaev",
"Bingham",
"Cressie",
"CCF09",
"FG01",
"Gine_Fn",
"Gine_Gn",
"Gini",
"Gini_squared",
"Greenwood",
"Hermans_Rasson",
"Hodges_Ajne",
"Kuiper",
"Log_gaps",
"Max_uncover",
"Num_uncover",
"PAD",
"PCvM",
"PRt",
"Pycke",
"Pycke_q",
"Range",
"Rao",
"Rayleigh",
"Riesz",
"Rothman",
"Vacancy",
"Watson",
"Watson_1976")
#' @rdname avail_tests
#' @export
avail_sph_tests <- c("Ajne",
"Bakshaev",
"Bingham",
"CJ12",
"CCF09",
"Gine_Fn",
"Gine_Gn",
"PAD",
"PCvM",
"PRt",
"Pycke",
"Rayleigh",
"Rayleigh_HD",
"Riesz")
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