```
#' Permutation procedure that calculates the squared 2-Wasserstein distances for
#' random shuffles of two input samples representing two conditions
#'
#' Permutation procedure that calculates the squared 2-Wasserstein distances for
#' random shuffles of two input samples representing two conditions \eqn{A} and \eqn{B}, respectively (i.e. the elements of the pooled sample vector are randomly assigned to either condition \eqn{A} or condition \eqn{B}, where the sample sizes are as in the original samples).
#'
#' @param x sample (vector) representing the distribution of
#' condition \eqn{A}
#' @param y sample (vector) representing the distribution of
#' condition \eqn{B}
#' @param permnum number of permutations to be
#' performed
#'
#' @return Vector with squared 2-Wasserstein distances computed for random
#' shuffles of the two input samples
#'
.wassPermProcedure <- function(x, y, permnum) {
z <- c(x,y)
shuffle <- permutations(z, num_permutations=permnum)
return(apply( shuffle, 2,
function (k) {
return( wasserstein_metric(
k[seq_len(length(x))],
k[seq((length(x)+1), length(z))],
p=2) **2)
}))
}
#' Semi-parametric test using the 2-Wasserstein distance to check for differential distributions
#'
#' Two-sample test to check for differences between two distributions
#' using the 2-Wasserstein distance: Semi-parametric
#' implementation using a permutation test with a generalized Pareto
#' distribution (GPD) approximation to estimate small p-values accurately
#'
#' This is the semi-parametric version of \code{wasserstein.test}, for the
#' asymptotic theory-based procedure see \code{.wassersteinTestAsy}.
#'
#'@details Details concerning the permutation testing procedure with GPD
#' approximation to estimate small p-values accurately can be found in Schefzik
#' et al. (2020).
#'
#'@param x sample (vector) representing the distribution of
#' condition \eqn{A}
#'@param y sample (vector) representing the distribution of
#' condition \eqn{B}
#'@param permnum number of permutations used in the permutation testing
#' procedure
#'@return A vector of 15, see Schefzik et al. (2020) for details:
#' \itemize{
#' \item d.wass: 2-Wasserstein distance between the two samples computed by
#' quantile approximation
#' \item d.wass^2: squared 2-Wasserstein distance between the two samples
#' computed by quantile approximation
#' \item d.comp^2: squared 2-Wasserstein distance between the two samples
#' computed by decomposition approximation
#' \item d.comp: 2-Wasserstein distance between the two samples computed by
#' decomposition approximation
#' \item location: location term in the decomposition of the squared
#' 2-Wasserstein distance between the two samples
#' \item size: size term in the decomposition of the squared 2-Wasserstein
#' distance between the two samples
#' \item shape: shape term in the decomposition of the squared 2-Wasserstein
#' distance between the two samples
#' \item rho: correlation coefficient in the quantile-quantile plot
#' \item pval: p-value of the semi-parametric 2-Wasserstein distance-based
#' test
#' \item p.ad.gpd: in case the GPD fitting is performed: p-value of the
#' Anderson-Darling test to check whether the GPD actually fits the data well
#' (otherwise NA).
#' \item N.exc: in case the GPD fitting is performed: number of exceedances
#' (starting with 250 and iteratively decreased by 10 if necessary) that are
#' required to obtain a good GPD fit, i.e. p-value of Anderson-Darling test
#' \eqn{\geq 0.05} (otherwise NA).
#' \item perc.loc: fraction (in \%) of the location part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item perc.size: fraction (in \%) of the size part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item perc.shape: fraction (in \%) of the shape part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item decomp.error: relative error between the squared 2-Wasserstein
#' distance obtained by the quantile approximation and the squared
#' 2-Wasserstein distance obtained by the decomposition approximation
#' }
#'
#'@references Schefzik, R., Flesch, J., and Goncalves, A. (2020). waddR: Using the 2-Wasserstein distance to identify differences between distributions in two-sample testing, with application to single-cell RNA-sequencing data.
.wassersteinTestSp <- function(x, y, permnum=10000){
stopifnot(permnum>0)
if (length(x) !=0 & length(y) != 0){
# wasserstein distance between the samples
value <- wasserstein_metric(x, y, p=2)
value.sq <- value**2
# permutation procedure to calculate the wasserstein distances of
# random shuffles of x and y
bsn <- permnum
wass.values <- .wassPermProcedure(x, y, bsn)
wass.values.ordered <- sort(wass.values, decreasing=TRUE)
# computation of an approximative p-value
num.extr <- sum(wass.values >= value.sq)
pvalue.ecdf <- num.extr/bsn
pvalue.ecdf.pseudo <- (1 + num.extr) / (bsn + 1)
# gpd fitting needed
pvalue.wass <- pvalue.ecdf
pvalue.gpdfit <- NA
N.exc <- NA
env <- environment()
if (num.extr < 10) {
tryCatch({
res <- .gpdFittedPValue(value.sq,
wass.values.ordered)
assign("pvalue.wass", unname(res["pvalue.gpd"]), env)
assign("pvalue.gpdfit", unname(res["ad.pval"]), env)
assign("N.exc", unname(res["N.exc"]), env)
}, error=function(...) {
assign("pvalue.wass", pvalue.ecdf.pseudo, env)
assign("pvalue.gpdfit", NA, env)
assign("N.exc", NA, env)
})
# For now, just use pseudo pvalues
#assign("pvalue.wass", pvalue.ecdf.pseudo, env)
}
# correlation of quantile-quantile plot
rho.xy <- .quantileCorrelation(x, y)
# decomposition of wasserstein distance
wass.comp <- squared_wass_decomp(x, y)
location <- wass.comp$location
size <- wass.comp$size
shape <- wass.comp$shape
d.comp.sq <- wass.comp$distance
if (is.na(d.comp.sq)) {
d.comp.sq <- sum(na.exclude(location, shape, size))
}
d.comp <- sqrt(d.comp.sq)
perc.loc <- round(((location / d.comp.sq) * 100), 2)
perc.size <- round(((size / d.comp.sq) * 100), 2)
perc.shape <- round(((shape / d.comp.sq)*100), 2)
decomp.error <- .relativeError(d.comp.sq, value.sq)
output <- c("d.wass"=value, "d.wass^2"=value.sq, "d.comp^2"=d.comp.sq,
"d.comp"=d.comp, "location"=location, "size"=size,
"shape"=shape, "rho"=rho.xy, "pval"=pvalue.wass,
"p.ad.gpd"=pvalue.gpdfit, "N.exc"=N.exc,
"perc.loc"=perc.loc, "perc.size"=perc.size,
"perc.shape"=perc.shape, "decomp.error"=decomp.error)
} else { output <- c("d.wass"=NA, "d.wass^2"=NA, "d.comp^2"=NA,
"d.comp"=NA, "location"=NA, "size"=NA,
"shape"=NA, "rho"=NA, "pval"=NA,
"p.ad.gpd"=NA, "N.exc"=NA,
"perc.loc"=NA, "perc.size"=NA,
"perc.shape"=NA, "decomp.error"=NA) }
return(output)
}
#'Asymptotic theory-based test using the 2-Wasserstein distance to check for differential distributions
#'
#' Two-sample test to check for differences between two distributions
#' using the 2-Wasserstein distance: Implementation using a test
#' based on asymptotic theory
#'
#' This is the asymptotic version of \code{wasserstein.test}, for the
#' semi-parametric procedure see \code{.wassersteinTestSp}.
#'
#'@details Details concerning the testing procedure based on asymptotic theory
#' can be found in Schefzik et al (2020).
#'
#' Note that the asymptotic theory-based test should only be employed when the two samples \eqn{x} and \eqn{y} can be assumed to come from continuous distributions.
#'
#'@param x sample (vector) representing the distribution of
#' condition \eqn{A}
#'@param y sample (vector) representing the distribution of
#' condition \eqn{B}
#'
#'@return A vector of 13, see Schefzik et al. (2020) for details:
#' \itemize{
#' \item d.wass: 2-Wasserstein distance between the two samples computed
#' by quantile approximation
#' \item d.wass^2: squared 2-Wasserstein distance between the two samples
#' computed by quantile approximation
#' \item d.comp^2: squared 2-Wasserstein distance between the two samples
#' computed by decomposition approximation
#' \item d.comp: 2-Wasserstein distance between the two samples computed by
#' decomposition approximation
#' \item location: location term in the decomposition of the squared
#' 2-Wasserstein distance between the two samples
#' \item size: size term in the decomposition of the squared 2-Wasserstein
#' distance between the two samples
#' \item shape: shape term in the decomposition of the squared 2-Wasserstein
#' distance between the two samples
#' \item rho: correlation coefficient in the quantile-quantile plot
#' \item pval: p-value of the 2-Wasserstein distance-based test using
#' asymptotic theory
#' \item perc.loc: fraction (in \%) of the location part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item perc.size: fraction (in \%) of the size part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item perc.shape: fraction (in \%) of the shape part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item decomp.error: relative error between the squared 2-Wasserstein
#' distance obtained by the quantile approximation and the squared
#' 2-Wasserstein distance obtained by the decomposition approximation
#' }
#'
#'@references Schefzik, R., Flesch, J., and Goncalves, A. (2020). waddR: Using the 2-Wasserstein distance to identify differences between distributions in two-sample testing, with application to single-cell RNA-sequencing data.
.wassersteinTestAsy <- function(x, y){
if (length(x) != 0 & length(y) != 0) {
value <- wasserstein_metric(x, y, p=2)
value.sq <- value **2
# compute p-value based on asymptotoc theory (brownian bridge)
pr <- seq(from=0, to=1, by=1/10000)
empir.cdf.y <- ecdf(y)
parts.trf <- ((empir.cdf.y(quantile(x, probs=pr, type=1))) - pr) **2
trf.int <- (1 / length(pr)) * sum(parts.trf)
test.stat <- (length(x) * length(y)
/ (length(x) + length(y))) * trf.int
# p-value
pvalue.wass <- 1 - .brownianBridgeEmpcdf(test.stat)
# correlation of quantile-quantile plot
rho.xy <- .quantileCorrelation(x, y)
# decomposition of wasserstein distance
wass.comp <- squared_wass_decomp(x, y)
location <- wass.comp$location
size <- wass.comp$size
shape <- wass.comp$shape
d.comp.sq <- wass.comp$distance
if (is.na(d.comp.sq)) {
d.comp.sq <- sum(na.exclude(location, shape, size))
}
d.comp <- sqrt(d.comp.sq)
perc.loc <- round(((location / d.comp.sq) * 100), 2)
perc.size <- round(((size / d.comp.sq) * 100), 2)
perc.shape <- round(((shape / d.comp.sq)*100), 2)
decomp.error <- .relativeError(value.sq, d.comp.sq)
output <- c("d.wass"=value, "d.wass^2"=value.sq, "d.comp^2"=d.comp.sq,
"d.comp"=d.comp, "location"=location, "size"=size,
"shape"=shape, "rho"=rho.xy, "pval"=pvalue.wass,
"perc.loc"=perc.loc, "perc.size"=perc.size,
"perc.shape"=perc.shape, "decomp.error"=decomp.error)
} else { output <-c("d.wass"=NA, "d.wass^2"=NA, "d.comp^2"=NA,
"d.comp"=NA, "location"=NA, "size"=NA,
"shape"=NA, "rho"=NA, "pval"=NA,
"perc.loc"=NA, "perc.size"=NA,
"perc.shape"=NA, "decomp.error"=NA)
}
return(output)
}
#'Two-sample test to check for differences between two distributions
#'using the 2-Wasserstein distance
#'
#'Two-sample test to check for differences between two distributions
#'using the 2-Wasserstein distance, either using the
#'semi-parametric permutation testing procedure with a generalized Pareto distribution (GPD) approximation to
#'estimate small p-values accurately or the test based on asymptotic theory
#'
#'@name wasserstein.test
#'@details Details concerning the two testing procedures (i.e. the semi-parametric permutation
#' testing procedure with GPD approximation and the test based on asymptotic theory) can be found in
#' Schefzik et al. (2020).
#'
#' Note that the asymptotic theory-based test (\code{method="ASY"}) should only be employed when the samples \eqn{x} and \eqn{y} can be assumed to come from continuous distributions. In contrast, the semi-parametric test (\code{method="SP"}) can be used for samples coming from continuous or discrete distributions.
#'
#'@param x sample (vector) representing the distribution of
#' condition \eqn{A}
#'@param y sample (vector) representing the distribution of
#' condition \eqn{B}
#'@param method testing procedure to be employed: "SP" for the semi-parametric
#' permutation testing procedure with GPD approximation, "ASY" for the test based on asymptotic theory; if no method is specified, "SP" will be used by default.
#'@param permnum number of permutations used in the permutation testing
#' procedure (if \code{method="SP"} is performed); default is 10000
#'
#'@return A vector, see Schefzik et al. (2020) for details:
#' \itemize{
#' \item d.wass: 2-Wasserstein distance between the two samples computed by
#' quantile approximation
#' \item d.wass^2: squared 2-Wasserstein distance between the two samples
#' computed by quantile approximation
#' \item d.comp^2: squared 2-Wasserstein distance between the two samples
#' computed by decomposition approximation
#' \item d.comp: 2-Wasserstein distance between the two samples computed by
#' decomposition approximation
#' \item location: location term in the decomposition of the squared
#' 2-Wasserstein distance between the two samples
#' \item size: size term in the decomposition of the squared 2-Wasserstein
#' distance between the two samples
#' \item shape: shape term in the decomposition of the squared 2-Wasserstein
#' distance between the two samples
#' \item rho: correlation coefficient in the quantile-quantile plot
#' \item pval: The p-value of the semi-parametric or the asymptotic theory-based test, depending on the specified method
#' \item p.ad.gpd: in case the GPD fitting is performed: p-value of the
#' Anderson-Darling test to check whether the GPD actually fits the data well
#' (otherwise NA). This output is only returned when performing the
#' semi-parametric test (method="SP")!
#' \item N.exc: in case the GPD fitting is performed: number of exceedances
#' (starting with 250 and iteratively decreased by 10 if necessary) that are
#' required to obtain a good GPD fit, i.e. p-value of Anderson-Darling test
#' \eqn{\geq 0.05} (otherwise NA). This output is only returned when
#' performing the semi-parametric test (method="SP")!
#' \item perc.loc: fraction (in \%) of the location part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item perc.size: fraction (in \%) of the size part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item perc.shape: fraction (in \%) of the shape part with respect to the
#' overall squared 2-Wasserstein distance obtained by the decomposition
#' approximation
#' \item decomp.error: relative error between the squared 2-Wasserstein
#' distance computed by the quantile approximation and the squared
#' 2-Wasserstein distance computed by the decomposition approximation
#' }
#'
#'@references Schefzik, R., Flesch, J., and Goncalves, A. (2020). waddR: Using the 2-Wasserstein distance to identify differences between distributions in two-sample testing, with application to single-cell RNA-sequencing data.
#'
#'@examples
#' set.seed(24)
#' x<-rnorm(100)
#' y1<-rnorm(150)
#' y2<-rexp(150,3)
#' y3<-rpois(150,2)
#'
#' #for reproducibility, set a seed for the semi-parametric, permutation-based test
#' set.seed(32)
#' wasserstein.test(x,y1,method="SP",permnum=10000)
#' wasserstein.test(x,y1,method="ASY")
#'
#' set.seed(33)
#' wasserstein.test(x,y2,method="SP",permnum=10000)
#' wasserstein.test(x,y2,method="ASY")
#'
#' set.seed(34)
#' #only consider SP method, as Poisson distribution is discrete
#' wasserstein.test(x,y3,method="SP",permnum=10000)
#'
#'@export
#'
wasserstein.test <- function(x, y, method=c("SP", "ASY"), permnum=10000){
method <- match.arg(method)
switch(method,
"SP"=.wassersteinTestSp(x, y, permnum),
"ASY"=.wassersteinTestAsy(x, y))
}
```

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