#' Non-Parametric Jackstraw for K-means Clustering
#'
#' Test the cluster membership for K-means clustering
#'
#' K-means clustering assign \code{m} rows into \code{K} clusters. This function enable statistical
#' evaluation if the cluster membership is correctly assigned. Each of \code{m} p-values refers to
#' the statistical test of that row with regard to its assigned cluster.
#' Its resampling strategy accounts for the over-fitting characteristics due to direct computation of clusters from the observed data
#' and protects against an anti-conservative bias.
#'
#' The input data (\code{dat}) must be of a class `matrix`.
#'
#' @param dat a matrix with \code{m} rows as variables and \code{n} columns as observations.
#' @param kmeans.dat an output from applying \code{kmeans()} onto \code{dat}.
#' @param s a number of ``synthetic'' null variables. Out of \code{m} variables, \code{s} variables are independently permuted.
#' @param B a number of resampling iterations.
#' @param center a logical specifying to center the rows of the null samples. By default, \code{TRUE}.
#' @param covariate a model matrix of covariates with \code{n} observations. Must include an intercept in the first column.
#' @param match a logical specifying to match the observed clusters and jackstraw clusters using minimum Euclidean distances.
#' @param pool a logical specifying to pool the null statistics across all clusters. By default, \code{TRUE}.
#' @param verbose a logical specifying to print the computational progress. By default, \code{FALSE}.
#' @param ... optional arguments to control the k-means clustering algorithm (refers to \code{kmeans}).
#'
#' @return \code{jackstraw_kmeans} returns a list consisting of
#' \item{F.obs}{\code{m} observed F statistics between variables and cluster centers.}
#' \item{F.null}{F null statistics between null variables and cluster centers, from the jackstraw method.}
#' \item{p.F}{\code{m} p-values of membership.}
#'
#' @author Neo Christopher Chung \email{nchchung@@gmail.com}
#' @references Chung (2020) Statistical significance of cluster membership for unsupervised evaluation of cell identities. Bioinformatics, 36(10): 3107–3114 \url{https://academic.oup.com/bioinformatics/article/36/10/3107/5788523}
#' @examples
#' \dontrun{
#' dat = t(scale(t(Jurkat293T), center=TRUE, scale=FALSE))
#' kmeans.dat <- kmeans(dat, centers=2, nstart = 10, iter.max = 100)
#' jackstraw.out <- jackstraw_kmeans(dat, kmeans.dat)
#' }
#'
#' @export
jackstraw_kmeans <- function(
dat,
kmeans.dat,
s = NULL,
B = NULL,
center = FALSE,
covariate = NULL,
match = TRUE,
pool = TRUE,
verbose = FALSE,
...
) {
# check mandatory data
if ( missing( dat ) )
stop( '`dat` is required!' )
if ( missing( kmeans.dat ) )
stop( '`kmeans.dat` is required!' )
if ( !is.matrix( dat ) )
stop( '`dat` must be a matrix!' )
if ( !methods::is( kmeans.dat, "kmeans" ) )
stop( "`kmeans.dat` must be an object of class `kmeans`. See ?kmeans." )
m <- nrow(dat)
n <- ncol(dat)
# if there are covariates, the dimensions must agree
# covariate can be either a vector or a matrix, test both cases
if ( !is.null( covariate ) ) {
if ( is.matrix( covariate ) ) {
if ( nrow( covariate ) != n )
stop( 'Matrix `covariate` must have `n` rows, has: ', nrow( covariate ), ', expected: ', n )
} else {
if ( length( covariate ) != n )
stop( 'Vector `covariate` must have `n` elements, has: ', length( covariate ), ', expected: ', n )
}
}
if (is.null(s)) {
s <- round(m/10)
if (verbose)
message( "A number of null variables (s) to be permuted is not specified: s=round(0.10*m)=", s, "." )
}
if (is.null(B)) {
B <- round(m * 10/s)
if (verbose)
message( "A number of resampling iterations (B) is not specified: B=round(m*10/s)=", B, "." )
}
k <- nrow(kmeans.dat$centers)
obs.centers <- kmeans.dat$centers
rownames(obs.centers) <- paste0("obs",1:k)
if (verbose)
cat(paste0("\nComputating null statistics (", B, " total iterations): "))
# compute the observed
# statistics between rows and
# cluster centers
F.obs <- vector("numeric", m)
for (i in 1:k) {
F.obs[kmeans.dat$cluster == i] <- FSTAT(
dat[kmeans.dat$cluster == i, , drop = FALSE],
LV = t(kmeans.dat$centers[i, , drop = FALSE]),
covariate = covariate
)$fstat
}
# set-up empty matrices for
# null statistics
F.null <- vector("list", length = k)
for (j in 1:B) {
if (verbose)
cat(paste(j, " "))
jackstraw.dat <- dat
# randomly choose s variables
# to permute
ind <- sample.int( m, s )
jackstraw.dat[ind, ] <- apply(
dat[ind, , drop = FALSE],
1,
function(x) sample(x, replace = TRUE)
)
if (center)
jackstraw.dat[ind, ] <- t(scale(
t( jackstraw.dat[ ind, , drop = FALSE ] ),
center = TRUE,
scale = FALSE
))
# re-cluster the jackstraw data
kmeans.null <- stats::kmeans(
jackstraw.dat,
centers = kmeans.dat$centers,
...
)
# with stable clusters, numeric identities of clusters are typically matched, after resampling s
if (match) {
jck.centers <- kmeans.null$centers
rownames(jck.centers) <- paste0("jck",1:k)
# min euclidean dist to match jck.centers with obs.centers
jck.clmatch <- apply(
jck.centers,
1,
function(y) which.min(apply(obs.centers, 1, function(x,y) stats::dist(rbind(x,y)),y))
)
if (verbose)
message("Numeric identities of clusters are matched according to min euclidean distances.")
for (i in 1:k) {
ind.i <- intersect(
ind,
which(kmeans.null$cluster == which(jck.clmatch == i) )
)
if (verbose)
message( "Iteration ", j, ": jackstraw cluster ", which(jck.clmatch == i), " matched with obs cluster ", i,"; # intersections ",length(ind.i))
if (length(ind.i) > 0) {
F.null[[i]] <- c(
F.null[[i]],
as.vector(
FSTAT(
dat = jackstraw.dat[ind.i, , drop = FALSE],
LV = t(kmeans.null$centers[which(jck.clmatch == i), , drop = FALSE]),
covariate = covariate
)$fstat
)
)
}
}
} else {
# no matching required
for (i in 1:k) {
ind.i <- intersect(ind, which(kmeans.null$cluster == i))
if (length(ind.i) > 0) {
F.null[[i]] <- c(
F.null[[i]],
as.vector(
FSTAT(
dat = jackstraw.dat[ind.i, , drop = FALSE],
LV = t(kmeans.null$centers[i, , drop = FALSE]),
covariate = covariate
)$fstat
)
)
}
}
}
}
# compute p-values
p.F <- vector("numeric", m)
if (pool) {
p.F <- empPvals( F.obs, unlist( F.null ) )
} else {
for (i in 1:k) {
# warn about a relatively low
# number of null statistics
if (length(F.null[[i]]) == 0)
stop( "There are no null statistics for the cluster ", i, ". Check if B is large enough, clusters stable, and k selected appropriately. Also consider running the algorithm with the pool option." )
if (length(F.null[[i]]) < (B * s/k * 0.1))
warning( "The number of empirical null statistics for the cluster [", i, "] is [", length(F.null[[i]]), "]." )
p.F[kmeans.dat$cluster == i] <- empPvals( F.obs[kmeans.dat$cluster == i], F.null[[i]] )
}
}
return(
list(
call = match.call(),
F.obs = F.obs,
F.null = F.null,
p.F = p.F
)
)
}
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