R/stabilityIndex.R
In neobernad/evaluomeR: Evaluation of Bioinformatics Metrics
Defines functions
runStabilityIndexMetric_IMG
runStabilityIndexK_IMG
runStabilityIndexTableRange
runStabilityIndex
stabilitySet
stabilityRange
stability
Documented in
stability
stabilityRange
stabilitySet
#' @title Stability index.
#' @name stability
#' @aliases stability
#' @description
#' This analysis permits to estimate whether the clustering is meaningfully
#' affected by small variations in the sample. First, a clustering using the
#' k-means algorithm is carried out. The value of \code{k} can be provided by the user.
#' Then, the stability index is the mean of the Jaccard coefficient
#' values of a number of \code{bs} bootstrap replicates. The values are in the range [0,1],
#' having the following meaning:
#' \itemize{
#' \item Unstable: [0, 0.60[.
#' \item Doubtful: [0.60, 0.75].
#' \item Stable: ]0.75, 0.85].
#' \item Highly Stable: ]0.85, 1].
#' }
#'
#' @param data A \code{\link{SummarizedExperiment}}.
#' The SummarizedExperiment must contain an assay with the following structure:
#' A valid header with names. The first column of the header is the ID or name
#' of the instance of the dataset (e.g., ontology, pathway, etc.) on which the
#' metrics are measured.
#' The other columns of the header contains the names of the metrics.
#' The rows contains the measurements of the metrics for each instance in the dataset.
#' @param k Positive integer. Number of clusters between [2,15] range.
#' @param bs Positive integer. Bootstrap value to perform the resampling.
#' @param cbi Clusterboot interface name (default: "kmeans"):
#' "kmeans", "clara", "clara_pam", "hclust", "pamk", "pamk_pam", "pamk".
#' Any CBI appended with '_pam' makes use of \code{\link{pam}}.
#' The method used in 'hclust' CBI is "ward.D2".
#' @param getImages Boolean. If true, a plot is displayed.
#' @param all_metrics Boolean. If true, clustering is performed upon all the dataset.
#' @param seed Positive integer. A seed for internal bootstrap.
#' @param gold_standard: Numeric vector. A vector of clusters from a gold standard classification, e.g. c(1,2,1,1,2).
#' Only applicable if parameter 'all_metrics' is set to TRUE.
#'
#' @return A \code{\link{ExperimentList}} containing the stability and cluster measurements
#' for k clusters.
#'
#' @examples
#' # Using example data from our package
#' data("ontMetrics")
#' result <- stability(ontMetrics, k=6, getImages=TRUE)
#'
#' @references
#' \insertRef{milligan1996measuring}{evaluomeR}
#'
#' \insertRef{jaccard1901distribution}{evaluomeR}
#'
#'
stability <- function(data, k=5, bs=100, cbi="kmeans",
getImages=FALSE, all_metrics=FALSE, seed=NULL,
gold_standard=NULL,...) {
data <- as.data.frame(assay(data))
checkKValue(k)
runStabilityIndex(data, k.min=k, k.max=k, bs=bs, cbi=cbi,
all_metrics=all_metrics, gold_standard=gold_standard, seed=seed, ...)
stabilityDataFrame <- suppressWarnings(
runStabilityIndexTableRange(data, k.min=k, k.max=k))
if (getImages == TRUE) {
suppressWarnings(
runStabilityIndexK_IMG(bs, k.min = k, k.max = k))
}
se <- createSEList(stabilityDataFrame)
return(se)
}
#' @title Stability index for a range of k clusters.
#' @name stabilityRange
#' @aliases stabilityRange
#' @description
#' This analysis permits to estimate whether the clustering is meaningfully
#' affected by small variations in the sample. For a range of k values (\code{k.range}),
#' a clustering using the k-means algorithm is carried out.
#' Then, the stability index is the mean of the Jaccard coefficient
#' values of a number of \code{bs} bootstrap replicates. The values are in the range [0,1],
#' having the following meaning:
#' \itemize{
#' \item Unstable: [0, 0.60[.
#' \item Doubtful: [0.60, 0.75].
#' \item Stable: ]0.75, 0.85].
#' \item Highly Stable: ]0.85, 1].
#' }
#'
#' @inheritParams stability
#' @param k.range Concatenation of two positive integers.
#' The first value \code{k.range[1]} is considered as the lower bound of the range,
#' whilst the second one, \code{k.range[2]}, as the higher. Both values must be
#' contained in [2,15] range.
#'
#' @return A \code{\link{ExperimentList}} containing the stability and cluster measurements
#' for 2 to \code{k} clusters.
#'
#' @examples
#' # Using example data from our package
#' data("ontMetrics")
#' result <- stabilityRange(ontMetrics, k.range=c(2,3))
#'
#' @references
#' \insertRef{milligan1996measuring}{evaluomeR}
#'
#' \insertRef{jaccard1901distribution}{evaluomeR}
#'
#'
stabilityRange <- function(data, k.range=c(2,15), bs=100, cbi="kmeans",
getImages=FALSE, all_metrics=FALSE, seed=NULL,
gold_standard=NULL,...) {
k.range.length = length(k.range)
if (k.range.length != 2) {
stop("k.range length must be 2")
}
k.min = k.range[1]
k.max = k.range[2]
checkKValue(k.min)
checkKValue(k.max)
if (k.max < k.min) {
stop("The first value of k.range cannot be greater than its second value")
}
data <- as.data.frame(SummarizedExperiment::assay(data))
runStabilityIndex(data, k.min=k.min, k.max=k.max, bs, cbi, all_metrics=all_metrics, gold_standard=gold_standard, seed=seed, ...)
stabilityDataFrame <- suppressWarnings(
runStabilityIndexTableRange(data, k.min=k.min, k.max=k.max))
if (getImages == TRUE) {
suppressWarnings(
runStabilityIndexK_IMG(bs, k.min=k.min, k.max=k.max))
suppressWarnings(
runStabilityIndexMetric_IMG(bs, k.min=k.min, k.max=k.max))
}
se <- createSEList(stabilityDataFrame)
return(se)
}
#' @title Stability index for a set of k clusters.
#' @name stabilitySet
#' @aliases stabilitySet
#' @description
#' This analysis permits to estimate whether the clustering is meaningfully
#' affected by small variations in the sample. For a set of k values (\code{k.set}),
#' a clustering using the k-means algorithm is carried out.
#' Then, the stability index is the mean of the Jaccard coefficient
#' values of a number of \code{bs} bootstrap replicates. The values are in the range [0,1],
#' having the following meaning:
#' \itemize{
#' \item Unstable: [0, 0.60[.
#' \item Doubtful: [0.60, 0.75].
#' \item Stable: ]0.75, 0.85].
#' \item Highly Stable: ]0.85, 1].
#' }
#'
#' @inheritParams stability
#' @param k.set A list of integer values of \code{k}, as in c(2,4,8).
#' The values must be contained in [2,15] range.
#'
#' @return A \code{\link{ExperimentList}} containing the stability and cluster measurements
#' of the list of \code{k} clusters.
#'
#' @examples
#' # Using example data from our package
#' data("rnaMetrics")
#' result <- stabilitySet(rnaMetrics, k.set=c(2,3))
#'
#' @references
#' \insertRef{milligan1996measuring}{evaluomeR}
#'
#' \insertRef{jaccard1901distribution}{evaluomeR}
#'
#'
stabilitySet <- function(data, k.set=c(2,3), bs=100, cbi="kmeans",
getImages=FALSE, all_metrics=FALSE, seed=NULL,
gold_standard=NULL, ...) {
k.set.length = length(k.set)
if (k.set.length == 0) {
stop("k.set list is empty")
} else if (k.set.length == 1) {
stop("k.set list contains only one element. For one K analysis use 'stability' method")
}
k.set = sort(k.set)
for (k in k.set) {
checkKValue(k)
}
data <- as.data.frame(SummarizedExperiment::assay(data))
runStabilityIndex(data, k.set = k.set, bs=bs, cbi=cbi, all_metrics=all_metrics, gold_standard=gold_standard, seed=seed, ...)
stabilityDataFrame <- suppressWarnings(
runStabilityIndexTableRange(data, k.set = k.set))
if (getImages == TRUE) {
suppressWarnings(
runStabilityIndexK_IMG(bs, k.set = k.set))
suppressWarnings(
runStabilityIndexMetric_IMG(bs, k.set = k.set))
}
se <- createSEList(stabilityDataFrame)
return(se)
}
runStabilityIndex <- function(data, k.min=NULL, k.max=NULL, bs,
cbi, all_metrics, seed, k.set=NULL,
gold_standard=NULL,...) {
if (is.null(seed)) {
seed = pkg.env$seed
}
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndex: All k parameters are null!")
}
if (!is.null(gold_standard)) {
if (all_metrics==FALSE) {
stop("Gold standard parameter can be set only if the clustering of all the metrics is selected (all_metrics = TRUE)")
}
if (bs != 0) {
message("Warning: 'gold_standard' parameter is set, argument 'bs' will be ignored.")
}
}
data <- removeNAValues(data)
dfStats(data)
inversa=NULL
m.stab.global = NULL
m.stab.global.csv = NULL # To store new CSV output measures without altering legacy code
todo.estable = NULL
datos.bruto=data
names.metr=names(datos.bruto)[-c(1)]
pkg.env$names.metr = names.metr
bs.values=c(bs)
contador=0
i.min=k.min
i.max=k.max
k.range = NULL
k.range.length = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
} else {
k.range = i.min:i.max
k.range.length = length(i.min:i.max)+1
}
if (all_metrics == TRUE) { # Processing all metrics as one
num.metrics = 1
} else {
num.metrics = length(names.metr)
}
for (i.metr in 1:num.metrics) {
if (all_metrics == TRUE) {
message("Processing all metrics, 'merge', in dataframe (", length(names.metr),")")
pkg.env$names.metr = c("all_metrics")
} else {
message("Processing metric: ", names.metr[i.metr],"(", i.metr,")")
}
m.stab.global[[i.metr]]=matrix(data=NA, nrow=1,
ncol=k.range.length)
m.stab.global.csv[[i.metr]]=matrix(data=NA, nrow=1,
ncol=k.range.length)
for (j.k in k.range) {
message("\tCalculation of k = ", j.k,"")
estable=NULL
contador=contador+1
i=i.metr+1
estable$n.metric=i.metr
estable$name.metric=names.metr[i.metr]
estable$n.k=j.k
estable$name.ontology=names(datos.bruto[1])
if (all_metrics == TRUE) { # Processing all metrics as one
data_to_cluster = datos.bruto[,-1] # Removing first column
} else {
data_to_cluster = datos.bruto[,i]
}
km5=NULL
v.size = 0
if (!all_metrics) {
v.size = length(levels(as.factor(data_to_cluster)))
} else {
v.size = j.k
}
if (v.size>=j.k) {
#km5$cluster=boot.cluster(data=datos.bruto[,i],
# nk=j.k, B=bs, seed=seed)
#km5$jac=km5$cluster$means
clusterbootData = tryCatch({
clusterbootWrapper(data=data_to_cluster, B=bs,
bootmethod="boot",
cbi=cbi,
gold_standard=gold_standard,
krange=j.k, seed=seed, ...)
}, error = function(error_condition) {
error_condition
})
if(inherits(clusterbootData, "error")) {
message(paste0("\t", clusterbootData))
message("\tWarning: Could not process data for k = ", j.k)
km5$bspart=rep(NA,length(datos.bruto[,i]))
km5$jac=rep(NA,j.k)
km5$centr=rep(NA,j.k)
km5$means=km5$bspart
km5$bspart.or=km5$bspart
km5$bspart.inv=km5$means
km5$jac.or=km5$jac
km5$jac.inv=km5$jac
km5$partition=km5$bspart.inv
km5$jac.stab=km5$jac.inv
km5$csv = NULL
km5$csv$cluster_partition = NULL
km5$csv$cluster_mean = NULL
km5$csv$cluster_centers = NULL
km5$csv$cluster_size = NULL
km5$csv$cluster_betweenss = NULL
km5$csv$cluster_totss = NULL
km5$csv$cluster_tot.withinss = NULL
km5$csv$cluster_anova = NULL
m.stab.global[[i.metr]][j.k] = mean(km5$jac.stab)
m.stab.global.csv[[i.metr]][j.k] = list(km5)
estable[[which(bs.values==bs)]] = km5
next
}
km5$cluster = clusterbootData
km5$jac=km5$cluster$bootmean
km5$bspart=km5$cluster$partition
km5$csv = NULL
km5$csv$cluster_partition = km5$cluster$partition
km5$csv$cluster_mean = km5$cluster$bootmean
km5$csv$cluster_centers = km5$cluster$result$result$centers
km5$csv$cluster_size = km5$cluster$result$result$size
km5$csv$cluster_betweenss = km5$cluster$result$result$betweenss
km5$csv$cluster_totss = km5$cluster$result$result$totss
km5$csv$cluster_tot.withinss = km5$cluster$result$result$tot.withinss
km5$csv$cluster_anova = fAnova(km5$cluster$result$result, j.k, num.metrics)
km5$centr=km5$cluster$result$result$centers
for (km5.i in 1:length(km5$centr)) {
km5$means[which(km5$bspart==km5.i)]=km5$centr[km5.i]
}
#km5$bspart.or=ordered(km5$means,labels=seq(1,length(km5$centr)))
#km5$bspart.inv=ordered(km5$means,labels=seq(length(km5$centr),1))
#km5$jac.or=km5$jac[order(km5$centr)]
#km5$jac.inv=km5$jac[order(km5$centr,decreasing=TRUE)]
#if (any(inversa==names.metr[i.metr])) {
# km5$partition=km5$bspart.inv
# km5$jac.stab=km5$jac.inv
#} else {
# km5$partition=km5$bspart.or
# km5$jac.stab=km5$jac.or
#}
#m.stab.global[[i.metr]][j.k] = mean(km5$jac.stab)
m.stab.global[[i.metr]][j.k] = mean(km5$jac)
m.stab.global.csv[[i.metr]][j.k] = list(km5)
estable[[which(bs.values==bs)]] = km5
} else {
message("\tWarning: Could not process data for k = ", j.k)
km5$bspart=rep(NA,length(datos.bruto[,i]))
km5$jac=rep(NA,j.k)
km5$centr=rep(NA,j.k)
km5$means=km5$bspart
km5$bspart.or=km5$bspart
km5$bspart.inv=km5$means
km5$jac.or=km5$jac
km5$jac.inv=km5$jac
km5$partition=km5$bspart.inv
km5$jac.stab=km5$jac.inv
km5$csv = NULL
km5$csv$cluster_partition = NULL
km5$csv$cluster_mean = NULL
km5$csv$cluster_centers = NULL
km5$csv$cluster_size = NULL
km5$csv$cluster_betweenss = NULL
km5$csv$cluster_totss = NULL
km5$csv$cluster_tot.withinss = NULL
km5$csv$cluster_anova = NULL
# m.stab.global[[i.metr]][j.k] = mean(km5$jac.stab)
m.stab.global[[i.metr]][j.k] = mean(km5$jac)
m.stab.global.csv[[i.metr]][j.k] = list(km5)
estable[[which(bs.values==bs)]] = km5
}
estable$km5.dynamic = km5$partition
todo.estable[[contador]]=estable
}
}
e.stab.global=NULL
for (j.k in k.range) {
#e.stab.global[[j.k]]=matrix(data=NA, nrow=length(names.metr), ncol=length(i.min:i.max))
e.stab.global[[j.k]]=matrix(data=NA, nrow=k.range.length, ncol=1)
for (i.metr in 1:length(pkg.env$names.metr)) {
e.stab.global[[j.k]][i.metr]=m.stab.global[[i.metr]][j.k]
}
}
pkg.env$m.stab.global.csv = m.stab.global.csv
pkg.env$m.stab.global = m.stab.global
pkg.env$e.stab.global = e.stab.global
#pkg.env$names.metr = names.metr
#return(stabilityDataFrame)
#return(NULL)
}
runStabilityIndexTableRange <- function(data, k.min=NULL, k.max=NULL, k.set=NULL) {
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndexTableRange: All k parameters are null!")
}
stabilityDataFrame = NULL
m.stab.global = pkg.env$m.stab.global
m.stab.global.csv = pkg.env$m.stab.global.csv
names.metr = pkg.env$names.metr
measures = NULL
# Key = Dataframe name - Value = Header name for each k
measures["stability_mean"]= c("Mean_stability_k_")
measures["cluster_partition"]= c("Cluster_partition_k_")
measures["cluster_mean"]= c("Cluster_mean_k_")
measures["cluster_centers"]= c("Cluster_centers_k_")
measures["cluster_size"]= c("Cluster_size_k_")
measures["cluster_betweenss"]= c("Cluster_betweenss_k_")
measures["cluster_totss"]= c("Cluster_totss_k_")
measures["cluster_tot.withinss"]= c("Cluster_tot.withinss_k_")
measures["cluster_anova"]= c("Cluster_anova_k_")
k.range = NULL
k.range.length = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
} else {
k.range = k.min:k.max
k.range.length = length(k.min:k.max)+1
}
for (measure in names(measures)) {
stabilityDataList = list()
# Build header
header <- list("Metric")
for (k in k.range) {
nextHeader <- paste(measures[measure], k, sep="")
header <- c(header, nextHeader)
}
header = unlist(header, use.names=FALSE)
# Build rows
for (i.metr in 1:length(names.metr)) {
measure.data.list = list()
wrapper=NULL # Wrapper object to extract the measure data
wrapper$metric=names.metr[i.metr]
if (measure == "stability_mean") {
jacMean = m.stab.global[[i.metr]]
cur.value = jacMean[c(k.range)]
measure.data.list = c(measure.data.list, cur.value)
} else {
km5.list = m.stab.global.csv[[i.metr]]
for (k in k.range) {
km5.cur = km5.list[[k]]
cur.value = getMeasureValue(km5.cur, measure)
measure.data.list = c(measure.data.list, cur.value)
}
}
wrapper[[measure]] = unlist(measure.data.list, use.names = FALSE)
stabilityDataList[[i.metr]] = unlist(wrapper, use.names=FALSE)
} # end for i.metr
# Transform into dataframe
# Add df if has data
if (ncol(t(data.frame(stabilityDataList))) != 1) {
stabilityDataFrame[[measure]] = t(data.frame(stabilityDataList))
colnames(stabilityDataFrame[[measure]]) = header
rownames(stabilityDataFrame[[measure]]) <- NULL
}
}
return(stabilityDataFrame)
}
# Stability index per K value (x values = matrics)
runStabilityIndexK_IMG <- function(bs, k.min = NULL, k.max = NULL, k.set = NULL) {
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndexK_IMG: All k parameters are null!")
}
ancho=6
alto=4
escala=0.6
escalax=escala
ajuste=0.5
escalat=0.5
escalap=0.4
contadorFiguras=1
#Pattern: GlobalStability_K_2, ..., GlobalStability_K_N
figurename="GlobalStability_K_"
colores <- c("black", "blue", "red", "darkgreen", "orange", "magenta", "gray")
ltype <- c(1, 2, 3, 4, 3, 4, 6)
i.min=k.min
i.max=k.max
k.range = NULL
k.range.length = NULL
g.main = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
setAsStrList = paste(as.character(k.set),collapse=", ",sep="")
g.main=paste(" St. Indices of the metrics for k in {", setAsStrList, "}",sep="")
} else {
k.range = i.min:i.max
k.range.length = length(k.min:k.max)+1
g.main=paste(" St. Indices of the metrics for k in [", i.min, ",", i.max, "]",sep="")
}
stype <- c(1:k.range.length)
e.stab.global = pkg.env$e.stab.global
names.metr = pkg.env$names.metr
margins <- par(mar=c(5,5,3,3))
on.exit(par(margins))
xnames=as.character(names.metr)
ynames="Global Stability Indices"
metrics_length = length(names.metr)
num_metrics_plot = 19
num_iterations = round(metrics_length/num_metrics_plot)
if (num_iterations > 0) {
num_iterations = num_iterations - 1
}
for (iteration in 0:num_iterations) {
i = 1
labels = list()
rangeStart = (iteration*num_metrics_plot)+1
rangeEnd = rangeStart+num_metrics_plot
if (rangeEnd > metrics_length) {
rangeEnd = metrics_length
}
new_xnames = xnames[rangeStart:rangeEnd]
for (j.k in k.range) {
cur.data = e.stab.global[[j.k]]
cur.data = cur.data[rangeStart:rangeEnd]
#cur.data = cur.data[!is.na(cur.data)]
#cur.data = cur.data[c(i.min:i.max)
plot(cur.data, main=g.main, axes=TRUE, col.axis="white",
xlim=c(0.75,length(new_xnames)+0.25), xlab="", ylim=c(0,1),
ylab=ynames, col="black", type="o", lwd=1, lty=stype[i], pch=stype[i])
labels = c(labels,(paste0("k=", j.k)))
i = i + 1
par(new=TRUE)
}
axis(1,at=1:length(new_xnames),labels=new_xnames,las=2,cex.axis=0.75)
axis(2,las=3,cex.axis=0.85)
legend("bottomright", legend=labels, inset=.01, lwd=1, lty=stype,
col="black", cex=0.7, pch=stype)
mtext(side=1, text="Metrics",line=4)
text(0.75, 0.9, "H.stab", cex=0.6, col = "black")
abline(h = 0.85, col="black", lwd=1, lty=1) # Highly Stable: (0.85, 1]
text(0.74, 0.8, "Stab", cex=0.6, col = "black")
abline(h = 0.75, col="black", lwd=1, lty=1) # Stable: (0.75, 0.85]
text(0.76, 0.65, "Doubt.", cex=0.6, col = "black")
abline(h = 0.60, col="black", lwd=1, lty=1) # Doubtful: [0.60, 0.75]
text(0.76, 0.05, "Unstab.", cex=0.6, col = "black")
#abline(h = 0, col="black", lwd=1, lty=4) # Unstable: [0, 0.60)
par(new=FALSE)
}
}
# Stability index per metric (x values = k range)
runStabilityIndexMetric_IMG <- function(bs, k.min=NULL, k.max=NULL, k.set=NULL) {
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndexMetric_IMG: All k parameters are null!")
}
ancho=6
alto=4
escala=0.9
escalax=escala
ajuste=0.5
escalat=0.5
escalap=0.4
contadorFiguras=1
#Pattern: GlobalStability_MetricX, ..., GlobalStability_MetricN
figurename="GlobalStability_"
colores <- c("black", "blue", "red", "darkgreen", "orange", "magenta", "gray")
ltype <- c(1, 2, 3, 4, 3, 4, 6)
stype <- c("o", "l")
pchtype <- c(1, 2, 3, 4, 5, 5)
k.range = NULL
k.range.length = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
} else {
k.range = k.min:k.max
k.range.length = length(k.min:k.max)
}
m.stab.global = pkg.env$m.stab.global
names.metr = pkg.env$names.metr
margins <- par(mar=c(5,5,3,3))
on.exit(par(margins))
for (i.metr in 1:length(names.metr)) {
cur.data = m.stab.global[[i.metr]]
cur.data = cur.data[!is.na(cur.data)]
ymin = min(cur.data)
if (is.na(ymin) || ymin == Inf) {
ymin = 0
}
if (!is.null(k.set)) {
setAsStrList = paste(as.character(k.set),collapse=", ",sep="")
g.main=paste(" St. Indices of '", names.metr[i.metr], "' for k in {",
setAsStrList,"}",sep="")
} else {
g.main=paste(" St. Indices of '", names.metr[i.metr], "' for k in [",
k.min, ",", k.max,"]",sep="")
}
xnames=c(k.range)
ynames="Global Stability Indices"
plot(cur.data, main=g.main, axes=TRUE, col.axis="white",
xlim=c(0.75,k.range.length+0.25), xlab="", ylim=c(ymin,1),
ylab=ynames, col=colores[1],type="o", lwd=1, lty=ltype[1])
axis(1,at=1:k.range.length,labels=xnames,las=1,cex.axis=escalax)
axis(2,las=3,cex.axis=0.85)
labels <- paste("b=", bs, sep = "")
legend("topright", legend=labels, inset=.01, lwd=1, lty=ltype[1:4], col=colores[1:4], cex=0.7, pch=pchtype[1:4])
mtext(side=1, text="K values",line=3)
}
}
neobernad/evaluomeR documentation built on Dec. 18, 2024, 5:34 a.m.
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Defines functions runStabilityIndexMetric_IMG runStabilityIndexK_IMG runStabilityIndexTableRange runStabilityIndex stabilitySet stabilityRange stability
Documented in stability stabilityRange stabilitySet
#' @title Stability index.
#' @name stability
#' @aliases stability
#' @description
#' This analysis permits to estimate whether the clustering is meaningfully
#' affected by small variations in the sample. First, a clustering using the
#' k-means algorithm is carried out. The value of \code{k} can be provided by the user.
#' Then, the stability index is the mean of the Jaccard coefficient
#' values of a number of \code{bs} bootstrap replicates. The values are in the range [0,1],
#' having the following meaning:
#' \itemize{
#' \item Unstable: [0, 0.60[.
#' \item Doubtful: [0.60, 0.75].
#' \item Stable: ]0.75, 0.85].
#' \item Highly Stable: ]0.85, 1].
#' }
#'
#' @param data A \code{\link{SummarizedExperiment}}.
#' The SummarizedExperiment must contain an assay with the following structure:
#' A valid header with names. The first column of the header is the ID or name
#' of the instance of the dataset (e.g., ontology, pathway, etc.) on which the
#' metrics are measured.
#' The other columns of the header contains the names of the metrics.
#' The rows contains the measurements of the metrics for each instance in the dataset.
#' @param k Positive integer. Number of clusters between [2,15] range.
#' @param bs Positive integer. Bootstrap value to perform the resampling.
#' @param cbi Clusterboot interface name (default: "kmeans"):
#' "kmeans", "clara", "clara_pam", "hclust", "pamk", "pamk_pam", "pamk".
#' Any CBI appended with '_pam' makes use of \code{\link{pam}}.
#' The method used in 'hclust' CBI is "ward.D2".
#' @param getImages Boolean. If true, a plot is displayed.
#' @param all_metrics Boolean. If true, clustering is performed upon all the dataset.
#' @param seed Positive integer. A seed for internal bootstrap.
#' @param gold_standard: Numeric vector. A vector of clusters from a gold standard classification, e.g. c(1,2,1,1,2).
#' Only applicable if parameter 'all_metrics' is set to TRUE.
#'
#' @return A \code{\link{ExperimentList}} containing the stability and cluster measurements
#' for k clusters.
#'
#' @examples
#' # Using example data from our package
#' data("ontMetrics")
#' result <- stability(ontMetrics, k=6, getImages=TRUE)
#'
#' @references
#' \insertRef{milligan1996measuring}{evaluomeR}
#'
#' \insertRef{jaccard1901distribution}{evaluomeR}
#'
#'
stability <- function(data, k=5, bs=100, cbi="kmeans",
getImages=FALSE, all_metrics=FALSE, seed=NULL,
gold_standard=NULL,...) {
data <- as.data.frame(assay(data))
checkKValue(k)
runStabilityIndex(data, k.min=k, k.max=k, bs=bs, cbi=cbi,
all_metrics=all_metrics, gold_standard=gold_standard, seed=seed, ...)
stabilityDataFrame <- suppressWarnings(
runStabilityIndexTableRange(data, k.min=k, k.max=k))
if (getImages == TRUE) {
suppressWarnings(
runStabilityIndexK_IMG(bs, k.min = k, k.max = k))
}
se <- createSEList(stabilityDataFrame)
return(se)
}
#' @title Stability index for a range of k clusters.
#' @name stabilityRange
#' @aliases stabilityRange
#' @description
#' This analysis permits to estimate whether the clustering is meaningfully
#' affected by small variations in the sample. For a range of k values (\code{k.range}),
#' a clustering using the k-means algorithm is carried out.
#' Then, the stability index is the mean of the Jaccard coefficient
#' values of a number of \code{bs} bootstrap replicates. The values are in the range [0,1],
#' having the following meaning:
#' \itemize{
#' \item Unstable: [0, 0.60[.
#' \item Doubtful: [0.60, 0.75].
#' \item Stable: ]0.75, 0.85].
#' \item Highly Stable: ]0.85, 1].
#' }
#'
#' @inheritParams stability
#' @param k.range Concatenation of two positive integers.
#' The first value \code{k.range[1]} is considered as the lower bound of the range,
#' whilst the second one, \code{k.range[2]}, as the higher. Both values must be
#' contained in [2,15] range.
#'
#' @return A \code{\link{ExperimentList}} containing the stability and cluster measurements
#' for 2 to \code{k} clusters.
#'
#' @examples
#' # Using example data from our package
#' data("ontMetrics")
#' result <- stabilityRange(ontMetrics, k.range=c(2,3))
#'
#' @references
#' \insertRef{milligan1996measuring}{evaluomeR}
#'
#' \insertRef{jaccard1901distribution}{evaluomeR}
#'
#'
stabilityRange <- function(data, k.range=c(2,15), bs=100, cbi="kmeans",
getImages=FALSE, all_metrics=FALSE, seed=NULL,
gold_standard=NULL,...) {
k.range.length = length(k.range)
if (k.range.length != 2) {
stop("k.range length must be 2")
}
k.min = k.range[1]
k.max = k.range[2]
checkKValue(k.min)
checkKValue(k.max)
if (k.max < k.min) {
stop("The first value of k.range cannot be greater than its second value")
}
data <- as.data.frame(SummarizedExperiment::assay(data))
runStabilityIndex(data, k.min=k.min, k.max=k.max, bs, cbi, all_metrics=all_metrics, gold_standard=gold_standard, seed=seed, ...)
stabilityDataFrame <- suppressWarnings(
runStabilityIndexTableRange(data, k.min=k.min, k.max=k.max))
if (getImages == TRUE) {
suppressWarnings(
runStabilityIndexK_IMG(bs, k.min=k.min, k.max=k.max))
suppressWarnings(
runStabilityIndexMetric_IMG(bs, k.min=k.min, k.max=k.max))
}
se <- createSEList(stabilityDataFrame)
return(se)
}
#' @title Stability index for a set of k clusters.
#' @name stabilitySet
#' @aliases stabilitySet
#' @description
#' This analysis permits to estimate whether the clustering is meaningfully
#' affected by small variations in the sample. For a set of k values (\code{k.set}),
#' a clustering using the k-means algorithm is carried out.
#' Then, the stability index is the mean of the Jaccard coefficient
#' values of a number of \code{bs} bootstrap replicates. The values are in the range [0,1],
#' having the following meaning:
#' \itemize{
#' \item Unstable: [0, 0.60[.
#' \item Doubtful: [0.60, 0.75].
#' \item Stable: ]0.75, 0.85].
#' \item Highly Stable: ]0.85, 1].
#' }
#'
#' @inheritParams stability
#' @param k.set A list of integer values of \code{k}, as in c(2,4,8).
#' The values must be contained in [2,15] range.
#'
#' @return A \code{\link{ExperimentList}} containing the stability and cluster measurements
#' of the list of \code{k} clusters.
#'
#' @examples
#' # Using example data from our package
#' data("rnaMetrics")
#' result <- stabilitySet(rnaMetrics, k.set=c(2,3))
#'
#' @references
#' \insertRef{milligan1996measuring}{evaluomeR}
#'
#' \insertRef{jaccard1901distribution}{evaluomeR}
#'
#'
stabilitySet <- function(data, k.set=c(2,3), bs=100, cbi="kmeans",
getImages=FALSE, all_metrics=FALSE, seed=NULL,
gold_standard=NULL, ...) {
k.set.length = length(k.set)
if (k.set.length == 0) {
stop("k.set list is empty")
} else if (k.set.length == 1) {
stop("k.set list contains only one element. For one K analysis use 'stability' method")
}
k.set = sort(k.set)
for (k in k.set) {
checkKValue(k)
}
data <- as.data.frame(SummarizedExperiment::assay(data))
runStabilityIndex(data, k.set = k.set, bs=bs, cbi=cbi, all_metrics=all_metrics, gold_standard=gold_standard, seed=seed, ...)
stabilityDataFrame <- suppressWarnings(
runStabilityIndexTableRange(data, k.set = k.set))
if (getImages == TRUE) {
suppressWarnings(
runStabilityIndexK_IMG(bs, k.set = k.set))
suppressWarnings(
runStabilityIndexMetric_IMG(bs, k.set = k.set))
}
se <- createSEList(stabilityDataFrame)
return(se)
}
runStabilityIndex <- function(data, k.min=NULL, k.max=NULL, bs,
cbi, all_metrics, seed, k.set=NULL,
gold_standard=NULL,...) {
if (is.null(seed)) {
seed = pkg.env$seed
}
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndex: All k parameters are null!")
}
if (!is.null(gold_standard)) {
if (all_metrics==FALSE) {
stop("Gold standard parameter can be set only if the clustering of all the metrics is selected (all_metrics = TRUE)")
}
if (bs != 0) {
message("Warning: 'gold_standard' parameter is set, argument 'bs' will be ignored.")
}
}
data <- removeNAValues(data)
dfStats(data)
inversa=NULL
m.stab.global = NULL
m.stab.global.csv = NULL # To store new CSV output measures without altering legacy code
todo.estable = NULL
datos.bruto=data
names.metr=names(datos.bruto)[-c(1)]
pkg.env$names.metr = names.metr
bs.values=c(bs)
contador=0
i.min=k.min
i.max=k.max
k.range = NULL
k.range.length = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
} else {
k.range = i.min:i.max
k.range.length = length(i.min:i.max)+1
}
if (all_metrics == TRUE) { # Processing all metrics as one
num.metrics = 1
} else {
num.metrics = length(names.metr)
}
for (i.metr in 1:num.metrics) {
if (all_metrics == TRUE) {
message("Processing all metrics, 'merge', in dataframe (", length(names.metr),")")
pkg.env$names.metr = c("all_metrics")
} else {
message("Processing metric: ", names.metr[i.metr],"(", i.metr,")")
}
m.stab.global[[i.metr]]=matrix(data=NA, nrow=1,
ncol=k.range.length)
m.stab.global.csv[[i.metr]]=matrix(data=NA, nrow=1,
ncol=k.range.length)
for (j.k in k.range) {
message("\tCalculation of k = ", j.k,"")
estable=NULL
contador=contador+1
i=i.metr+1
estable$n.metric=i.metr
estable$name.metric=names.metr[i.metr]
estable$n.k=j.k
estable$name.ontology=names(datos.bruto[1])
if (all_metrics == TRUE) { # Processing all metrics as one
data_to_cluster = datos.bruto[,-1] # Removing first column
} else {
data_to_cluster = datos.bruto[,i]
}
km5=NULL
v.size = 0
if (!all_metrics) {
v.size = length(levels(as.factor(data_to_cluster)))
} else {
v.size = j.k
}
if (v.size>=j.k) {
#km5$cluster=boot.cluster(data=datos.bruto[,i],
# nk=j.k, B=bs, seed=seed)
#km5$jac=km5$cluster$means
clusterbootData = tryCatch({
clusterbootWrapper(data=data_to_cluster, B=bs,
bootmethod="boot",
cbi=cbi,
gold_standard=gold_standard,
krange=j.k, seed=seed, ...)
}, error = function(error_condition) {
error_condition
})
if(inherits(clusterbootData, "error")) {
message(paste0("\t", clusterbootData))
message("\tWarning: Could not process data for k = ", j.k)
km5$bspart=rep(NA,length(datos.bruto[,i]))
km5$jac=rep(NA,j.k)
km5$centr=rep(NA,j.k)
km5$means=km5$bspart
km5$bspart.or=km5$bspart
km5$bspart.inv=km5$means
km5$jac.or=km5$jac
km5$jac.inv=km5$jac
km5$partition=km5$bspart.inv
km5$jac.stab=km5$jac.inv
km5$csv = NULL
km5$csv$cluster_partition = NULL
km5$csv$cluster_mean = NULL
km5$csv$cluster_centers = NULL
km5$csv$cluster_size = NULL
km5$csv$cluster_betweenss = NULL
km5$csv$cluster_totss = NULL
km5$csv$cluster_tot.withinss = NULL
km5$csv$cluster_anova = NULL
m.stab.global[[i.metr]][j.k] = mean(km5$jac.stab)
m.stab.global.csv[[i.metr]][j.k] = list(km5)
estable[[which(bs.values==bs)]] = km5
next
}
km5$cluster = clusterbootData
km5$jac=km5$cluster$bootmean
km5$bspart=km5$cluster$partition
km5$csv = NULL
km5$csv$cluster_partition = km5$cluster$partition
km5$csv$cluster_mean = km5$cluster$bootmean
km5$csv$cluster_centers = km5$cluster$result$result$centers
km5$csv$cluster_size = km5$cluster$result$result$size
km5$csv$cluster_betweenss = km5$cluster$result$result$betweenss
km5$csv$cluster_totss = km5$cluster$result$result$totss
km5$csv$cluster_tot.withinss = km5$cluster$result$result$tot.withinss
km5$csv$cluster_anova = fAnova(km5$cluster$result$result, j.k, num.metrics)
km5$centr=km5$cluster$result$result$centers
for (km5.i in 1:length(km5$centr)) {
km5$means[which(km5$bspart==km5.i)]=km5$centr[km5.i]
}
#km5$bspart.or=ordered(km5$means,labels=seq(1,length(km5$centr)))
#km5$bspart.inv=ordered(km5$means,labels=seq(length(km5$centr),1))
#km5$jac.or=km5$jac[order(km5$centr)]
#km5$jac.inv=km5$jac[order(km5$centr,decreasing=TRUE)]
#if (any(inversa==names.metr[i.metr])) {
# km5$partition=km5$bspart.inv
# km5$jac.stab=km5$jac.inv
#} else {
# km5$partition=km5$bspart.or
# km5$jac.stab=km5$jac.or
#}
#m.stab.global[[i.metr]][j.k] = mean(km5$jac.stab)
m.stab.global[[i.metr]][j.k] = mean(km5$jac)
m.stab.global.csv[[i.metr]][j.k] = list(km5)
estable[[which(bs.values==bs)]] = km5
} else {
message("\tWarning: Could not process data for k = ", j.k)
km5$bspart=rep(NA,length(datos.bruto[,i]))
km5$jac=rep(NA,j.k)
km5$centr=rep(NA,j.k)
km5$means=km5$bspart
km5$bspart.or=km5$bspart
km5$bspart.inv=km5$means
km5$jac.or=km5$jac
km5$jac.inv=km5$jac
km5$partition=km5$bspart.inv
km5$jac.stab=km5$jac.inv
km5$csv = NULL
km5$csv$cluster_partition = NULL
km5$csv$cluster_mean = NULL
km5$csv$cluster_centers = NULL
km5$csv$cluster_size = NULL
km5$csv$cluster_betweenss = NULL
km5$csv$cluster_totss = NULL
km5$csv$cluster_tot.withinss = NULL
km5$csv$cluster_anova = NULL
# m.stab.global[[i.metr]][j.k] = mean(km5$jac.stab)
m.stab.global[[i.metr]][j.k] = mean(km5$jac)
m.stab.global.csv[[i.metr]][j.k] = list(km5)
estable[[which(bs.values==bs)]] = km5
}
estable$km5.dynamic = km5$partition
todo.estable[[contador]]=estable
}
}
e.stab.global=NULL
for (j.k in k.range) {
#e.stab.global[[j.k]]=matrix(data=NA, nrow=length(names.metr), ncol=length(i.min:i.max))
e.stab.global[[j.k]]=matrix(data=NA, nrow=k.range.length, ncol=1)
for (i.metr in 1:length(pkg.env$names.metr)) {
e.stab.global[[j.k]][i.metr]=m.stab.global[[i.metr]][j.k]
}
}
pkg.env$m.stab.global.csv = m.stab.global.csv
pkg.env$m.stab.global = m.stab.global
pkg.env$e.stab.global = e.stab.global
#pkg.env$names.metr = names.metr
#return(stabilityDataFrame)
#return(NULL)
}
runStabilityIndexTableRange <- function(data, k.min=NULL, k.max=NULL, k.set=NULL) {
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndexTableRange: All k parameters are null!")
}
stabilityDataFrame = NULL
m.stab.global = pkg.env$m.stab.global
m.stab.global.csv = pkg.env$m.stab.global.csv
names.metr = pkg.env$names.metr
measures = NULL
# Key = Dataframe name - Value = Header name for each k
measures["stability_mean"]= c("Mean_stability_k_")
measures["cluster_partition"]= c("Cluster_partition_k_")
measures["cluster_mean"]= c("Cluster_mean_k_")
measures["cluster_centers"]= c("Cluster_centers_k_")
measures["cluster_size"]= c("Cluster_size_k_")
measures["cluster_betweenss"]= c("Cluster_betweenss_k_")
measures["cluster_totss"]= c("Cluster_totss_k_")
measures["cluster_tot.withinss"]= c("Cluster_tot.withinss_k_")
measures["cluster_anova"]= c("Cluster_anova_k_")
k.range = NULL
k.range.length = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
} else {
k.range = k.min:k.max
k.range.length = length(k.min:k.max)+1
}
for (measure in names(measures)) {
stabilityDataList = list()
# Build header
header <- list("Metric")
for (k in k.range) {
nextHeader <- paste(measures[measure], k, sep="")
header <- c(header, nextHeader)
}
header = unlist(header, use.names=FALSE)
# Build rows
for (i.metr in 1:length(names.metr)) {
measure.data.list = list()
wrapper=NULL # Wrapper object to extract the measure data
wrapper$metric=names.metr[i.metr]
if (measure == "stability_mean") {
jacMean = m.stab.global[[i.metr]]
cur.value = jacMean[c(k.range)]
measure.data.list = c(measure.data.list, cur.value)
} else {
km5.list = m.stab.global.csv[[i.metr]]
for (k in k.range) {
km5.cur = km5.list[[k]]
cur.value = getMeasureValue(km5.cur, measure)
measure.data.list = c(measure.data.list, cur.value)
}
}
wrapper[[measure]] = unlist(measure.data.list, use.names = FALSE)
stabilityDataList[[i.metr]] = unlist(wrapper, use.names=FALSE)
} # end for i.metr
# Transform into dataframe
# Add df if has data
if (ncol(t(data.frame(stabilityDataList))) != 1) {
stabilityDataFrame[[measure]] = t(data.frame(stabilityDataList))
colnames(stabilityDataFrame[[measure]]) = header
rownames(stabilityDataFrame[[measure]]) <- NULL
}
}
return(stabilityDataFrame)
}
# Stability index per K value (x values = matrics)
runStabilityIndexK_IMG <- function(bs, k.min = NULL, k.max = NULL, k.set = NULL) {
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndexK_IMG: All k parameters are null!")
}
ancho=6
alto=4
escala=0.6
escalax=escala
ajuste=0.5
escalat=0.5
escalap=0.4
contadorFiguras=1
#Pattern: GlobalStability_K_2, ..., GlobalStability_K_N
figurename="GlobalStability_K_"
colores <- c("black", "blue", "red", "darkgreen", "orange", "magenta", "gray")
ltype <- c(1, 2, 3, 4, 3, 4, 6)
i.min=k.min
i.max=k.max
k.range = NULL
k.range.length = NULL
g.main = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
setAsStrList = paste(as.character(k.set),collapse=", ",sep="")
g.main=paste(" St. Indices of the metrics for k in {", setAsStrList, "}",sep="")
} else {
k.range = i.min:i.max
k.range.length = length(k.min:k.max)+1
g.main=paste(" St. Indices of the metrics for k in [", i.min, ",", i.max, "]",sep="")
}
stype <- c(1:k.range.length)
e.stab.global = pkg.env$e.stab.global
names.metr = pkg.env$names.metr
margins <- par(mar=c(5,5,3,3))
on.exit(par(margins))
xnames=as.character(names.metr)
ynames="Global Stability Indices"
metrics_length = length(names.metr)
num_metrics_plot = 19
num_iterations = round(metrics_length/num_metrics_plot)
if (num_iterations > 0) {
num_iterations = num_iterations - 1
}
for (iteration in 0:num_iterations) {
i = 1
labels = list()
rangeStart = (iteration*num_metrics_plot)+1
rangeEnd = rangeStart+num_metrics_plot
if (rangeEnd > metrics_length) {
rangeEnd = metrics_length
}
new_xnames = xnames[rangeStart:rangeEnd]
for (j.k in k.range) {
cur.data = e.stab.global[[j.k]]
cur.data = cur.data[rangeStart:rangeEnd]
#cur.data = cur.data[!is.na(cur.data)]
#cur.data = cur.data[c(i.min:i.max)
plot(cur.data, main=g.main, axes=TRUE, col.axis="white",
xlim=c(0.75,length(new_xnames)+0.25), xlab="", ylim=c(0,1),
ylab=ynames, col="black", type="o", lwd=1, lty=stype[i], pch=stype[i])
labels = c(labels,(paste0("k=", j.k)))
i = i + 1
par(new=TRUE)
}
axis(1,at=1:length(new_xnames),labels=new_xnames,las=2,cex.axis=0.75)
axis(2,las=3,cex.axis=0.85)
legend("bottomright", legend=labels, inset=.01, lwd=1, lty=stype,
col="black", cex=0.7, pch=stype)
mtext(side=1, text="Metrics",line=4)
text(0.75, 0.9, "H.stab", cex=0.6, col = "black")
abline(h = 0.85, col="black", lwd=1, lty=1) # Highly Stable: (0.85, 1]
text(0.74, 0.8, "Stab", cex=0.6, col = "black")
abline(h = 0.75, col="black", lwd=1, lty=1) # Stable: (0.75, 0.85]
text(0.76, 0.65, "Doubt.", cex=0.6, col = "black")
abline(h = 0.60, col="black", lwd=1, lty=1) # Doubtful: [0.60, 0.75]
text(0.76, 0.05, "Unstab.", cex=0.6, col = "black")
#abline(h = 0, col="black", lwd=1, lty=4) # Unstable: [0, 0.60)
par(new=FALSE)
}
}
# Stability index per metric (x values = k range)
runStabilityIndexMetric_IMG <- function(bs, k.min=NULL, k.max=NULL, k.set=NULL) {
if (is.null(k.min) && is.null(k.max) && is.null(k.set)) {
stop("runStabilityIndexMetric_IMG: All k parameters are null!")
}
ancho=6
alto=4
escala=0.9
escalax=escala
ajuste=0.5
escalat=0.5
escalap=0.4
contadorFiguras=1
#Pattern: GlobalStability_MetricX, ..., GlobalStability_MetricN
figurename="GlobalStability_"
colores <- c("black", "blue", "red", "darkgreen", "orange", "magenta", "gray")
ltype <- c(1, 2, 3, 4, 3, 4, 6)
stype <- c("o", "l")
pchtype <- c(1, 2, 3, 4, 5, 5)
k.range = NULL
k.range.length = NULL
if (!is.null(k.set)) {
k.range = k.set
k.range.length = length(k.set)
} else {
k.range = k.min:k.max
k.range.length = length(k.min:k.max)
}
m.stab.global = pkg.env$m.stab.global
names.metr = pkg.env$names.metr
margins <- par(mar=c(5,5,3,3))
on.exit(par(margins))
for (i.metr in 1:length(names.metr)) {
cur.data = m.stab.global[[i.metr]]
cur.data = cur.data[!is.na(cur.data)]
ymin = min(cur.data)
if (is.na(ymin) || ymin == Inf) {
ymin = 0
}
if (!is.null(k.set)) {
setAsStrList = paste(as.character(k.set),collapse=", ",sep="")
g.main=paste(" St. Indices of '", names.metr[i.metr], "' for k in {",
setAsStrList,"}",sep="")
} else {
g.main=paste(" St. Indices of '", names.metr[i.metr], "' for k in [",
k.min, ",", k.max,"]",sep="")
}
xnames=c(k.range)
ynames="Global Stability Indices"
plot(cur.data, main=g.main, axes=TRUE, col.axis="white",
xlim=c(0.75,k.range.length+0.25), xlab="", ylim=c(ymin,1),
ylab=ynames, col=colores[1],type="o", lwd=1, lty=ltype[1])
axis(1,at=1:k.range.length,labels=xnames,las=1,cex.axis=escalax)
axis(2,las=3,cex.axis=0.85)
labels <- paste("b=", bs, sep = "")
legend("topright", legend=labels, inset=.01, lwd=1, lty=ltype[1:4], col=colores[1:4], cex=0.7, pch=pchtype[1:4])
mtext(side=1, text="K values",line=3)
}
}
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