/
omics_network.R
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omics_network.R
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#' @rdname show-methods
setMethod(f= "show", signature = "omics_network", definition = function(object) {
cat(
paste(
"This is a S4 class with : \n - (@omics_network) a matrix of dimension ",
dim(object@omics_network)[1],
"*",
dim(object@omics_network)[2],
" .... [the omics_network] \n - (@name) a vector of length ",
length(object@name),
" .... [omics names] \n",
"- (@F) a array of dimension ",
dim(object@F)[1],
"*",
dim(object@F)[2],
"*",
dim(object@F)[3],
" .... [F matrices] \n",
"- (@convF) a matrix of dimension ",
dim(object@convF)[1],
"*",
dim(object@convF)[2],
" .... [convergence (L1 norm) of array F] \n",
"- (@convO)a vector of length ",
length(object@convO),
" .... [convergence (L1 norm) of matrix Omega]\n",
"- (@time_pt) an vector of length",
length(object@time_pt),
" .... [time points]"
)
)
invisible(object)
})
#' Analysing the network
#'
#' Calculates some indicators for each node in the network.
#'
#'
#' @aliases analyze_network analyze_network-methods
#' analyze_network,omics_network-method
#' @param Omega a omics_network object
#' @param nv the level of cutoff at which the analysis should be done
#' @param label_v (optionnal) the name of the genes
#' @return A matrix containing, for each node, its betweenness,its degree, its
#' output, its closeness.
#' @author Bertrand Frederic, Myriam Maumy-Bertrand.
#' @keywords methods
#' @examples
#'
#' data(network)
#' analyze_network(network,nv=0)
#'
setMethod("analyze_network", "omics_network", function(Omega, nv, label_v = NULL) {
require(tnet)
if (is.null(label_v)) {
label_v <- 1:dim(Omega@omics_network)[1]
}
O <- Omega@omics_network
Omega <- Omega@omics_network
O[abs(O) <= nv] <- 0
G <- graph.adjacency(O, weighted = TRUE)
get.edgelist(G) -> Q
weight <- rep(0, dim(Q)[1])
for (i in 1:dim(Q)[1]) {
weight[i] <- Omega[Q[i, 1], Q[i, 2]]
}
R <- cbind(Q, abs(weight))
G <- as.tnet(R,type="weighted one-mode tnet")
C <-
data.frame(label_v,
betweenness_w(G)[, 2],
degree_w(G)[, 2:3],
closeness_w(G, gconly = FALSE)[, 2])
colnames(C) <- c("node", "betweenness", "degree", "output", "closeness")
return(C)
})
#' See the evolution of the network with change of cutoff
#'
#' See the evolution of the network with change of cutoff
#'
#' Several types of outputs are available using the type.ani option. \itemize{
#' \item html \item latex (requires latex) \item swf (requires swftools)
#' \item video (requires ffmpeg) \item gif \item manual_gif }
#'
#' @aliases evolution evolution-methods evolution,omics_network-method
#' @param net a omics_network object
#' @param list_nv a vector of cutoff at which the network should be shown
#' @param gr a vector giving the group of each genee. Defaults to NULL
#' @param color.vertex a vector giving the color of each nodee. Defaults to NULL
#' @param color.edge a vector giving the color of each edge. Defaults to NULL
#' @param fix logical, should the position of the node in the network be calculated once at the beginning ? Defaults to TRUE.
#' @param size vector giving the size of the plot. Defaults to c(2000,1000)
#' @param label_v vector giving the labels of each vertex. Defaults to 1:dim(net@omics_network)[1]
#' @param legend string giving the position of the legend. Defaults to "topleft"
#' @param legend.position string giving the position of the legend. Defaults to "topleft"
#' @param frame.color string giving the color of the frame of the plot. Defaults to "black"
#' @param label.hub label hubs. Defaults to FALSE
#' @param outdir Directory to save the animation. No default value since it must be specified by the user.
#' @param type.ani Type of animation. Defaults to "html"
#' @return A HTML page with the evolution of the network.
#' @author Bertrand Frederic, Myriam Maumy-Bertrand.
#' @keywords methods
#' @examples
#'
#' \donttest{
#' data(network)
#' sequence<-seq(0,0.2,length.out=20)
#'
#' #Change the destdir to have the animation created where you want.
#' destdir = tempdir()
#'
#' #Example of use of the evolution method with an html output.
#' evolution(network,sequence,type.ani = "html", outdir=destdir)
#' }
#'
#' \dontrun{
#' #Example of use of the evolution method with an animated gif output.
#' evolution(network,sequence,type.ani = "gif", outdir=destdir)
#' }
#'
setMethod("evolution", "omics_network",
function(net
,
list_nv
,
gr = NULL
,
color.vertex = NULL
,
color.edge = NULL
,
fix = TRUE
,
size = c(2000, 1000)
,
label_v = 1:dim(net@omics_network)[1]
,
legend.position = "topleft"
,
frame.color = "black"
,
label.hub = FALSE
,
outdir
,
type.ani = "html"
#,edge.arrow.size=0.6*(1+size.ed)
#,edge.thickness=1
)
{
if(missing(outdir)){stop("An output directory is required")}
#We save the user working directory.
orig_working_directory <- getwd()
#We restore the user working directory on function exit.
on.exit(setwd(orig_working_directory))
#Set the working directory to the outdir directory
setwd(outdir)
Omega <- net
if (length(list_nv) == 1) {
list_nv = seq(0, max(net@omics_network) - 0.05, length.out = list_nv)
}
if (type.ani == "html") {
if (!requireNamespace("animation", quietly = TRUE))
stop ("The 'animation' package is not installed installed.", call. = FALSE)
#require(animation)
animation::ani.options(ani.height = size[2],
ani.width = size[1])
animation::saveHTML({
POS <- position(net, nv = list_nv[1])
for (i in list_nv) {
if (fix == TRUE) {
plot(
Omega
,
nv = i
,
gr = gr
,
ini = POS
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
else{
plot(
Omega
,
nv = i
,
gr = gr
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
text(-1, 1, round(i, 3))
}
})
}
if (type.ani == "latex") {
if (!requireNamespace("animation", quietly = TRUE))
stop ("The 'animation' package is not installed installed.", call. = FALSE)
#require(animation)
animation::ani.options(ani.height = size[2],
ani.width = size[1])#, qpdf = '/opt/local/bin/qpdf'
animation::saveLatex({
POS <- position(net, nv = list_nv[1])
for (i in list_nv) {
if (fix == TRUE) {
plot(
Omega
,
nv = i
,
gr = gr
,
ini = POS
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
else{
plot(
Omega
,
nv = i
,
gr = gr
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
text(-1, 1, round(i, 3))
}
})
}
if (type.ani == "swf") {
if (!requireNamespace("animation", quietly = TRUE))
stop ("The 'animation' package is not installed installed.", call. = FALSE)
#require(animation)
animation::ani.options(ani.height = size[2],
ani.width = size[1])#, qpdf = '/opt/local/bin/qpdf'
animation::saveSWF({
POS <- position(net, nv = list_nv[1])
for (i in list_nv) {
if (fix == TRUE) {
plot(
Omega
,
nv = i
,
gr = gr
,
ini = POS
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
else{
plot(
Omega
,
nv = i
,
gr = gr
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
text(-1, 1, round(i, 3))
}
})
}
if (type.ani == "video") {
if (!requireNamespace("animation", quietly = TRUE))
stop ("The 'animation' package is not installed installed.", call. = FALSE)
#require(animation)
animation::ani.options(ani.height = size[2],
ani.width = size[1])#, qpdf = '/opt/local/bin/qpdf'
animation::saveVideo({
POS <- position(net, nv = list_nv[1])
for (i in list_nv) {
if (fix == TRUE) {
plot(
Omega
,
nv = i
,
gr = gr
,
ini = POS
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
else{
plot(
Omega
,
nv = i
,
gr = gr
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
text(-1, 1, round(i, 3))
}
})
}
if (type.ani == "gif") {
setwd(orig_working_directory)
if (!requireNamespace("animation", quietly = TRUE))
stop ("The 'animation' package is not installed installed.", call. = FALSE)
#require(animation)
animation::ani.options(ani.height = size[2],
ani.width = size[1])#, qpdf = '/opt/local/bin/qpdf'
animation::saveGIF({
POS <- position(net, nv = list_nv[1])
for (i in list_nv) {
if (fix == TRUE) {
plot(
Omega
,
nv = i
,
gr = gr
,
ini = POS
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub,
movie.name=paste(outdir,'animation.gif',sep='/')
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
else{
plot(
Omega
,
nv = i
,
gr = gr
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
text(-1, 1, round(i, 3))
}
})
}
if (type.ani == "manual_gif") {
par(ask = TRUE)
POS <-
position(net, nv = list_nv[1])
if (is.null(gr)) {
gr <- rep(1, dim(Omega@omics_network)[1])
}
for (i in list_nv) {
plot(
Omega
,
nv = i
,
gr = gr
,
ini = POS
,
color.vertex = color.vertex
,
color.edge = color.edge
,
label_v = label_v
,
legend.position = legend.position
,
frame.color = frame.color
,
label.hub = label.hub
#,edge.arrow.size=edge.arrow.size
#,edge.thickness=edge.thickness
)
}
par(ask = FALSE)
}
}
)
#' Returns the position of edges in the network
#'
#' Returns the position of edges in the network
#' Retrieve network position for consistent plotting.
#' Utility function to plot networks.
#'
#' @name position-methods
#' @aliases position position-methods position,omics_network-method
#' @docType methods
#' @section Methods: \describe{
#'
#' \item{list("signature(net = \"omics_network\")")}{ Returns a matrix with the
#' position of the node. This matrix can then be used as an argument in the
#' plot function. } }
#'
#' @param net a omics_network object
#' @param nv the level of cutoff at which the analysis should be done
#' @return Matrix with as many rows as the number of edges of the network and three
#' columns (name, xcoord, ycoord).
#' @author Bertrand Frederic, Myriam Maumy-Bertrand.
#' @keywords methods dplots
#' @examples
#'
#' data(network)
#' position(network)
#'
setMethod("position", "omics_network", function(net, nv = 0) {
require(igraph, quietly = TRUE, warn.conflicts = FALSE);on.exit(unloadNamespace("package:igraph"))
O <- net@omics_network
Omega <- net@omics_network
O[abs(O) <= nv] <- 0
O[abs(O) > nv] <- 1
nom <- 1:dim(O)[1]
enle <- which(apply(O, 1, sum) + apply(O, 2, sum) == 0)
if (length(enle) != 0) {
nom <- nom[-enle]
O <- O[-enle, -enle]
}
G <- graph.adjacency(O, weighted = TRUE)
# if(.Platform$OS.type=="unix"){
# get.edgelist(G)+1->Q
# }else{
get.edgelist(G) -> Q
# }
Q[, 1] <- nom[Q[, 1]]
Q[, 2] <- nom[Q[, 2]]
# dev.new(width=150,heigth=300,xlim=c(min(L[,1]),max(L[,1])))
#par(mar=c(0,0,0,0), oma=c(0,0,0,0),mai=c(0,0,0,0))
L <-
layout.fruchterman.reingold(
G,
minx = rep(0, vcount(G)),
maxx = rep(150, vcount(G)),
miny = rep(0, vcount(G)),
maxy = rep(300, vcount(G))
)
return(cbind(nom, L))
})
#######################################
#######################################
#######################################
#' Find the neighborhood of a set of nodes.
#'
#' Find the neighborhood of a set of nodes.
#'
#'
#' @aliases geneNeighborhood geneNeighborhood-methods
#' geneNeighborhood,omics_network-method
#' @param net a omics_network object
#' @param targets a vector containing the set of nodes
#' @param nv the level of cutoff. Defaut to 0.
#' @param order of the neighborhood. Defaut to `length(net@time_pt)-1`.
#' @param label_v vector defining the vertex labels.
#' @param ini using the ``position'' function, you can
#' fix the position of the nodes.
#' @param frame.color color of the frames.
#' @param label.hub logical ; if TRUE only the hubs are labeled.
#' @param graph plot graph of the network. Defaults to `TRUE`.
#' @param names return names of the neighbors. Defaults to `FALSE`.
#' @return The neighborhood of the targeted genes.
#' @author Bertrand Frederic, Myriam Maumy-Bertrand.
#' @keywords methods
#' @examples
#'
#' data(Selection)
#' data(infos)
#' #Find probesets for EGR1
#' pbst_EGR1 = infos[infos$hgnc_symbol=="EGR1", "affy_hg_u133_plus_2"]
#'
#' gene_IDs = infos[match(Selection@name, infos$affy_hg_u133_plus_), "hgnc_symbol"]
#'
#' data(network)
#' #A nv value can chosen using the cutoff function
#' nv=.11
#' EGR1<-which(is.element(Selection@name,pbst_EGR1))
#' P<-position(network,nv=nv)
#'
#' geneNeighborhood(network,targets=EGR1,nv=nv,ini=P,
#' label_v=gene_IDs)
#'
setMethod("geneNeighborhood", "omics_network"
, function(net
,
targets
,
nv = 0
,
order = length(net@time_pt) - 1
,
label_v = NULL
,
ini = NULL
,
frame.color = "white"
,
label.hub = FALSE
#,edge.arrow.size=0.7
#,edge.thickness=1
,
graph = TRUE
,
names = F) {
require(igraph, quietly = TRUE, warn.conflicts = FALSE);on.exit(unloadNamespace("package:igraph"))
O <- net@omics_network
Omega <- net@omics_network
O[abs(O) <= nv] <- 0
O[abs(O) > nv] <- 1
G <- graph.adjacency(O, weighted = TRUE)
coloring <- colorRamp(c("red", "blue"))
color <- rep(grey(0.92), length(V(G)))
if (is.null(label_v)) {
label_v <- 1:dim(O)[1]
}
K <- vector("list", order)
for (j in order:1) {
# if(.Platform$OS.type=="unix"){
# N<-neighborhood(G, order=j, nodes=targets-1, mode=c( "out"))
# }else{
N <-
neighborhood(G,
order = j,
nodes = targets,
mode = c("out"))
# }
K[[j]] <- N
gg <- length(targets)
for (i in 1:gg) {
#if(.Platform$OS.type=="unix"){
#color[N[[i]]+1]<-rgb(coloring((j-1)*1/(order-1)),alpha=255,maxColorValue=255)
#}else{
color[N[[i]]] <-
rgb(coloring((j - 1) * 1 / (order - 1)),
alpha = 255,
maxColorValue = 255)
#}
}
}
if (graph == TRUE) {
color[targets] <- "green"
plot(
net
,
nv = nv
,
ini = ini
,
color.vertex = color
,
frame.color = frame.color
,
label.hub = label.hub
,
label_v = label_v
)
legend(
"topright"
,
legend = paste("order", 1:order)
,
pch = 16
,
col = rgb(coloring((1:order - 1) * 1 / (order - 1)), maxColorValue = 255)
)
}
ind1 <- K[[1]][[1]]
neighbors <- list(net@name[ind1])
for (ord in 2:order) {
neighbors[[ord]] <- list()
for (nod in 1:gg) {
neighbPrec <- K[[ord - 1]][[nod]]
neighbCur <- K[[ord]][[nod]]
ind <- neighbCur[!(neighbCur %in% neighbPrec)]
neighbors[[ord]][[nod]] <- net@name[ind]
}
}
if (names) {
return(neighbors)
}
else{
return(K)
}
})
#' Choose the best cutoff
#'
#' Allows estimating the best cutoff. For a sequence of cutoff, the p value
#' corresponding to each cutoff value of the sequence. Mainly recommended for
#' single time cascade networks. To achieve more sparsity in other settings,
#' please use a fiiting function based on the stability selection or
#' selectboost algorithms.
#'
#'
#' @aliases cutoff cutoff-methods cutoff,omics_network-method
#' @param Omega a omics_network object
#' @param sequence a vector corresponding to the sequence of cutoffs that will be tested.
#' @param x_min an integer ; only values over x_min are further retained for performing the test.
#'
#' @return A list containing two objects : \item{p.value}{the p values
#' corresponding to the sequence of cutoff} \item{p.value.inter}{the smoothed p
#' value vector, using the loess function}
#' @author Bertrand Frederic, Myriam Maumy-Bertrand.
#' @keywords methods
#' @examples
#'
#' \donttest{
#' data(network)
#' cutoff(network)
#' #See vignette for more details
#' }
#'
setMethod("cutoff", "omics_network", function(Omega,
sequence = NULL,
x_min = 0) {
plfit <- function(x = VGAM::rpareto(1000, 10, 2.5)
,
method = "limit"
,
value = c()
,
finite = FALSE
,
nowarn = FALSE
,
nosmall = FALSE) {
#init method value to NULL
vec <- c()
sampl <- c()
limit <- c()
fixed <- c()
###########################################################################################
#
# test and trap for bad input
#
switch(
method
,
range = vec <- value
,
sample = sampl <- value
,
limit = limit <- value
,
fixed = fixed <- value
,
argok <- 0
)
if (exists("argok")) {
stop("(plfit) Unrecognized method")
}
if (!is.null(vec) &&
(!is.vector(vec) || min(vec) <= 1 || length(vec) <= 1)) {
print(
paste(
"(plfit) Error: ''range'' argument must contain a vector > 1; using default."
)
)
vec <- c()
}
if (!is.null(sampl) &&
(!(sampl == floor(sampl)) || length(sampl) > 1 || sampl < 2)) {
print(
paste(
"(plfit) Error: ''sample'' argument must be a positive integer > 2; using default."
)
)
sample <- c()
}
if (!is.null(limit) && (length(limit) > 1 || limit < 1)) {
print(paste(
"(plfit) Error: ''limit'' argument must be a positive >=1; using default."
))
limit <- c()
}
if (!is.null(fixed) && (length(fixed) > 1 || fixed <= 0)) {
print(paste(
"(plfit) Error: ''fixed'' argument must be a positive >0; using default."
))
fixed <- c()
}
# select method (discrete or continuous) for fitting and test if x is a vector
fdattype <- "unknow"
if (is.vector(x, "numeric")) {
fdattype <- "real"
}
if (all(x == floor(x)) && is.vector(x)) {
fdattype <- "integer"
}
if (all(x == floor(x)) &&
min(x) > 1000 && length(x) > 100) {
fdattype <- "real"
}
if (fdattype == "unknow") {
stop("(plfit) Error: x must contain only reals or only integers.")
}
#
# end test and trap for bad input
#
###########################################################################################
###########################################################################################
# CONTINUOUS CASE
# estimate xmin and alpha in the continuous case
#
if (fdattype == "real") {
xmins <- sort(unique(x))
xmins <- xmins[-length(xmins)]
if (!is.null(limit)) {
xmins <- xmins[xmins >= limit]
}
if (!is.null(fixed)) {
xmins <- fixed
}
if (!is.null(sampl)) {
xmins <- xmins[unique(round(seq(1, length(xmins), length.out = sampl)))]
}
#Vecteur pour stocker les valeurs de D
dat <- rep(0, length(xmins))
#Echantillon trie
z <- sort(x)
#Boucle sur toutes les valeurs de xmin pour lesquelles on calcule D
for (xm in 1:length(xmins)) {
#xmin courant
xmin <- xmins[xm]
#Exces
z <- z[z >= xmin]
n <- length(z)
# estimate alpha-1 using direct MLE
a <- n / sum(log(z / xmin))
# truncate search if nosmall is selected
if (nosmall) {
if ((a - 1) / sqrt(n) > 0.1) {
dat <- dat[1:(xm - 1)]
print(paste(
"(plfit) Warning : xmin search truncated beyond",
xmins[xm - 1]
))
break
}
}
# compute KS statistic
cx <- c(0:(n - 1)) / n
cf <- 1 - (xmin / z) ^ a
dat[xm] <- max(abs(cf - cx))
}
#min de D
D <- min(dat)
#argmin de D
xmin <- xmins[min(which(dat <= D))]
#exces et estimations finaux
z <- x[x >= xmin]
n <- length(z)
alpha <- 1 + n / sum(log(z / xmin))
if (finite) {
# finite-size correction
alpha <- alpha * (n - 1) / n + 1 / n
}
}
#
# end continuous case
#
###########################################################################################
###########################################################################################
# DISCRETE CASE
# estimate xmin and alpha in the discrete case
#
if (fdattype == "integer") {
if (is.null(vec)) {
vec <-
seq(1.5, 3.5, .01)
} # covers range of most practical scaling parameters
zvec <- VGAM::zeta(vec)
#On enleve la plus grande valeur
xmins <- sort(unique(x))
xmins <- xmins[-length(xmins)]
if (!is.null(limit)) {
limit <- round(limit)
xmins <- xmins[xmins >= limit]
}
if (!is.null(fixed)) {
xmins <- fixed
}
if (!is.null(sampl)) {
xmins <- xmins[unique(round(seq(1, length(xmins), length.out = sampl)))]
}
if (is.null(xmins) || length(xmins) < 2) {
stop("(plfit) error: x must contain at least two unique values.")
}
if (length(which(xmins == 0) > 0)) {
stop("(plfit) error: x must not contain the value 0.")
}
xmax <- max(x)
dat <- matrix(0, nrow = length(xmins), ncol = 2)
z <- x
for (xm in 1:length(xmins)) {
xmin <- xmins[xm]
z <- z[z >= xmin]
n <- length(z)
# estimate alpha via direct maximization of likelihood function
# vectorized version of numerical computation
# matlab: zdiff = sum( repmat((1:xmin-1)',1,length(vec)).^-repmat(vec,xmin-1,1) ,1);
if (xmin == 1) {
zdiff <- rep(0, length(vec))
} else{
zdiff <-
apply(rep(t(1:(xmin - 1)), length(vec)) ^ -t(kronecker(t(
array(1, xmin - 1)
), vec)), 2, sum)
}
# matlab: L = -vec.*sum(log(z)) - n.*log(zvec - zdiff);
L <- -vec * sum(log(z)) - n * log(zvec - zdiff)
I <- which.max(L)
# compute KS statistic
fit <-
cumsum((((xmin:xmax) ^ -vec[I])) / (zvec[I] - sum((1:(
xmin - 1
)) ^ -vec[I])))
cdi <-
cumsum(hist(z, c(
min(z) - 1, (xmin + .5):xmax, max(z) + 1
), plot = FALSE)$counts / n)
dat[xm, ] <- c(max(abs(fit - cdi)), vec[I])
}
D <- min(dat[, 1])
I <- which.min(dat[, 1])
xmin <- xmins[I]
n <- sum(x >= xmin)
alpha <- dat[I, 2]
if (finite) {
alpha <- alpha * (n - 1) / n + 1 / n # finite-size correction