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R/semimetric.NPFDA.r
In fda.usc: Functional Data Analysis and Utilities for Statistical Computing
Defines functions
hshift
semimetric.pca
semimetric.mplsr
semimetric.hshift
semimetric.fourier
semimetric.deriv
Documented in
semimetric.deriv
semimetric.fourier
semimetric.hshift
semimetric.mplsr
semimetric.pca
#' @name semimetric.NPFDA
#' @title Proximities between functional data (semi-metrics)
#'
#' @description Computes semi-metric distances of functional data based on Ferraty F and
#' Vieu, P. (2006).
#'
#' @details
#' \code{semimetric.deriv}: approximates \eqn{L_2} metric
#' between derivatives of the curves based on ther B-spline representation. The
#' derivatives set with the argument \code{nderiv}.\cr
#' \code{semimetric.fourier}: approximates \eqn{L_2} metric between the curves
#' based on ther B-spline representation. The derivatives set with the argument
#' \code{nderiv}.\cr \code{semimetric.hshift}: computes distance between curves
#' taking into account an horizontal shift effect.\cr \code{semimetric.mplsr}:
#' computes distance between curves based on the partial least squares
#' method.\cr \code{semimetric.pca}: computes distance between curves based on
#' the functional principal components analysis method.
#'
#' In the next semi-metric functions the functional data \eqn{X} is
#' approximated by \eqn{k_n} elements of the Fourier, B--spline, PC or PLS basis
#' using, \eqn{\hat{X_i} =\sum_{k=1}^{k_n}\nu_{k,i}\xi_k}, where \eqn{\nu_k}
#' are the coefficient of the expansion on the basis function
#' \eqn{\left\{\xi_k\right\}_{k=1}^{\infty}}.\cr The distances between the q-order derivatives of two curves \eqn{X_{1}} and
#' \eqn{X_2} is,
#' \deqn{d_{2}^{(q)}\left(X_1,X_2\right)_{k_n}=\sqrt{\frac{1}{T}\int_{T}\left(X_{1}^{(q)}(t)-X_{2}^{(q)}(t)\right)^2
#' dt}} where \eqn{X_{i}^{(q)}\left(t\right)} denot the \eqn{q} derivative of
#' \eqn{X_i}.
#'
#' \code{semimetric.deriv} and \code{semimetric.fourier} function use a
#' B-spline and Fourier approximation respectively for each curve and the
#' derivatives are directly computed by differentiating several times their
#' analytic form, by default \code{q=1} and \code{q=0} respectively.
#' \code{semimetric.pca} and \code{semimetric.mprls} function compute
#' proximities between curves based on the functional principal components
#' analysis (FPCA) and the functional partial least square analysis (FPLS),
#' respectively. The FPC and FPLS reduce the functional data in a reduced
#' dimensional space (q components). \code{semimetric.mprls} function requires
#' a scalar response.
#'
#' \deqn{d_{2}^{(q)}\left(X_1,X_2\right)_{k_n}\approx\sqrt{\sum_{k=1}^{k_n}\left(\nu_{k,1}-\nu_{k,2}\right)^2\left\|\xi_k^{(q)}\right\|dt}}
#' \code{semimetric.hshift} computes proximities between curves taking into
#' account an horizontal shift effect.
#'
#' \deqn{d_{hshift}\left(X_1,X_2\right)=\min_{h\in\left[-mh,mh\right]}d_2(X_1(t),X_2(t+h))}
#' where \eqn{mh} is the maximum horizontal shifted allowed.
#'
#' @aliases semimetric.NPFDA semimetric.deriv semimetric.fourier
#' semimetric.hshift semimetric.mplsr semimetric.pca
#' @param fdata1 Functional data 1 or curve 1. \code{DATA1} with dimension
#' (\code{n1} x \code{m}), where \code{n1} is the number of curves and \code{m}
#' are the points observed in each curve.
#' @param fdata2 Functional data 2 or curve 2. \code{DATA1} with dimension
#' (\code{n2} x \code{m}), where \code{n2} is the number of curves and \code{m}
#' are the points observed in each curve.
#' @param q If \code{semimetric.pca}: the retained number of principal
#' components.\cr If \code{semimetric.mplsr}: the retained number of factors.
#' @param nknot semimetric.deriv argument: number of interior knots (needed for
#' defining the B-spline basis).
#' @param nderiv Order of derivation, used in \code{semimetric.deriv} and \cr
#' \code{semimetric.fourier}
#' @param nbasis \code{semimetric.fourier}: size of the basis.
#' @param period \code{semimetric.fourier}:allows to select the period for the
#' fourier expansion.
#' @param t \code{semimetric.hshift}: vector which defines \code{t} (one can
#' choose \code{1,2,...,nbt} where \code{nbt} is the number of points of the
#' discretization)
#' @param class1 \code{semimetric.mplsr}: vector containing a categorical
#' response which corresponds to class number for units stored in \code{DATA1}.
#' @param \dots Further arguments passed to or from other methods.
#' @return Returns a proximities matrix between two functional datasets.
#' @seealso See also \code{\link{metric.lp}} and \code{\link{semimetric.basis}}
#' @references Ferraty, F. and Vieu, P. (2006). \emph{Nonparametric functional
#' data analysis.} Springer Series in Statistics, New York.
#'
#' Ferraty, F. and Vieu, P. (2006). \emph{NPFDA in practice}. Free access on
#' line at \url{https://www.math.univ-toulouse.fr/~ferraty/SOFTWARES/NPFDA/}
#' @source \url{https://www.math.univ-toulouse.fr/~ferraty/SOFTWARES/NPFDA/}
#' @keywords cluster
#' @examples
#' \dontrun{
#' # INFERENCE PHONDAT
#' data(phoneme)
#' ind=1:100 # 2 groups
#' mlearn<-phoneme$learn[ind,]
#' mtest<-phoneme$test[ind,]
#' n=nrow(mlearn[["data"]])
#' np=ncol(mlearn[["data"]])
#' mdist1=semimetric.pca(mlearn,mtest)
#' mdist2=semimetric.pca(mlearn,mtest,q=2)
#' mdist3=semimetric.deriv(mlearn,mtest,nderiv=0)
#' mdist4=semimetric.fourier(mlearn,mtest,nderiv=2,nbasis=21)
#' #uses hshift function
#' #mdist5=semimetric.hshift(mlearn,mtest) #takes a lot
#' glearn<-phoneme$classlearn[ind]
#' #uses mplsr function
#' mdist6=semimetric.mplsr(mlearn,mtest,5,glearn)
#' mdist0=metric.lp(mlearn,mtest)
#' b=as.dist(mdist6)
#' c2=hclust(b)
#' plot(c2)
#' memb <- cutree(c2, k = 2)
#' table(memb,phoneme$classlearn[ind])
#' }
#'
#' @rdname semimetric.NPFDA
#' @export
semimetric.deriv <- function(fdata1,fdata2=fdata1, nderiv=1,
nknot=ifelse(floor(ncol(DATA1)/3) > floor((ncol(DATA1) - nderiv - 4)/2),
floor((ncol(DATA1) - nderiv - 4)/2), floor(ncol(DATA1)/3)),...)
{
###############################################################
# Computes a semimetric between curves based on their derivatives.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "nderiv" order of derivation
# "nknot" number of interior knots (needed for defining the B-spline basis)
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
#####################################################################
# B-spline approximation of the curves containing in DATASET :
# -----------------------------------------------------------
# "knot" and "x" allow to define the B-spline basis
# "coef.mat1[, i]" corresponds to the B-spline expansion
# of the discretized curve contained in DATASET[i, ].
# The B-spline approximation of the curve contained in "DATA1[i, ]"
# is given by "Bspline %*% coef.mat1[, i]"
#####################################################################
p <- ncol(DATA1)
a <- range.t[1]
b <- range.t[2]
x <- seq(a, b, length = p)
order.Bspline <- nderiv + 3
nknotmax <- (p - order.Bspline - 1)%/%2
if(nknot > nknotmax){
stop(paste("give a number nknot smaller than ",nknotmax, " for avoiding ill-conditioned matrix"))
}
Knot <- seq(a, b, length = nknot + 2)[ - c(1, nknot + 2)]
delta <- sort(c(rep(c(a, b), order.Bspline), Knot))
Bspline <- splineDesign(delta, x, order.Bspline)
Cmat <- crossprod(Bspline)
Dmat1 <- crossprod(Bspline, t(DATA1))
coef.mat1 <- symsolve(Cmat, Dmat1)
#######################################################################
# Numerical integration by the Gauss method :
# -------------------------------------------
# The objects ending by "gauss" allow us to compute numerically
# integrals by means the "Gauss method" (lx.gauss=6 ==> the computation
# of the integral is exact for polynom of degree less or equal to 11).
#######################################################################
point.gauss <- c(-0.9324695142, -0.6612093865, -0.2386191861,
0.2386191861, 0.6612093865, 0.9324695142)
weight.gauss <- c(0.1713244924, 0.360761573, 0.4679139346, 0.4679139346,0.360761573, 0.1713244924)
x.gauss <- 0.5 * ((b + a) + (b - a) * point.gauss)
lx.gauss <- length(x.gauss)
Bspline.deriv <- splineDesign(delta, x.gauss, order.Bspline, rep(nderiv, lx.gauss))
H <- t(Bspline.deriv) %*% (Bspline.deriv * (weight.gauss * 0.5 * (b - a)))
eigH <- eigen(H, symmetric = TRUE)
eigH$values[eigH$values < 0] <- 0
Hhalf <- t(eigH$vectors %*% (t(eigH$vectors) * sqrt(eigH$values)))
COEF1 <- t(Hhalf %*% coef.mat1)
if(twodatasets){
Dmat2 <- crossprod(Bspline, t(DATA2))
coef.mat2 <- symsolve(Cmat, Dmat2)
COEF2 <- t(Hhalf %*% coef.mat2)
} else {
COEF2 <- COEF1
}
SEMIMETRIC <- 0
nbasis <- nrow(H)
for(f in 1:nbasis)
SEMIMETRIC <- SEMIMETRIC + outer(COEF1[, f], COEF2[, f], "-")^2
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.deriv"
attr(mdist,"par.metric")<-list("nderiv"=nderiv,"nknot"=nknot,"range.t"=range.t)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.fourier <- function(fdata1,fdata2=fdata1, nderiv=0,
nbasis=ifelse(floor(ncol(DATA1)/3) > floor((ncol(DATA1) - nderiv - 4)/2),
floor((ncol(DATA1) - nderiv - 4)/2), floor(ncol(DATA1)/3)), period=NULL,...)
{
###############################################################
# Computes a semimetric between curves based on their Fourier expansion.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "nderiv" order of derivation
# "nbasis" size of the basis
# "period" allows to select the period for the fourier expansion
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
p <- ncol(DATA1)
nbasismax <- (p - nbasis)%/%2
if(nbasis > nbasismax){
stop(paste("give a number nbasis smaller than ",nbasismax, " for avoiding ill-conditioned matrix"))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
a <- range.t[1]
b <- range.t[2]
Eval <- seq(a, b, length = p)
#####################################################################
# Fourier approximation of the curves containing in DATA1 :
# -----------------------------------------------------------
# "fourier" allows to define the Fourier basis
# "COEF1[, i]" corresponds to the Fourier expansion
# of the discretized curve contained in DATA1[i, ].
# The Fourier approximation of the curve contained in "DATA1[i, ]"
# is given by "FOURIER %*% COEF1[, i]"
#####################################################################
if(is.null(period)) period <- b - a
FOURIER <- fourier(Eval, nbasis, period)
CMAT <- crossprod(FOURIER)
DMAT1 <- crossprod(FOURIER, t(DATA1))
COEF1 <- symsolve(CMAT, DMAT1)
#######################################################################
# Numerical integration by the Gauss method :
# -------------------------------------------
# The objects ending by "gauss" allow us to compute numerically
# integrals by means the "Gauss method" (Leval.gauss=6 ==> the computation
# of the integral is exact for polynom of degree less or equal to 11).
#######################################################################
Point.gauss <- c(-0.9324695142, -0.6612093865, -0.2386191861,
0.2386191861, 0.6612093865, 0.9324695142)
Weight.gauss <- c(0.1713244924, 0.360761573, 0.4679139346, 0.4679139346,
0.360761573, 0.1713244924)
Eval.gauss <- 0.5 * (b - a) * (1 + Point.gauss)
Leval.gauss <- length(Eval.gauss)
FOURIER.DERIV <- fourier(Eval.gauss, nbasis, period, nderiv)
H <- t(FOURIER.DERIV) %*% (FOURIER.DERIV * (Weight.gauss * 0.5 * (b - a
)))
eigH <- eigen(H, symmetric = TRUE)
eigH$values[eigH$values < 0] <- 0
HALF <- t(eigH$vectors %*% (t(eigH$vectors) * sqrt(eigH$values)))
COEF1 <- t(HALF %*% COEF1)
if(twodatasets) {
DMAT2 <- crossprod(FOURIER, t(DATA2))
COEF2 <- symsolve(CMAT, DMAT2)
COEF2 <- t(HALF %*% COEF2)
}
else {
COEF2 <- COEF1
}
SEMIMETRIC <- 0
for(f in 1:nbasis)
SEMIMETRIC <- SEMIMETRIC + outer(COEF1[, f], COEF2[, f], "-")^2
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.fourier"
attr(mdist,"par.metric")<-list("nderiv"=nderiv,"nbasis"=nbasis,"range.t"=range.t,"period"=period)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.hshift <- function(fdata1,fdata2=fdata1, t=1:ncol(DATA1),...)
{
###############################################################
# Computes between curves a semimetric taking into account an
# horizontal shift effect.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "t" vector which defines the grid (one can choose 1,2,...,nbgrid
# where nbgrid is the number of points of the discretization)
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
n1 <- nrow(DATA1)
if(twodatasets) n2 <- nrow(DATA2) else n2 <- n1
SEMIMETRIC <- matrix(0, nrow=n1, ncol=n2)
if(!twodatasets){
for(i in 1:(n1-1)){
for(j in (i+1):n2){
SEMIMETRIC[i,j] <- hshift(DATA1[i,], DATA2[j,], t)$dist
}
}
SEMIMETRIC <- SEMIMETRIC + t(SEMIMETRIC)
}else{
for(i in 1:n1){
for(j in 1:n2){
SEMIMETRIC[i,j] <- hshift(DATA1[i,], DATA2[j,], t)$dist
}
}
}
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.hshift"
attr(mdist,"par.metric")<-list("t"=t)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.mplsr <- function(fdata1,fdata2=fdata1, q=2, class1,...)
{
###############################################################
# Computes between curves a semimetric based on the partial least
# squares method.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "q" the retained number of factors
# "class1" vector containing a categorical response which
# corresponds to class number for units stored in DATA1
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
qmax <- ncol(DATA1)
if(q > qmax) stop(paste("give a integer q smaller than ", qmax))
n1 <- nrow(DATA1)
if (is.factor(class1)) {
class1=as.numeric(class1)
nbclass <- length(table(class1))#max(class1)
BINARY1 <- matrix(0, nrow = n1, ncol = nbclass)
for(g in 1:nbclass) {
BINARY1[, g] <- as.numeric(class1 == g)
}
}
else {
BINARY1<- class1
if (!is.matrix(class1)) class1<-as.matrix(class1,ncol=1)
nbclass<-ncol(class1)
}
mplsr.res <- mplsr(DATA1, BINARY1, q)
COMPONENT1 <- DATA1 %*% mplsr.res$COEF
COMPONENT1 <- outer(rep(1, n1), as.vector(mplsr.res$b0)) + COMPONENT1
if(twodatasets) {
n2 <- nrow(DATA2)
COMPONENT2 <- DATA2 %*% mplsr.res$COEF
COMPONENT2 <- outer(rep(1, n2), as.vector(mplsr.res$b0)) +
COMPONENT2
}
else {
COMPONENT2 <- COMPONENT1
}
SEMIMETRIC <- 0
for(g in 1:nbclass)
SEMIMETRIC <- SEMIMETRIC + outer(COMPONENT1[, g], COMPONENT2[,
g], "-")^2
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.mplsr"
attr(mdist,"par.metric")<-list("q"=q,"class1"=class1)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.pca <- function(fdata1, fdata2=fdata1, q=1,...)
{
###############################################################
# Computes between curves a pca-type semimetric based on the
# functional principal components analysis method.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "q" the retained number of principal components
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
qmax <- ncol(DATA1)
if(q > qmax) stop(paste("give a integer q smaller than ", qmax))
n <- nrow(DATA1)
COVARIANCE <- t(DATA1) %*% DATA1/n
ei=eigen(COVARIANCE, symmetric = TRUE)
EIGENVECTORS <- matrix(ei$vectors[, 1:q],ncol=q)
COMPONENT1 <- DATA1 %*% EIGENVECTORS
if(twodatasets) { COMPONENT2 <- DATA2 %*% EIGENVECTORS }
else { COMPONENT2 <- COMPONENT1 }
SEMIMETRIC <- 0
for(qq in 1:q)
SEMIMETRIC <- SEMIMETRIC + outer(COMPONENT1[, qq], COMPONENT2[,qq], "-")^2
mdist<-sqrt(SEMIMETRIC)
# attr(mdist,"call")<-C1
attr(mdist,"call")<-"semimetric.pca"
attr(mdist,"par.metric")<-list("q"=q)
return(mdist)
}
hshift <- function(x,y, t=1:ncol(x),...)
{
####################################################################
# Returns the "horizontal shifted proximity" between two discretized
# curves "x" and "y" (vectors of same length).
# The user has to choose a "t".
#####################################################################
lgrid <- length(t)
a <- t[1]
b <- t[lgrid]
rang <- b - a
lagmax <- floor(0.2 * rang)
integrand <- (x-y)^2
Dist1 <- sum(integrand[-1] + integrand[-lgrid])/(2 * rang)
Dist2 <- Dist1
for(i in 1:lagmax){
xlag <- x[-(1:i)]
xforward <- x[-((lgrid-i+1):lgrid)]
ylag <- y[-(1:i)]
yforward <- y[-((lgrid-i+1):lgrid)]
integrand1 <- (xlag-yforward)^2
integrand2 <- (xforward-ylag)^2
lintegrand <- length(integrand1)
rescaled.range <- 2 * (rang - 2 * i)
Dist1[i+1] <- sum(integrand1[-1] + integrand1[-lintegrand])/rescaled.range
Dist2[i+1] <- sum(integrand2[-1] + integrand2[-lintegrand])/rescaled.range
}
lag1 <- (0:lagmax)[order(Dist1)[1]]
lag2 <- (0:lagmax)[order(Dist2)[1]]
distmin1 <- min(Dist1)
distmin2 <- min(Dist2)
if(distmin1 < distmin2){
distmin <- distmin1
lagopt <- lag1
}else{
distmin <- distmin2
lagopt <- lag2
}
return(list(dist=sqrt(distmin),lag=lagopt))
}
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Defines functions hshift semimetric.pca semimetric.mplsr semimetric.hshift semimetric.fourier semimetric.deriv
Documented in semimetric.deriv semimetric.fourier semimetric.hshift semimetric.mplsr semimetric.pca
#' @name semimetric.NPFDA
#' @title Proximities between functional data (semi-metrics)
#'
#' @description Computes semi-metric distances of functional data based on Ferraty F and
#' Vieu, P. (2006).
#'
#' @details
#' \code{semimetric.deriv}: approximates \eqn{L_2} metric
#' between derivatives of the curves based on ther B-spline representation. The
#' derivatives set with the argument \code{nderiv}.\cr
#' \code{semimetric.fourier}: approximates \eqn{L_2} metric between the curves
#' based on ther B-spline representation. The derivatives set with the argument
#' \code{nderiv}.\cr \code{semimetric.hshift}: computes distance between curves
#' taking into account an horizontal shift effect.\cr \code{semimetric.mplsr}:
#' computes distance between curves based on the partial least squares
#' method.\cr \code{semimetric.pca}: computes distance between curves based on
#' the functional principal components analysis method.
#'
#' In the next semi-metric functions the functional data \eqn{X} is
#' approximated by \eqn{k_n} elements of the Fourier, B--spline, PC or PLS basis
#' using, \eqn{\hat{X_i} =\sum_{k=1}^{k_n}\nu_{k,i}\xi_k}, where \eqn{\nu_k}
#' are the coefficient of the expansion on the basis function
#' \eqn{\left\{\xi_k\right\}_{k=1}^{\infty}}.\cr The distances between the q-order derivatives of two curves \eqn{X_{1}} and
#' \eqn{X_2} is,
#' \deqn{d_{2}^{(q)}\left(X_1,X_2\right)_{k_n}=\sqrt{\frac{1}{T}\int_{T}\left(X_{1}^{(q)}(t)-X_{2}^{(q)}(t)\right)^2
#' dt}} where \eqn{X_{i}^{(q)}\left(t\right)} denot the \eqn{q} derivative of
#' \eqn{X_i}.
#'
#' \code{semimetric.deriv} and \code{semimetric.fourier} function use a
#' B-spline and Fourier approximation respectively for each curve and the
#' derivatives are directly computed by differentiating several times their
#' analytic form, by default \code{q=1} and \code{q=0} respectively.
#' \code{semimetric.pca} and \code{semimetric.mprls} function compute
#' proximities between curves based on the functional principal components
#' analysis (FPCA) and the functional partial least square analysis (FPLS),
#' respectively. The FPC and FPLS reduce the functional data in a reduced
#' dimensional space (q components). \code{semimetric.mprls} function requires
#' a scalar response.
#'
#' \deqn{d_{2}^{(q)}\left(X_1,X_2\right)_{k_n}\approx\sqrt{\sum_{k=1}^{k_n}\left(\nu_{k,1}-\nu_{k,2}\right)^2\left\|\xi_k^{(q)}\right\|dt}}
#' \code{semimetric.hshift} computes proximities between curves taking into
#' account an horizontal shift effect.
#'
#' \deqn{d_{hshift}\left(X_1,X_2\right)=\min_{h\in\left[-mh,mh\right]}d_2(X_1(t),X_2(t+h))}
#' where \eqn{mh} is the maximum horizontal shifted allowed.
#'
#' @aliases semimetric.NPFDA semimetric.deriv semimetric.fourier
#' semimetric.hshift semimetric.mplsr semimetric.pca
#' @param fdata1 Functional data 1 or curve 1. \code{DATA1} with dimension
#' (\code{n1} x \code{m}), where \code{n1} is the number of curves and \code{m}
#' are the points observed in each curve.
#' @param fdata2 Functional data 2 or curve 2. \code{DATA1} with dimension
#' (\code{n2} x \code{m}), where \code{n2} is the number of curves and \code{m}
#' are the points observed in each curve.
#' @param q If \code{semimetric.pca}: the retained number of principal
#' components.\cr If \code{semimetric.mplsr}: the retained number of factors.
#' @param nknot semimetric.deriv argument: number of interior knots (needed for
#' defining the B-spline basis).
#' @param nderiv Order of derivation, used in \code{semimetric.deriv} and \cr
#' \code{semimetric.fourier}
#' @param nbasis \code{semimetric.fourier}: size of the basis.
#' @param period \code{semimetric.fourier}:allows to select the period for the
#' fourier expansion.
#' @param t \code{semimetric.hshift}: vector which defines \code{t} (one can
#' choose \code{1,2,...,nbt} where \code{nbt} is the number of points of the
#' discretization)
#' @param class1 \code{semimetric.mplsr}: vector containing a categorical
#' response which corresponds to class number for units stored in \code{DATA1}.
#' @param \dots Further arguments passed to or from other methods.
#' @return Returns a proximities matrix between two functional datasets.
#' @seealso See also \code{\link{metric.lp}} and \code{\link{semimetric.basis}}
#' @references Ferraty, F. and Vieu, P. (2006). \emph{Nonparametric functional
#' data analysis.} Springer Series in Statistics, New York.
#'
#' Ferraty, F. and Vieu, P. (2006). \emph{NPFDA in practice}. Free access on
#' line at \url{https://www.math.univ-toulouse.fr/~ferraty/SOFTWARES/NPFDA/}
#' @source \url{https://www.math.univ-toulouse.fr/~ferraty/SOFTWARES/NPFDA/}
#' @keywords cluster
#' @examples
#' \dontrun{
#' # INFERENCE PHONDAT
#' data(phoneme)
#' ind=1:100 # 2 groups
#' mlearn<-phoneme$learn[ind,]
#' mtest<-phoneme$test[ind,]
#' n=nrow(mlearn[["data"]])
#' np=ncol(mlearn[["data"]])
#' mdist1=semimetric.pca(mlearn,mtest)
#' mdist2=semimetric.pca(mlearn,mtest,q=2)
#' mdist3=semimetric.deriv(mlearn,mtest,nderiv=0)
#' mdist4=semimetric.fourier(mlearn,mtest,nderiv=2,nbasis=21)
#' #uses hshift function
#' #mdist5=semimetric.hshift(mlearn,mtest) #takes a lot
#' glearn<-phoneme$classlearn[ind]
#' #uses mplsr function
#' mdist6=semimetric.mplsr(mlearn,mtest,5,glearn)
#' mdist0=metric.lp(mlearn,mtest)
#' b=as.dist(mdist6)
#' c2=hclust(b)
#' plot(c2)
#' memb <- cutree(c2, k = 2)
#' table(memb,phoneme$classlearn[ind])
#' }
#'
#' @rdname semimetric.NPFDA
#' @export
semimetric.deriv <- function(fdata1,fdata2=fdata1, nderiv=1,
nknot=ifelse(floor(ncol(DATA1)/3) > floor((ncol(DATA1) - nderiv - 4)/2),
floor((ncol(DATA1) - nderiv - 4)/2), floor(ncol(DATA1)/3)),...)
{
###############################################################
# Computes a semimetric between curves based on their derivatives.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "nderiv" order of derivation
# "nknot" number of interior knots (needed for defining the B-spline basis)
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
#####################################################################
# B-spline approximation of the curves containing in DATASET :
# -----------------------------------------------------------
# "knot" and "x" allow to define the B-spline basis
# "coef.mat1[, i]" corresponds to the B-spline expansion
# of the discretized curve contained in DATASET[i, ].
# The B-spline approximation of the curve contained in "DATA1[i, ]"
# is given by "Bspline %*% coef.mat1[, i]"
#####################################################################
p <- ncol(DATA1)
a <- range.t[1]
b <- range.t[2]
x <- seq(a, b, length = p)
order.Bspline <- nderiv + 3
nknotmax <- (p - order.Bspline - 1)%/%2
if(nknot > nknotmax){
stop(paste("give a number nknot smaller than ",nknotmax, " for avoiding ill-conditioned matrix"))
}
Knot <- seq(a, b, length = nknot + 2)[ - c(1, nknot + 2)]
delta <- sort(c(rep(c(a, b), order.Bspline), Knot))
Bspline <- splineDesign(delta, x, order.Bspline)
Cmat <- crossprod(Bspline)
Dmat1 <- crossprod(Bspline, t(DATA1))
coef.mat1 <- symsolve(Cmat, Dmat1)
#######################################################################
# Numerical integration by the Gauss method :
# -------------------------------------------
# The objects ending by "gauss" allow us to compute numerically
# integrals by means the "Gauss method" (lx.gauss=6 ==> the computation
# of the integral is exact for polynom of degree less or equal to 11).
#######################################################################
point.gauss <- c(-0.9324695142, -0.6612093865, -0.2386191861,
0.2386191861, 0.6612093865, 0.9324695142)
weight.gauss <- c(0.1713244924, 0.360761573, 0.4679139346, 0.4679139346,0.360761573, 0.1713244924)
x.gauss <- 0.5 * ((b + a) + (b - a) * point.gauss)
lx.gauss <- length(x.gauss)
Bspline.deriv <- splineDesign(delta, x.gauss, order.Bspline, rep(nderiv, lx.gauss))
H <- t(Bspline.deriv) %*% (Bspline.deriv * (weight.gauss * 0.5 * (b - a)))
eigH <- eigen(H, symmetric = TRUE)
eigH$values[eigH$values < 0] <- 0
Hhalf <- t(eigH$vectors %*% (t(eigH$vectors) * sqrt(eigH$values)))
COEF1 <- t(Hhalf %*% coef.mat1)
if(twodatasets){
Dmat2 <- crossprod(Bspline, t(DATA2))
coef.mat2 <- symsolve(Cmat, Dmat2)
COEF2 <- t(Hhalf %*% coef.mat2)
} else {
COEF2 <- COEF1
}
SEMIMETRIC <- 0
nbasis <- nrow(H)
for(f in 1:nbasis)
SEMIMETRIC <- SEMIMETRIC + outer(COEF1[, f], COEF2[, f], "-")^2
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.deriv"
attr(mdist,"par.metric")<-list("nderiv"=nderiv,"nknot"=nknot,"range.t"=range.t)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.fourier <- function(fdata1,fdata2=fdata1, nderiv=0,
nbasis=ifelse(floor(ncol(DATA1)/3) > floor((ncol(DATA1) - nderiv - 4)/2),
floor((ncol(DATA1) - nderiv - 4)/2), floor(ncol(DATA1)/3)), period=NULL,...)
{
###############################################################
# Computes a semimetric between curves based on their Fourier expansion.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "nderiv" order of derivation
# "nbasis" size of the basis
# "period" allows to select the period for the fourier expansion
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
p <- ncol(DATA1)
nbasismax <- (p - nbasis)%/%2
if(nbasis > nbasismax){
stop(paste("give a number nbasis smaller than ",nbasismax, " for avoiding ill-conditioned matrix"))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
a <- range.t[1]
b <- range.t[2]
Eval <- seq(a, b, length = p)
#####################################################################
# Fourier approximation of the curves containing in DATA1 :
# -----------------------------------------------------------
# "fourier" allows to define the Fourier basis
# "COEF1[, i]" corresponds to the Fourier expansion
# of the discretized curve contained in DATA1[i, ].
# The Fourier approximation of the curve contained in "DATA1[i, ]"
# is given by "FOURIER %*% COEF1[, i]"
#####################################################################
if(is.null(period)) period <- b - a
FOURIER <- fourier(Eval, nbasis, period)
CMAT <- crossprod(FOURIER)
DMAT1 <- crossprod(FOURIER, t(DATA1))
COEF1 <- symsolve(CMAT, DMAT1)
#######################################################################
# Numerical integration by the Gauss method :
# -------------------------------------------
# The objects ending by "gauss" allow us to compute numerically
# integrals by means the "Gauss method" (Leval.gauss=6 ==> the computation
# of the integral is exact for polynom of degree less or equal to 11).
#######################################################################
Point.gauss <- c(-0.9324695142, -0.6612093865, -0.2386191861,
0.2386191861, 0.6612093865, 0.9324695142)
Weight.gauss <- c(0.1713244924, 0.360761573, 0.4679139346, 0.4679139346,
0.360761573, 0.1713244924)
Eval.gauss <- 0.5 * (b - a) * (1 + Point.gauss)
Leval.gauss <- length(Eval.gauss)
FOURIER.DERIV <- fourier(Eval.gauss, nbasis, period, nderiv)
H <- t(FOURIER.DERIV) %*% (FOURIER.DERIV * (Weight.gauss * 0.5 * (b - a
)))
eigH <- eigen(H, symmetric = TRUE)
eigH$values[eigH$values < 0] <- 0
HALF <- t(eigH$vectors %*% (t(eigH$vectors) * sqrt(eigH$values)))
COEF1 <- t(HALF %*% COEF1)
if(twodatasets) {
DMAT2 <- crossprod(FOURIER, t(DATA2))
COEF2 <- symsolve(CMAT, DMAT2)
COEF2 <- t(HALF %*% COEF2)
}
else {
COEF2 <- COEF1
}
SEMIMETRIC <- 0
for(f in 1:nbasis)
SEMIMETRIC <- SEMIMETRIC + outer(COEF1[, f], COEF2[, f], "-")^2
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.fourier"
attr(mdist,"par.metric")<-list("nderiv"=nderiv,"nbasis"=nbasis,"range.t"=range.t,"period"=period)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.hshift <- function(fdata1,fdata2=fdata1, t=1:ncol(DATA1),...)
{
###############################################################
# Computes between curves a semimetric taking into account an
# horizontal shift effect.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "t" vector which defines the grid (one can choose 1,2,...,nbgrid
# where nbgrid is the number of points of the discretization)
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
n1 <- nrow(DATA1)
if(twodatasets) n2 <- nrow(DATA2) else n2 <- n1
SEMIMETRIC <- matrix(0, nrow=n1, ncol=n2)
if(!twodatasets){
for(i in 1:(n1-1)){
for(j in (i+1):n2){
SEMIMETRIC[i,j] <- hshift(DATA1[i,], DATA2[j,], t)$dist
}
}
SEMIMETRIC <- SEMIMETRIC + t(SEMIMETRIC)
}else{
for(i in 1:n1){
for(j in 1:n2){
SEMIMETRIC[i,j] <- hshift(DATA1[i,], DATA2[j,], t)$dist
}
}
}
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.hshift"
attr(mdist,"par.metric")<-list("t"=t)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.mplsr <- function(fdata1,fdata2=fdata1, q=2, class1,...)
{
###############################################################
# Computes between curves a semimetric based on the partial least
# squares method.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "q" the retained number of factors
# "class1" vector containing a categorical response which
# corresponds to class number for units stored in DATA1
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
qmax <- ncol(DATA1)
if(q > qmax) stop(paste("give a integer q smaller than ", qmax))
n1 <- nrow(DATA1)
if (is.factor(class1)) {
class1=as.numeric(class1)
nbclass <- length(table(class1))#max(class1)
BINARY1 <- matrix(0, nrow = n1, ncol = nbclass)
for(g in 1:nbclass) {
BINARY1[, g] <- as.numeric(class1 == g)
}
}
else {
BINARY1<- class1
if (!is.matrix(class1)) class1<-as.matrix(class1,ncol=1)
nbclass<-ncol(class1)
}
mplsr.res <- mplsr(DATA1, BINARY1, q)
COMPONENT1 <- DATA1 %*% mplsr.res$COEF
COMPONENT1 <- outer(rep(1, n1), as.vector(mplsr.res$b0)) + COMPONENT1
if(twodatasets) {
n2 <- nrow(DATA2)
COMPONENT2 <- DATA2 %*% mplsr.res$COEF
COMPONENT2 <- outer(rep(1, n2), as.vector(mplsr.res$b0)) +
COMPONENT2
}
else {
COMPONENT2 <- COMPONENT1
}
SEMIMETRIC <- 0
for(g in 1:nbclass)
SEMIMETRIC <- SEMIMETRIC + outer(COMPONENT1[, g], COMPONENT2[,
g], "-")^2
mdist<-sqrt(SEMIMETRIC)
attr(mdist,"call")<-"semimetric.mplsr"
attr(mdist,"par.metric")<-list("q"=q,"class1"=class1)
return(mdist)
}
#' @rdname semimetric.NPFDA
#' @export
semimetric.pca <- function(fdata1, fdata2=fdata1, q=1,...)
{
###############################################################
# Computes between curves a pca-type semimetric based on the
# functional principal components analysis method.
# "DATA1" matrix containing a first set of curves stored row by row
# "DATA2" matrix containing a second set of curves stored row by row
# "q" the retained number of principal components
# Returns a "semimetric" matrix containing the semimetric computed
# between the curves lying to the first sample and the curves lying
# to the second one.
###############################################################
C1<-match.call()
if (is.fdata(fdata1)) {
tt<-fdata1[["argvals"]]
rtt<-fdata1[["rangeval"]]
nas1<-is.na.fdata(fdata1)
if (any(nas1)) stop("fdata1 contain ",sum(nas1)," curves with some NA value \n")
else if (!is.fdata(fdata2)) {fdata2<-fdata(fdata2,tt,rtt) }
nas2<-is.na.fdata(fdata2)
if (any(nas2)) stop("fdata2 contain ",sum(nas2)," curves with some NA value \n")
DATA1<-fdata1[["data"]]
DATA2<-fdata2[["data"]]
range.t<-rtt
}
else {
if(is.vector(fdata1)) fdata1 <- as.matrix(t(fdata1))
if(is.vector(fdata2)) fdata2 <- as.matrix(t(fdata2))
DATA1<-fdata1
DATA2<-fdata2
range.t<-c(1,ncol(DATA1))
}
testfordim <- sum(dim(DATA1)==dim(DATA2))==2
twodatasets <- TRUE
if(testfordim) twodatasets <- sum(DATA1==DATA2)!=prod(dim(DATA1))
qmax <- ncol(DATA1)
if(q > qmax) stop(paste("give a integer q smaller than ", qmax))
n <- nrow(DATA1)
COVARIANCE <- t(DATA1) %*% DATA1/n
ei=eigen(COVARIANCE, symmetric = TRUE)
EIGENVECTORS <- matrix(ei$vectors[, 1:q],ncol=q)
COMPONENT1 <- DATA1 %*% EIGENVECTORS
if(twodatasets) { COMPONENT2 <- DATA2 %*% EIGENVECTORS }
else { COMPONENT2 <- COMPONENT1 }
SEMIMETRIC <- 0
for(qq in 1:q)
SEMIMETRIC <- SEMIMETRIC + outer(COMPONENT1[, qq], COMPONENT2[,qq], "-")^2
mdist<-sqrt(SEMIMETRIC)
# attr(mdist,"call")<-C1
attr(mdist,"call")<-"semimetric.pca"
attr(mdist,"par.metric")<-list("q"=q)
return(mdist)
}
hshift <- function(x,y, t=1:ncol(x),...)
{
####################################################################
# Returns the "horizontal shifted proximity" between two discretized
# curves "x" and "y" (vectors of same length).
# The user has to choose a "t".
#####################################################################
lgrid <- length(t)
a <- t[1]
b <- t[lgrid]
rang <- b - a
lagmax <- floor(0.2 * rang)
integrand <- (x-y)^2
Dist1 <- sum(integrand[-1] + integrand[-lgrid])/(2 * rang)
Dist2 <- Dist1
for(i in 1:lagmax){
xlag <- x[-(1:i)]
xforward <- x[-((lgrid-i+1):lgrid)]
ylag <- y[-(1:i)]
yforward <- y[-((lgrid-i+1):lgrid)]
integrand1 <- (xlag-yforward)^2
integrand2 <- (xforward-ylag)^2
lintegrand <- length(integrand1)
rescaled.range <- 2 * (rang - 2 * i)
Dist1[i+1] <- sum(integrand1[-1] + integrand1[-lintegrand])/rescaled.range
Dist2[i+1] <- sum(integrand2[-1] + integrand2[-lintegrand])/rescaled.range
}
lag1 <- (0:lagmax)[order(Dist1)[1]]
lag2 <- (0:lagmax)[order(Dist2)[1]]
distmin1 <- min(Dist1)
distmin2 <- min(Dist2)
if(distmin1 < distmin2){
distmin <- distmin1
lagopt <- lag1
}else{
distmin <- distmin2
lagopt <- lag2
}
return(list(dist=sqrt(distmin),lag=lagopt))
}
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