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#' Improved Local Tangent Space Alignment
#'
#' Conventional LTSA method relies on PCA for approximating local tangent spaces.
#' Improved LTSA (ILTSA) provides a remedy that can efficiently recover the geometric
#' structure of data manifolds even when data are sparse or non-uniformly distributed.
#'
#' @param X an \eqn{(n\times p)} matrix or data frame whose rows are observations
#' and columns represent independent variables.
#' @param ndim an integer-valued target dimension.
#' @param type a vector of neighborhood graph construction. Following types are supported;
#' \code{c("knn",k)}, \code{c("enn",radius)}, and \code{c("proportion",ratio)}.
#' Default is \code{c("proportion",0.1)}, connecting about 1/10 of nearest data points
#' among all data points. See also \code{\link{aux.graphnbd}} for more details.
#' @param symmetric one of \code{"intersect"}, \code{"union"} or \code{"asymmetric"} is supported. Default is \code{"union"}. See also \code{\link{aux.graphnbd}} for more details.
#' @param preprocess an additional option for preprocessing the data.
#' Default is "center". See also \code{\link{aux.preprocess}} for more details.
#' @param t heat kernel bandwidth parameter in \eqn{(0,\infty)}.
#'
#' @return a named list containing
#' \describe{
#' \item{Y}{an \eqn{(n\times ndim)} matrix whose rows are embedded observations.}
#' \item{trfinfo}{a list containing information for out-of-sample prediction.}
#' }
#'
#' @examples
#' \donttest{
#' ## load iris data
#' data(iris)
#' set.seed(100)
#' subid = sample(1:150,50)
#' X = as.matrix(iris[subid,1:4])
#' label = as.factor(iris[subid,5])
#'
#' ## try different bandwidth size
#' out1 <- do.iltsa(X, t=1)
#' out2 <- do.iltsa(X, t=10)
#' out3 <- do.iltsa(X, t=100)
#'
#' ## Visualize two comparisons
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(out1$Y, pch=19, col=label, main="ILTSA::t=1")
#' plot(out2$Y, pch=19, col=label, main="ILTSA::t=10")
#' plot(out3$Y, pch=19, col=label, main="ILTSA::t=100")
#' par(opar)
#' }
#'
#' @references
#' \insertRef{zhang_improved_2011}{Rdimtools}
#'
#' @author Kisung You
#' @rdname nonlinear_ILTSA
#' @concept nonlinear_methods
#' @export
do.iltsa <- function(X, ndim=2, type=c("proportion",0.25),
symmetric=c("union","intersect","asymmetric"),
preprocess=c("center","scale","cscale","decorrelate","whiten"),
t=10.0){
#------------------------------------------------------------------------
## PREPROCESSING
# 1. data matrix
aux.typecheck(X)
n = nrow(X)
p = ncol(X)
# 2. ndim
ndim = as.integer(ndim)
if (!check_ndim(ndim,p)){stop("* do.iltsa : 'ndim' is a positive integer in [1,#(covariates)).")}
# 3. nbd setup
nbdtype = type
if (missing(symmetric)){
nbdsymmetric = "union"
} else {
nbdsymmetric = match.arg(symmetric)
}
# 4. preprocess
if (missing(preprocess)){
algpreprocess = "center"
} else {
algpreprocess = match.arg(preprocess)
}
# 5. t : kernel bandwidth
t = as.double(t)
if (!check_NumMM(t,.Machine$double.eps,Inf)){stop("* do.iltsa : 't' should be in (0,Inf).")}
#------------------------------------------------------------------------
## COMPUTATION : PRELIMINARY
# 1. preprocess of data
tmplist = (X,type=algpreprocess,algtype="nonlinear")
trfinfo = tmplist$info
pX = tmplist$pX
# 2. build neighborhood information
nbdstruct = aux.graphnbd(pX,method="euclidean",
type=nbdtype,symmetric=nbdsymmetric)
nbdmask = nbdstruct$mask
#------------------------------------------------------------------------
## COMPUTATION : IMPROVED LTSA IN A DESCRIPTIVE ORDER
# 1. main outer iterations
C = array(0,c(n,n))
for (i in 1:n){
# 1-1. select the index
Ii = which(nbdmask[i,])
ki = length(Ii)
if (ki <= 1){
stop("* do.iltsa : select the larger neighborhood.")
}
# 1-2. compute Xi and wi
tgtvec = as.vector(pX[i,])
tgtmat = pX[Ii,]
Xitilde = iltsa_Xtilde(tgtvec, tgtmat)
vecWi = iltsa_Wi(tgtvec, tgtmat, t)
Wi = diag(vecWi)
# 1-3. compute orthonormal columns
XWi = (Xitilde%*%Wi)
XWtXWi = (t(XWi)%*%XWi)
eigXWi = RSpectra::eigs(XWtXWi, ndim)
matEi = aux.adjprojection(eigXWi$vectors)
vecdi = sqrt(eigXWi$values)
# 1-4. Ti, Vi, and Ci
Ti = (diag(vecdi)%*%t(matEi)%*%diag(1/vecWi))
invTi = aux.pinv(Ti)
Vi = (diag(ki)-(invTi%*%Ti))
Ci = (Vi%*%t(Vi))
# 1-5. update scheme
ek = matrix(rep(1,ki),nrow=ki)
C[Ii,Ii] = C[Ii,Ii]+Ci
C[Ii,i] = C[Ii,i]-as.vector(Ci%*%ek)
C[i,Ii] = C[i,Ii]-as.vector(t(ek)%*%Ci)
C[i,i] = C[i,i]+ as.double(t(ek)%*%Ci%*%ek)
}
# 2. compute (ndim+1) lowest eigenvectors
if (ndim==(p-1)){
Youtput = aux.adjprojection(base::eigen(C)$vectors[,p:2])
} else {
Youtput = aux.adjprojection(RSpectra::eigs(C, (ndim+1), which="SR")$vectors[,(ndim+1):2])
}
#------------------------------------------------------------------------
## RETURN OUTPUT
result = list()
result$Y = Youtput
result$trfinfo = trfinfo
return(result)
}
# . -----------------------------------------------------------------------
#' @keywords internal
#' @noRd
iltsa_Xtilde <- function(vec, mat){
k = nrow(mat)
n = ncol(mat)
X = array(0,c(n,k))
vec1 = as.vector(vec)
for (i in 1:k){
vec2 = as.vector(mat[i,])
X[,i] = vec2-vec1
}
return(X)
}
#' @keywords internal
#' @noRd
iltsa_Wi <- function(vec, mat, t){
k = nrow(mat)
n = ncol(mat)
output = rep(0,k)
for (i in 1:k){
vecdiff = as.vector(vec)-as.vector(mat[i,])
output[i] = exp(-sum(vecdiff*vecdiff)/t)
}
return(output)
}
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