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#' Locality Preserving Fisher Discriminant Analysis
#'
#' Locality Preserving Fisher Discriminant Analysis (LPFDA) is a supervised variant of LPP.
#' It can also be seemed as an improved version of LDA where the locality structure of the data
#' is preserved. The algorithm aims at getting a subspace projection matrix by solving a generalized
#' eigenvalue problem.
#'
#' @param X an \eqn{(n\times p)} matrix or data frame whose rows are observations
#' and columns represent independent variables.
#' @param label a length-\eqn{n} vector of data class labels.
#' @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 preprocess an additional option for preprocessing the data.
#' Default is "center". See also \code{\link{aux.preprocess}} for more details.
#' @param t bandwidth parameter for heat kernel 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.}
#' \item{projection}{a \eqn{(p\times ndim)} whose columns are basis for projection.}
#' }
#'
#' @examples
#' ## generate data of 3 types with clear difference
#' set.seed(100)
#' dt1 = aux.gensamples(n=20)-50
#' dt2 = aux.gensamples(n=20)
#' dt3 = aux.gensamples(n=20)+50
#'
#' ## merge the data and create a label correspondingly
#' X = rbind(dt1,dt2,dt3)
#' label = rep(1:3, each=20)
#'
#' ## try different proportion of connected edges
#' out1 = do.lpfda(X, label, type=c("proportion",0.10))
#' out2 = do.lpfda(X, label, type=c("proportion",0.25))
#' out3 = do.lpfda(X, label, type=c("proportion",0.50))
#'
#' ## visualize
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(out1$Y, pch=19, col=label, main="10% connectivity")
#' plot(out2$Y, pch=19, col=label, main="25% connectivity")
#' plot(out3$Y, pch=19, col=label, main="50% connectivity")
#' par(opar)
#'
#' @references
#' \insertRef{zhao_locality_2009}{Rdimtools}
#'
#' @rdname linear_LPFDA
#' @author Kisung You
#' @concept linear_methods
#' @export
do.lpfda <- function(X, label, ndim=2, type=c("proportion",0.1),
preprocess=c("center","scale","cscale","whiten","decorrelate"), t=10.0){
#------------------------------------------------------------------------
## PREPROCESSING
# 1. data matrix
aux.typecheck(X)
n = nrow(X)
p = ncol(X)
# 2. label
label = as.numeric(as.factor(label))
ulabel = unique(label)
K = length(ulabel)
if (K==1){
stop("* do.lpfda : 'label' should have at least 2 unique labelings.")
}
if (K==n){
stop("* do.lpfda : given 'label' has all unique elements.")
}
if (any(is.na(label))||(any(is.infinite(label)))){
stop("* Supervised Learning : any element of 'label' as NA or Inf will simply be considered as a class, not missing entries.")
}
# 3. ndim
ndim = as.integer(ndim)
if (!check_ndim(ndim,p)){stop("* do.lpfda : 'ndim' is a positive integer in [1,#(covariates)).")}
# 4. type
nbdtype = type
nbdsymmetric = "union"
# 5. preprocess
if (missing(preprocess)){
algpreprocess = "center"
} else {
algpreprocess = match.arg(preprocess)
}
# 6. t : kernel bandwidth
t = as.double(t)
if (!check_NumMM(t, .Machine$double.eps*10, Inf)){stop("* do.lpfda : 't' is a positive kernel bandwidth parameter.")}
#------------------------------------------------------------------------
## COMPUTATION : PRELIMINARY
# 1. preprocessing
tmplist = (X,type=algpreprocess,algtype="linear")
trfinfo = tmplist$info
pX = tmplist$pX
# 2. process : neighborhood selection
nbdstruct = aux.graphnbd(pX,method="euclidean",
type=nbdtype,symmetric=nbdsymmetric)
nbdmask = nbdstruct$mask
#------------------------------------------------------------------------
## COMPUTATION : MAIN PART FOR LPFDA
# 1. LPP part : compute weight matrix W and other auxiliary's
Dsqmat = exp(-(as.matrix(dist(pX))^2)/t)
W = Dsqmat*nbdmask
L = diag(rowSums(W))-W
# 2. FDA part : compute two scatter matrices
# 2-1. compute S_W : within-group variance for multiclss problem
Sw = array(0,c(p,p))
for (i in 1:K){
idxnow = which(label==ulabel[i])
Sw = Sw + lda_outer(pX[idxnow,])
}
# 2-2. compute S_B : between-group variance for multiclass problem
Sb = array(0,c(p,p))
m = colMeans(pX)
for (i in 1:K){
idxnow = which(label==ulabel[i])
Nk = length(idxnow)
mdiff = (colMeans(pX[idxnow,])-m)
Sb = Sb + Nk*outer(mdiff,mdiff)
}
# 3. cost function for geigen : use Maximal
costL = Sb - t(pX)%*%L%*%pX
projection = aux.geigen(costL, Sw, ndim, maximal=TRUE)
#------------------------------------------------------------------------
## RETURN
result = list()
result$Y = pX%*%projection
result$trfinfo = trfinfo
result$projection = projection
return(result)
}
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