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#' Kernel-Weighted Maximum Variance Projection
#'
#' Kernel-Weighted Maximum Variance Projection (KMVP) is a generalization of
#' Maximum Variance Projection (MVP). Even though its name contains \emph{kernel}, it is
#' not related to kernel trick well known in the machine learning community. Rather, it
#' generalizes the binary penalization on class discrepancy,
#' \deqn{S_{ij} = \exp(-\|x_i-x_j\|^2/t) \quad\textrm{if}\quad C_i \ne C_j}
#' where \eqn{x_i} is an \eqn{i}-th data point and \eqn{t} a kernel bandwidth (\code{bandwidth}). \bold{Note} that
#' when the bandwidth value is too small, it might suffer from numerical instability and rank deficiency due to its formulation.
#'
#' @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 preprocess an additional option for preprocessing the data.
#' Default is "center". See also \code{\link{aux.preprocess}} for more details.
#' @param bandwidth bandwidth parameter for heat kernel as the equation above.
#'
#' @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
#' ## use 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])
#'
#' ## perform KMVP with different bandwidths
#' out1 = do.kmvp(X, label, bandwidth=0.1)
#' out2 = do.kmvp(X, label, bandwidth=1)
#' out3 = do.kmvp(X, label, bandwidth=10)
#'
#' ## visualize
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,3))
#' plot(out1$Y, main="bandwidth=0.1", col=label, pch=19)
#' plot(out2$Y, main="bandwidth=1", col=label, pch=19)
#' plot(out3$Y, main="bandwidth=10", col=label, pch=19)
#' par(opar)
#'
#' @references
#' \insertRef{zhang_maximum_2007}{Rdimtools}
#'
#' @seealso \code{\link{do.mvp}}
#' @author Kisung You
#' @rdname linear_KMVP
#' @concept linear_methods
#' @export
do.kmvp <- function(X, label, ndim=2,
preprocess=c("center","scale","cscale","decorrelate","whiten"),
bandwidth=1.0){
#------------------------------------------------------------------------
## PREPROCESSING
# 1. data matrix
aux.typecheck(X)
n = nrow(X)
p = ncol(X)
# 2. label : check and return a de-factored vector
# For this example, there should be no degenerate class of size 1.
label = check_label(label, n)
ulabel = unique(label)
for (i in 1:length(ulabel)){
if (sum(label==ulabel[i])==1){
stop("* do.kmvp : no degerate class of size 1 is allowed.")
}
}
N = length(ulabel)
if (any(is.na(label))||(any(is.infinite(label)))){
stop("* Supervised Learning : any element of 'label' as NA or Inf will simply be considered as a class, not missing entries.")
}
# 3. ndim
ndim = as.integer(ndim)
if (!check_ndim(ndim,p)){stop("* do.kmvp : 'ndim' is a positive integer in [1,#(covariates)).")}
if (ndim>=N){
stop("* do.kmvp : the method requires {ndim <= N-1}, where N is the number of classes.")
}
# 4. preprocess
if (missing(preprocess)){
algpreprocess = "center"
} else {
algpreprocess = match.arg(preprocess)
}
# 5. bandwidth
bandwidth = as.double(bandwidth)
if (!check_NumMM(bandwidth,0,1e+10,compact=TRUE)){stop("* do.kmvp : 'bandwidth' should be a nonnegative real number.")}
#------------------------------------------------------------------------
## MAIN COMPUTATION
# 1. preprocess of data
tmplist = (X,type=algpreprocess,algtype="linear")
trfinfo = tmplist$info
pX = tmplist$pX
# 2. perform PCA onto (N-1) dimensional space
outPCA = do.pca(pX,ndim=(N-1))
projection_first = aux.adjprojection(outPCA$projection)
ppX = outPCA$Y
# 3. Start of MVP algorithm here.
# 3-1. Laplacian Part
S = kmvp_S(ppX, label, bandwidth) # S_ij = 1 if Ci != Cj
D = diag(rowSums(S))
L = D-S
# 3-2. LLE approximation part # we need to construct W !!
W = array(0,c(n,n))
for (i in 1:n){
# 1. the data
dataw = ppX[i,]
# 2. target index and corresponding data
# note that the index 'i' should be removed
tgtidx = setdiff(which(label==label[i]),i)
tgtdata = ppX[tgtidx,]
# 3. compute w and assign
W[i,tgtidx] = kmvp_Gvec(dataw, tgtdata)
}
# 3-3. compute M
Mhalf = diag(n)-W
M = (t(Mhalf)%*%Mhalf)
# 4. solve geigen
LHS = t(ppX)%*%M%*%ppX
RHS = t(ppX)%*%L%*%ppX
projection_second = aux.geigen(LHS,RHS,ndim,maximal=FALSE)
projection_all = projection_first%*%projection_second
#------------------------------------------------------------------------
## RETURN THE RESULTS
result = list()
result$Y = pX%*%projection_all
result$trfinfo = trfinfo
result$projection = projection_all
return(result)
}
# ------------------------------------------------------------------------
#' @keywords internal
#' @noRd
kmvp_S <- function(X, label, bd){
n = nrow(X)
S = array(0,c(n,n))
for (i in 1:(n-1)){
for (j in (i+1):n){
if (label[i]!=label[j]){
diffvec = as.vector(X[i,]-X[j,])
thevalue = exp(-sum(diffvec*diffvec)/bd)
S[i,j] = thevalue
S[j,i] = thevalue
}
}
}
return(S)
}
#' @keywords internal
#' @noRd
kmvp_Gvec <- function(vec, mat){
# 1. settings
n = nrow(mat)
G = array(0,c(n,n))
# 2. iterate to construct
for (i in 1:n){
veci = mat[i,]
for (j in i:n){
vecj = mat[j,]
diffi = vec-veci
diffj = vec-vecj
if (i==j){
G[i,i] = sum(diffi*diffj)
} else {
valueout = sum(diffi*diffj)
G[i,j] = valueout
G[j,i] = valueout
}
}
}
# 3. compute the vector
Ginv = 1/G
if (any(is.infinite(Ginv))){
Ginv[which(is.infinite(Ginv))] = 0
}
denominator = sum(Ginv)
woutput = (rowSums(Ginv))/(sum(Ginv))
return(woutput)
}
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