#' Linear Quadratic Mutual Information
#'
#' Linear Quadratic Mutual Information (LQMI) is a supervised linear dimension reduction method.
#' Quadratic Mutual Information is an efficient nonparametric estimation method for Mutual Information
#' for class labels not requiring class priors. For the KQMI formulation, LQMI is a linear equivalent.
#'
#' @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.
#'
#' @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])
#'
#' ## compare against LDA
#' out1 = do.lda(X, label)
#' out2 = do.lqmi(X, label)
#'
#' ## visualize
#' opar <- par(no.readonly=TRUE)
#' par(mfrow=c(1,2))
#' plot(out1$Y, col=label, main="LDA projection")
#' plot(out2$Y, col=label, main="LQMI projection")
#' par(opar)
#'
#' @references
#' \insertRef{bouzas_graph_2015}{Rdimtools}
#'
#' @seealso \code{\link{do.kqmi}}
#' @author Kisung You
#' @rdname linear_LQMI
#' @concept linear_methods
#' @export
do.lqmi <- function(X, label, ndim=2, preprocess=c("center","scale","cscale","whiten","decorrelate")){
#------------------------------------------------------------------------
## PREPROCESSING
# 1. data matrix
aux.typecheck(X)
n = nrow(X)
p = ncol(X)
# 2. label information
label = as.numeric(as.factor(label))
ulabel = unique(label)
nlabel = length(ulabel)
if ((ndim+1)!=nlabel){
message("* do.lqmi : the method is intended for case when ndim=(C-1).")
}
if (nlabel==1){
stop("* do.lqmi : 'label' should have at least 2 unique labelings.")
}
if (nlabel==n){
stop("* do.lqmi : 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.lqmi : 'ndim' is a positive integer in [1,#(covariates)].")
}
# 4. preprocess
if (missing(preprocess)){
algpreprocess = "center"
} else {
algpreprocess = match.arg(preprocess)
}
#------------------------------------------------------------------------
## COMPUTATION : PRELIMINARY
tmplist = (X,type=algpreprocess,algtype="linear")
trfinfo = tmplist$info
pX = tmplist$pX
#------------------------------------------------------------------------
## COMPUTATION : MAIN STEPS FOR LQMI
# 1. compute M and symmetrize
M = qmi_M(pX, label)
symM = ((M+t(M))/2)
# 2. eigendecomposition for achieving W
termA = t(pX)%*%pX
termB = t(pX)%*%symM%*%pX
linsol = aux.bicgstab(termA, termB, verbose=FALSE)$x
W = base::eigen(linsol)$vectors
# 3. compute projection
projection = qr.Q(qr(W))[,1:ndim]
#------------------------------------------------------------------------
## RETURN
result = list()
result$Y = pX%*%projection
result$trfinfo = trfinfo
result$projection = projection
return(result)
}
# auxiliary for LQMI and KQMI ---------------------------------------------
#' @keywords internal
#' @noRd
qmi_M <- function(X, label){
n = nrow(X)
ulabel = unique(label)
C = length(ulabel)
onesN = rep(1,n)
# constants
C_all = 0
C_btw = rep(0,C)
for (i in 1:C){
idxlength = as.double(length(which(label==ulabel[i])))
C_all = C_all + (idxlength^2)
C_btw[i] = (idxlength/(n^3))
}
C_all = C_all/(n^4)
C_in = (1/(n^2))
# matrix
M = C_all*outer(onesN,onesN)
for (i in 1:C){
onesC = (as.vector((label==ulabel[i])*1.0))
M = M + (C_in*outer(onesC,onesC)) -(2*outer(onesN,as.double(C_btw[i])*onesC))
}
return(M)
}
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