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#' @title Flury's Common Principal Component Analysis
#'
#' @description
#' Common principal component Analysis
#'
#' @param Data a numeric matrix or data frame
#' @param Group a vector of factors associated with group structure
#' @param Scale scaling variables, by default is False. By default data are centered within groups.
#' @param graph should loading and component be plotted
#' @return list with the following results:
#' @return \item{Data}{Original data}
#' @return \item{Con.Data}{Concatenated centered data}
#' @return \item{split.Data}{Group centered data}
#' @return \item{Group}{Group as a factor vector}
#' @return \item{loadings.common}{Matrix of common loadings}
#' @return \item{lambda}{The specific variances of group}
#' @return \item{exp.var}{Percentages of total variance recovered associated with each dimension }
#' @seealso \code{\link{mgPCA}}, \code{\link{DGPA}}, \code{\link{DCCSWA}}, \code{\link{DSTATIS}}, \code{\link{BGC}}, \code{\link{summarize}}, \code{\link{TBWvariance}}, \code{\link{loadingsplot}}, \code{\link{scoreplot}}, \code{\link{iris}}
#' @export
#' @references B. N. Flury (1984). Common principal components in k groups.
#' \emph{Journal of the American Statistical Association}, 79, 892-898.
#'
#' A. Eslami, E. M. Qannari, A. Kohler and S. Bougeard (2013). General overview
#' of methods of analysis of multi-group datasets,
#' \emph{Revue des Nouvelles Technologies de l'Information}, 25, 108-123.
#'
#'
#' @examples
#' Data = iris[,-5]
#' Group = iris[,5]
#' res.FCPCA = FCPCA(Data, Group, graph=TRUE)
#' loadingsplot(res.FCPCA, axes=c(1,2))
#' scoreplot(res.FCPCA, axes=c(1,2))
FCPCA <-function(Data, Group, Scale=FALSE, graph=FALSE){
#============================================================================
# 1. Checking the inputs
#============================================================================
check(Data, Group)
#============================================================================
# 2. preparing Data
#============================================================================
if (is.data.frame(Data) == TRUE) {
Data=as.matrix(Data)
}
if(is.null(colnames(Data))) {
colnames(Data) = paste('V', 1:ncol(Data), sep='')
}
Group = as.factor(Group)
rownames(Data) = Group #---- rownames of data=groups
M = length(levels(Group)) #----number of groups: M
P = dim(Data)[2] #----number of variables: P
n = as.vector(table(Group)) #----number of individuals in each group
N = sum(n) #----number of individuals
split.Data = split(Data,Group) #----split Data to M parts
# centering and scaling if TRUE
for(m in 1:M){
split.Data[[m]] = matrix(split.Data[[m]], nrow=n[m])
split.Data[[m]] = scale(split.Data[[m]], center=TRUE, scale=Scale)
}
# concatinated dataset by row as groups
Con.Data = split.Data[[1]]
for(m in 2:M) {
Con.Data = rbind(Con.Data, split.Data[[m]])
}
rownames(Con.Data) = Group
colnames(Con.Data) = colnames(Data)
# Variance-covariance matrix for each group
cov.Group = vector("list", M)
for(m in 1:M){
cov.Group[[m]] = t(split.Data[[m]]) %*% split.Data[[m]] / n[m]
}
#============================================================================
# 3. FG algorithm
#============================================================================
FG <- function(cov.Group,L,P,M){
#------ F step
k = M
K = M
B = diag(P)
Bold = diag(P)
C = cov.Group
T = vector("list",k)
d1 = c(rep(0,k))
d2 = c(rep(0,k))
for(l in 1:L){
for(p in 1:(P-1)){
for(e in (p+1):P){
Q=diag(2)
#------ G step
M=diag(2)
for(k in 1:K){
H=B[,c(p,e)]
T[[k]]=t(H)%*%C[[k]]%*%H
d1[k]=t(Q[,1])%*%T[[k]]%*%Q[,1]
d2[k]=t(Q[,2])%*%T[[k]]%*%Q[,2]
M = M + (d1[k]-d2[k])/(d1[k]*d2[k])*T[[k]]
}
eig <- eigen(M)
Q=eig$vectors
B[,c(p,e)]=H%*%Q
}
}
}
W=B
res=W
return(W)
}
W =FG(cov.Group,15,P,M)
#============================================================================
# 4. Explained variance
# variance of each loading
# lambda = t(common loading)*(t(Xm)* Xm) * common loading
#============================================================================
lambda = matrix(0, nrow=M, ncol=P)
for(m in 1:M){
lambda[m,] = round(diag(t(W) %*% cov.Group[[m]] %*% W),3)
}
#============================================================================
# 5. Outputs
#============================================================================
res <- list(
Data = Data,
Con.Data = Con.Data,
split.Data = split.Data,
Group=Group)
res$loadings.common = W
rownames(res$loadings.common) = colnames(Data)
colnames(res$loadings.common) = paste("Dim", 1:P, sep="")
res$lambda = lambda
rownames(res$lambda) = levels(Group)
colnames(res$lambda) = paste("Dim", 1:P, sep="")
ncomp = ncol(res$lambda)
exp.var = matrix(0,M,ncomp)
for(m in 1:M){
exp.var[m,] = 100 * lambda[m,]/ sum(diag(cov.Group[[m]]))
}
res$exp.var = exp.var
rownames(res$exp.var) = levels(Group)
colnames(res$exp.var) = paste("Dim", 1:ncomp, sep="")
if(graph) {plot.mg(res)}
class(res) = c("FCPCA", "mg")
return(res)
}
#' @S3method print FCPCA
print.FCPCA <- function(x, ...)
{
cat("\nCommon Principal Component Analysis\n")
cat(rep("-",43), sep="")
cat("\n$lambda ", "variance for each group")
cat("\n$loadings.common ", "common loadings")
cat("\n$Data ", "Data set")
cat("\n")
invisible(x)
}
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