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#' Iterative EM PCA imputation
#'
#' Greedy algorithm for EM-PCA including robust methods
#'
#' @param x data.frame or matrix
#' @param method `"classical"` or `"mcd"` (robust estimation)
#' @param eps threshold for convergence
#' @param m number of multiple imputations (only if parameter `boot` equals `TRUE`)
#' @param k number of principal components for reconstruction of `x`
#' @param maxit maximum number of iterations
#' @param boot residual bootstrap (if `TRUE`)
#' @param verbose TRUE/FALSE if additional information about the imputation
#' process should be printed
#' @return the imputed data set. If `boot = FALSE` this is a data.frame.
#' If `boot = TRUE` this is a list where each list element contains a data.frame.
#' @author Matthias Templ
#' @references Serneels, Sven and Verdonck, Tim (2008).
#' Principal component analysis for data containing outliers and missing elements.
#' Computational Statistics and Data Analysis, Elsevier, vol. 52(3), pages 1712-1727
#' @keywords manip
#' @family imputation methods
#' @examples
#'
#' data(Animals, package = "MASS")
#' Animals$brain[19] <- Animals$brain[19] + 0.01
#' Animals <- log(Animals)
#' colnames(Animals) <- c("log(body)", "log(brain)")
#' Animals_na <- Animals
#' probs <- abs(Animals$`log(body)`^2)
#' probs <- rep(0.5, nrow(Animals))
#' probs[c(6,16,26)] <- 0
#' set.seed(1234)
#' Animals_na[sample(1:nrow(Animals), 10, prob = probs), "log(brain)"] <- NA
#' w <- is.na(Animals_na$`log(brain)`)
#' impPCA(Animals_na)
#' impPCA(Animals_na, method = "mcd")
#' impPCA(Animals_na, boot = TRUE, m = 10)
#' impPCA(Animals_na, method = "mcd", boot = TRUE)[[1]]
#' plot(`log(brain)` ~ `log(body)`, data = Animals, type = "n", ylab = "", xlab="")
#' mtext(text = "impPCA robust", side = 3)
#' points(Animals$`log(body)`[!w], Animals$`log(brain)`[!w])
#' points(Animals$`log(body)`[w], Animals$`log(brain)`[w], col = "grey", pch = 17)
#' imputed <- impPCA(Animals_na, method = "mcd", boot = TRUE)[[1]]
#' colnames(imputed) <- c("log(body)", "log(brain)")
#' points(imputed$`log(body)`[w], imputed$`log(brain)`[w], col = "red", pch = 20, cex = 1.4)
#' segments(x0 = Animals$`log(body)`[w], x1 = imputed$`log(body)`[w], y0 = Animals$`log(brain)`[w],
#' y1 = imputed$`log(brain)`[w], lty = 2, col = "grey")
#' legend("topleft", legend = c("non-missings", "set to missing", "imputed values"),
#' pch = c(1,17,20), col = c("black","grey","red"), cex = 0.7)
#' mape <- round(100* 1/sum(is.na(Animals_na$`log(brain)`)) * sum(abs((Animals$`log(brain)` -
#' imputed$`log(brain)`) / Animals$`log(brain)`)), 2)
#' s2 <- var(Animals$`log(brain)`)
#' nrmse <- round(sqrt(1/sum(is.na(Animals_na$`log(brain)`)) * sum(abs((Animals$`log(brain)` -
#' imputed$`log(brain)`) / s2))), 2)
#' text(x = 8, y = 1.5, labels = paste("MAPE =", mape))
#' text(x = 8, y = 0.5, labels = paste("NRMSE =", nrmse))
#'
#' @export
impPCA <- function(x, method = "classical", m = 1, eps = 0.5,
k = ncol(x) - 1, maxit = 100, boot = FALSE,
verbose = TRUE){
### x ... Matrix or data.frame with missings
### method ... mcd, classical, gridMAD
### eps ... convergence criteria
### all.obs ... TRUE, if the whole observation should be replaced with the pca-estimate
###
### Matthias Templ
### original from 10.09.2006
### adapted 31.10.2006
### Imputation nach Sven
### Statistics Austria
# classx <- class(x)
classes <- lapply(x, class)
if(!all(classes %in% c("numeric", "integer"))) stop("only numeric variables allowed")
xorig <- x
indexMiss <- is.na(x)
#P <- dim(x)[2]
## erste Schaetzung mit Median:
#w <- which(is.na(x), arr.ind=TRUE)
# cm <- colMeans(x, na.rm=TRUE) ## fuers Ruecktransf.
# cmed <- apply(x, 2, median, na.rm = TRUE)
# csd <- apply(x, 2, sd, na.rm = TRUE) ## fuers Ruecktransf.
# csdrob <- apply(x, 2, mad, na.rm = TRUE) ## fuers Ruecktransf.
# xorig <- x
# x <- apply(x, 2, function(x) (x - mean(x, na.rm=TRUE))/sd(x, na.rm=TRUE))
## centering and initialisierung
# if(method == "classical"){
cm <- colMeans(x, na.rm=TRUE) ## fuers Ruecktransf.
csd <- apply(x, 2, sd, na.rm = TRUE) ## fuers Ruecktransf.
x <- apply(x, 2, function(x) (x - mean(x, na.rm=TRUE))/sd(x, na.rm=TRUE))
# initialise missing values
for(j in 1:ncol(x)){
x[indexMiss[,j], j] <- mean(x[, j], na.rm = TRUE)
}
# } else if(method == "mcd"){
# cmed <- apply(x, 2, median, na.rm = TRUE)
# csdrob <- apply(x, 2, mad, na.rm = TRUE) ## fuers Ruecktransf.
# x <- apply(x, 2, function(x) (x - median(na.omit(x))/mad(na.omit(x))))
# # initialise missing values
# for(j in 1:ncol(x)){
# x[indexMiss[,j], j] <- median(na.omit(x[, j]))
# }
# }
### PCA, Iteration:
d <- 1000000
it <- 0
if(!boot){
while(d > eps & it <= maxit){
it <- it + 1
if( method == "mcd" ){
xMcd <- robustbase::covMcd(x)
p <- princomp(x, covmat=xMcd)
}
if( method == "classical" ){
p <- princomp(x)
}
#if( method == "gridMAD" ){ p <- PCAgrid(x, method="mad", k=ncol(x)) }
xneu <- p$sco[,1:k] %*% t(p$load[,1:k]) #(p$load[,1:P])
##+rep(1,dim(x)[1])%*%t(xMcd$center) # p-dim???
d <- sum(abs(x[indexMiss] - xneu[indexMiss])) ## Konvergenzkriterium
x[indexMiss] <- xneu[indexMiss]
}
### Ruecktrans:
# if(method == "classical"){
for( i in 1: dim(x)[2] ){
x[,i] <- (x[,i] * csd[i]) + cm[i]
}
# } else {
# for( i in 1: dim(x)[2] ){
# x[,i] <- (x[,i] * csdrob[i]) + cmed[i]
# }
# }
if(verbose) message("\nIterations: ", it)
return(data.frame(x))
} else { # boot
# create bootstrap samples
if( method == "mcd" ){
xMcd <- robustbase::covMcd(x)
p <- princomp(x, covmat=xMcd)
}
if( method == "classical" ){
p <- princomp(x)
}
xneu <- p$sco[,1:k] %*% t(p$load[,1:k])
residuals <- x - xneu
xboot <- x
residualboot <- function(){
for(j in 1:ncol(x)){
xboot[, j] <- x[, j] + sample(residuals[,j], replace = TRUE)
}
### PCA, Iteration:
d <- 1000000
it <- 0
while(d > eps & it <= maxit){
it <- it + 1
if( method == "mcd" ){
xMcd <- robustbase::covMcd(xboot)
p <- princomp(xboot, covmat=xMcd)
}
if( method == "classical" ){
p <- princomp(xboot)
}
#if( method == "gridMAD" ){ p <- PCAgrid(x, method="mad", k=ncol(x)) }
xneu <- p$sco[,1:k] %*% t(p$load[,1:k])
d <- sum(abs(xboot[indexMiss] - xneu[indexMiss])) ## Konvergenzkriterium
xboot[indexMiss] <- xneu[indexMiss]
}
### Ruecktrans:
# if(method == "classical"){
for( i in 1: dim(x)[2] ){
xboot[,i] <- (xboot[,i] * csd[i]) + cm[i]
}
# } else {
# for( i in 1: dim(x)[2] ){
# xboot[,i] <- (xboot[,i] * csdrob[i]) + cmed[i]
# }
# }
#
xorig[indexMiss] <- xboot[indexMiss]
return(xorig)
}
res <- list()
for(r in 1:m){
res[[r]] <- residualboot()
}
if(verbose) message("\nIterations: ", it)
return(res)
}
}
# impPCA_boot <- function(x, method = "classical", m = 10 , eps = 0.5,
# k = ncol(x) - 1, maxit = 10){
# ### x ... Matrix or data.frame with missings
# ### method ... mcd, classical, gridMAD
# ### eps ... convergence criteria
# ### all.obs ... TRUE, if the whole observation should be replaced with the pca-estimate
# ###
# ### Matthias Templ
# ### original from 10.09.2006
# ### adapted 31.10.2006
# ### Imputation nach Sven
# ### Statistics Austria
#
# # classx <- class(x)
# classes <- lapply(x, class)
# if(!all(classes %in% c("numeric", "integer"))) stop("only numeric variables allowed")
#
# indexMiss <- is.na(x)
#
# #P <- dim(x)[2]
#
# ## erste Schaetzung mit Median:
# #w <- which(is.na(x), arr.ind=TRUE)
# cm <- colMeans(x, na.rm=TRUE) ## fuers Ruecktransf.
# cmed <- apply(x, 2, median, na.rm = TRUE)
# csd <- apply(x, 2, sd, na.rm = TRUE) ## fuers Ruecktransf.
# xorig <- x
# x <- apply(x, 2, function(x) (x - mean(x, na.rm=TRUE))/sd(x, na.rm=TRUE))
#
# ## Initialisierung
# if(method == "classical"){
# for(j in 1:ncol(x)){
# x[indexMiss[,j], j] <- cm[j]
# }
# } else if(method == "mcd"){
# for(j in 1:ncol(x)){
# x[indexMiss[,j], j] <- cmmed[j]
# }
# }
#
# # create bootstrap samples
#
# if( method == "mcd" ){
# xMcd <- robustbase::covMcd(x)
# p <- princomp(x, covmat=xMcd)
# }
# if( method == "classical" ){
# p <- princomp(x)
# }
#
# xneu <- p$sco[,1:k] %*% t(p$load[,1:k])
# residuals <- x - xneu
# xboot <- x
# residualboot <- function(){
# for(j in 1:ncol(x)){
# xboot[, j] <- x[, j] + sample(residuals[,j], replace = TRUE)
# }
#
# ### PCA, Iteration:
# d <- 1000000
# it <- 0
#
# while(d > eps & it <= maxit){
# it <- it + 1
# if( method == "mcd" ){
# xMcd <- robustbase::covMcd(xboot)
# p <- princomp(xboot, covmat=xMcd)
# }
# if( method == "classical" ){
# p <- princomp(xboot)
# }
# #if( method == "gridMAD" ){ p <- PCAgrid(x, method="mad", k=ncol(x)) }
# xneu <- p$sco[,1:k] %*% t(p$load[,1:k])
# d <- sum(abs(xboot[indexMiss] - xneu[indexMiss])) ## Konvergenzkriterium
# xboot[indexMiss] <- xneu[indexMiss]
# }
#
# ### Ruecktrans:
#
# for( i in 1: dim(x)[2] ){
# xboot[,i] <- (xboot[,i] * csd[i]) + cm[i]
# }
# xorig[indexMiss] <- xboot[indexMiss]
# return(xorig)
# }
#
# res <- list()
# for(r in 1:m){
# res[[r]] <- residualboot()
# }
#
# return(res)
# }
#
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