#' Robust principal component analysis for compositional data
#' 
#' This function applies robust principal component analysis for compositional
#' data.
#' 
#' The compositional data set is expressed in isometric logratio coordinates.
#' Afterwards, robust principal component analysis is performed.  Resulting
#' loadings and scores are back-transformed to the clr space where the
#' compositional biplot can be shown.
#' 
#' \code{mult_comp} is used when there are more than one group of compositional
#' parts in the data. To give an illustrative example, lets assume that one
#' variable group measures angles of the inner ear-bones of animals which sum
#' up to 100 and another one having percentages of a whole on the thickness of
#' the inner ear-bones included. Then two groups of variables exists which are
#' both compositional parts. The isometric logratio coordinates are then internally applied
#' to each group independently whenever the \code{mult_comp} is set correctly.
#' 
#' @aliases pcaCoDa print.pcaCoDa 
#' @param x compositional data
#' @param method must be either \dQuote{robust} (default) or \dQuote{classical}
#' @param mult_comp a list of numeric vectors holding the indices of linked
#' compositions
#' @param external external non-compositional variables
#' @return \item{scores }{scores in clr space} \item{loadings }{loadings in clr
#' space} \item{eigenvalues }{eigenvalues of the clr covariance matrix}
#' \item{method }{method} \item{princompOutputClr }{output of \code{princomp}
#' needed in \code{plot.pcaCoDa}}
#' @author Karel Hron, Peter Filzmoser, Matthias Templ and a contribution for dimnames in external variables by Amelia Landre.
#' @seealso \code{\link{print.pcaCoDa}}, \code{\link{summary.pcaCoDa}}, \code{\link{biplot.pcaCoDa}}, \code{\link{plot.pcaCoDa}}
#' @importFrom stats princomp
#' @references Filzmoser, P., Hron, K., Reimann, C. (2009) Principal component
#' analysis for compositional data with outliers. \emph{Environmetrics},
#' \bold{20}, 621-632.
#' 
#' Kynclova, P., Filzmoser, P., Hron, K. (2016) Compositional biplots including external non-compositional variables. 
#' \emph{Statistics: A Journal of Theoretical and Applied Statistics},
#' \bold{50}, 1132-1148.
#' @keywords multivariate
#' @export
#' @importFrom MASS cov.mve
#' @examples
#' 
#' data(arcticLake)
#' 
#' ## robust estimation (default):
#' res.rob <- pcaCoDa(arcticLake)
#' res.rob
#' summary(res.rob)
#' plot(res.rob)
#' 
#' ## classical estimation:
#' res.cla <- pcaCoDa(arcticLake, method="classical")
#' biplot(res.cla)
#' 
#' ## just for illustration how to set the mult_comp argument:
#' data(expenditures)
#' p1 <- pcaCoDa(expenditures, mult_comp=list(c(1,2,3),c(4,5)))
#' p1
#' 
#' ## example with external variables:
#' data(election)
#' # transform external variables
#' election$unemployment <- log((election$unemployment/100)/(1-election$unemployment/100))
#' election$income <- scale(election$income)
#' 
#' res <- pcaCoDa(election[,1:6], method="classical", external=election[,7:8])
#' res
#' biplot(res, scale=0)

pcaCoDa <- function (x, method = "robust", mult_comp = NULL, external = NULL) 
{
  if (is.vector(external) & length(external) != nrow(x)) {
    stop("external and x must have the same number of observations")
  }
  if (!is.null(mult_comp) & !is.list(mult_comp)) 
    stop("if specified, mult_comp must be a list")
  ilrV <- function(x) {
    x.ilr = matrix(NA, nrow = nrow(x), ncol = ncol(x) - 1)
    for (i in 1:ncol(x.ilr)) {
      x.ilr[, i] = sqrt((i)/(i + 1)) * log(((apply(as.matrix(x[, 
                                                               1:i]), 1, prod))^(1/i))/(x[, i + 1]))
    }
    return(x.ilr)
  }
  if (is.null(mult_comp)) {
    xilr <- ilrV(x)
  }
  else {
    xilr <- do.call("cbind", lapply(mult_comp, function(xx) ilrV(x[, 
                                                                   xx])))
  }
  if (!is.null(external)) {
    xilr <- cbind(xilr, external)
  }
  if (method == "robust") {
    cv <- robustbase::covMcd(xilr, cor = FALSE)
    pcaIlr <- suppressWarnings(princomp(xilr, covmat = cv, 
                                        cor = FALSE))
    eigenvalues <- eigen(cv$cov)$values
  }
  else if (method == "mve") {
    cv <- MASS::cov.mve(xilr)
    pcaIlr <- suppressWarnings(princomp(xilr, covmat = cv, 
                                        cor = FALSE))
    eigenvalues <- eigen(cv$cov)$values
  }
  else {
    pcaIlr <- princomp(xilr, cor = FALSE)
    eigenvalues <- eigen(cov(xilr))$values
  }
  if (is.null(mult_comp)) {
    V <- matrix(0, nrow = ncol(x), ncol = ncol(x) - 1)
    for (i in 1:ncol(V)) {
      V[1:i, i] <- 1/i
      V[i + 1, i] <- (-1)
      V[, i] <- V[, i] * sqrt(i/(i + 1))
    }
  }
  else {
    V <- matrix(0, nrow = length(unlist(mult_comp)), ncol = length(unlist(mult_comp)) - 
                  length(mult_comp))
    l <- sapply(mult_comp, length)
    start <- c(1, cumsum(l[-length(l)]))
    cumsum(l[-length(l)])
    end <- cumsum(l - 1)
    start2 <- c(1, cumsum(l[-length(l)]) + 1)
    end2 <- cumsum(l)
    for (j in 1:length(mult_comp)) {
      ind <- start[j]:end[j]
      ind2 <- start2[j]:end2[j]
      for (i in 1:length(ind)) {
        V[ind2[1:i], ind[i]] <- 1/i
        V[ind2[i] + 1, ind[i]] <- (-1)
        V[, ind[i]] <- V[, ind[i]] * sqrt(i/(i + 1))
      }
    }
  }
  if (!is.null(external)) {
    nload <- nrow(pcaIlr$loadings)
    if (is.null(dim(external)[2])){ 
      index <- 1
      nam <- "ext"
    }
    else{ 
      index <- ncol(external)
      nam <- names(external)
    }
    loadings <- V %*% pcaIlr$loadings[-c((nload - index + 
                                            1):nload), ]
    loadings <- rbind(loadings, pcaIlr$loadings[(nload - 
                                                   index + 1):nload, ])
  }
  else {
    loadings <- V %*% pcaIlr$loadings
  }
  if (is.null(mult_comp)) {
    if (!is.null(names(x)) & !is.null(external)) 
      dimnames(loadings)[[1]] <- c(names(x), nam)
    else if (!is.null(names(x))) 
      dimnames(loadings)[[1]] <- names(x)
  }
  else {
    ## modifications added :
    if (!is.null(names(x)) & !is.null(external)) 
      dimnames(loadings)[[1]] <- c(colnames(x)[unlist(mult_comp)], names(external))
    else if (!is.null(names(x))) 
      dimnames(loadings)[[1]] <- colnames(x)[unlist(mult_comp)]
  }
  pcaClr <- pcaIlr
  pcaClr$scores <- pcaIlr$scores
  pcaClr$loadings <- loadings
  res <- list(scores = pcaClr$scores, loadings = loadings, 
              eigenvalues = eigenvalues, method = method, princompOutputClr = pcaClr, 
              mult_comp = mult_comp)
  class(res) <- "pcaCoDa"
  invisible(res)
}

#' @rdname pcaCoDa
#' @export
#' @method print pcaCoDa
#' @param ... additional parameters for print method passed through
print.pcaCoDa <- function(x, ...){
  ## percentage of explained variability for clr transformed data
  eV <- x$eigenvalues / sum(x$eigenvalues)
  eVcum <- cumsum(x$eigenvalues) / sum(x$eigenvalues)
  cat("\n-------------------")
  cat("\n Percentages of explained variability for compositional data \n after clr transformation \n")
  print(eVcum)
  cat("\n-------------------\n\n")	
}

#' @rdname pcaCoDa
#' @method summary pcaCoDa
#' @param object object of class pcaCoDa
#' @export

summary.pcaCoDa <- function(object, ...){
    stopifnot(inherits(object, "pcaCoDa"))
    summary(object$princompOutputClr)
}