R/NIPALS.R

In DTSR: Distributed Trimmed Scores Regression for Handling Missing Data

#' NIPALS Algorithm with RPCA and Clustering
#'
#' This function performs the NIPALS (Nonlinear Iterative Partial Least Squares) algorithm to handle missing data by imputing the missing values based on the correlation structure within the data. It also calculates the RMSE and Consistency Proportion Index (CPP) using different hierarchical clustering methods.
#'
#' @param data0 The original dataset containing the response variable and features.
#' @param data.sample The dataset used for sampling, which may contain missing values.
#' @param data.copy A copy of the original dataset, used for comparison or validation.
#' @param mr Indices of the rows with missing values that need to be predicted.
#' @param km The number of clusters for k-means clustering.
#' @return A list containing:
#' \item{Xnew}{The imputed dataset.}
#' \item{RMSE}{The Root Mean Squared Error.}
#' \item{CPP1}{The K-means clustering Consistency Proportion Index.}
#' \item{CPP2}{The Hierarchical Clustering Complete Linkage Consistency Proportion Index.}
#' \item{CPP3}{The Hierarchical Clustering Single Linkage Consistency Proportion Index.}
#' \item{CPP4}{The Hierarchical Clustering Average Linkage Consistency Proportion Index.}
#' \item{CPP5}{The Hierarchical Clustering Centroid linkage Consistency Proportion Index.}
#' \item{CPP6}{The Hierarchical Clustering Median Linkage Consistency Proportion Index.}
#' \item{CPP7}{The Hierarchical Clustering Ward's Method Consistency Proportion Index.}
#' \item{timeNIPALS}{The NIPALS algorithm execution time.}
#' @export
#'
#' @examples
#' # Create a sample matrix with random values and introduce missing values
#' set.seed(123)
#' n <- 100
#' p <- 5
#' data.sample <- matrix(rnorm(n * p), nrow = n)
#' data.sample[sample(1:(n*p), 20)] <- NA
#' data.copy <- data.sample
#' data0 <- data.frame(data.sample, response = rnorm(n))
#' mr <- sample(1:n, 10)  # Sample rows for evaluation
#' km <- 3  # Number of clusters
#' # Perform NIPALS imputation
#' result <- NIPALS(data0, data.sample, data.copy, mr, km)
#' # Print the results
#' print(result$RMSE)
#' print(result$CPP1)
#' print(result$Xnew)
#'
#' @seealso \code{\link{princomp}} and \code{\link{svd}} for more information on PCA and SVD.
#' @keywords imputation NIPALS RPCA PCA SVD
#' @importFrom stats princomp kmeans hclust cutree dist
#' @importFrom MASS ginv
NIPALS<- function(data0, data.sample, data.copy, mr, km) {
  X0 <- data.sample
  n <- nrow(X0); p <- ncol(X0)
  cm0 <- colMeans(X0, na.rm = TRUE)
  data.sample[is.na(data.sample)] <- cm0[ceiling(which(is.na(data.sample)) / n)]
  Xm <- X <- as.matrix(data.sample)

 # Record the execution time
  timeNIPALS <- system.time({
  # PCA
  pca <- princomp(Xm, cor = TRUE)
  PCA <- summary(pca, loadings = TRUE)
  D <- (pca$sdev)^2
  A <- PCA$loadings
  l <- D / sum(D)
  J <- rep(l, times = p); dim(J) <- c(p, p)
  upper.tri(J, diag = TRUE); J[lower.tri(J)] <- 0
  ll <- matrix(colSums(J), nrow = 1, ncol = p, byrow = FALSE)
  ww <- which(ll >= 0.7)
  k <- ww[1]

  Z <- scale(X, center = TRUE, scale = FALSE)
  tol <- 1e-10; nb <- 10; niter <- 0; niter1 <- 0; d <- 1

  # NIPALS iterations
  while ((d >= tol) & (niter <= nb)) {
    niter <- niter + 1
    Zold <- Z
    lambda <- svd(Z)$d
    A <- svd(Z)$v
    Ak <- matrix(A[, 1:k], p, k)
    for (i in 1:n) {
      niter1 <- niter1 + 1
      M <- is.na(X0[i, ])
      job <- which(M == FALSE); jna <- which(M == TRUE)
      piob <- nrow(as.matrix(job)); pina <- nrow(as.matrix(jna))
      while ((piob > 0) & (pina > 0)) {
        Qi <- matrix(0, p, p)
        for (u in 1:piob) {
          Qi[job[u], u] <- 1
        }
        for (v in 1:pina) {
          Qi[jna[v], v + piob] <- 1
        }
        zi <- Z[i, ]
        zQi <- zi %*% Qi
        ZQi <- Z %*% Qi
        AQi <- t(t(Ak) %*% Qi)
        ziob <- matrix(zQi[, 1:piob], 1, piob)
        zina <- matrix(zQi[, piob + (1:pina)], 1, pina)
        Ziob <- matrix(ZQi[, 1:piob], n, piob, byrow = FALSE)
        Zina <- matrix(ZQi[, piob + (1:pina)], n, pina, byrow = FALSE)
        Aiob <- matrix(AQi[1:piob, ], piob, k, byrow = FALSE)
        Aina <- matrix(AQi[piob + (1:pina), ], pina, k, byrow = FALSE)
        Ti <- Ziob %*% Aiob
        betaihat <- ginv(t(Ti) %*% Ti) %*% t(Ti) %*% Zina
        zinahat <- ziob %*% Aiob %*% betaihat
        ZQi[i, piob + (1:pina)] <- zinahat
        Zi <- ZQi %*% t(Qi)
        Z <- Zi
        pina <- 0
      }
    }
    Znew <- Z
    d <- sqrt(sum(diag((t(Zold - Znew) %*% (Zold - Znew)))))
  }

  XNIPALS <- Z + matrix(1, n, p) %*% diag(cm0)
  predicteds <- XNIPALS[mr]
  actuals <- data.copy[mr]

  # Calculate RMSE
  RMSE <- sqrt(base::mean((actuals - predicteds)^2))

  # Calculate MMAE
  MMAE <- base::mean(abs(predicteds - actuals))

  # Calculate RRE
  RRE <- RMSE / (max(actuals) - min(actuals))
  # Calculate CPP Index


  # K-means clustering
  s <- scale(XNIPALS)
  km <- kmeans(s, km)
  I1 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I1[g, 1] <- g
  }
  I1[, 2] <- km$cluster
  I1[, 3] <- data0[, p + 1]
  CPP1 <- IndexCPP(I1)

  # Hierarchical clustering
  HCdata <- XNIPALS
  distance <- dist(HCdata)

  # Complete linkage
  HCdata.hc <- hclust(distance)
  HCdata.id <- cutree(HCdata.hc, 3)
  I2 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I2[g, 1] <- g
  }
  I2[, 2] <- HCdata.id
  I2[, 3] <- data0[, p + 1]
  CPP2 <- IndexCPP(I2)

  # Single linkage
  HCdata.single <- hclust(distance, method = "single")
  HCdatasingle.id <- cutree(HCdata.single, 3)
  I3 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I3[g, 1] <- g
  }
  I3[, 2] <- HCdatasingle.id
  I3[, 3] <- data0[, p + 1]
  CPP3 <- IndexCPP(I3)

  # Average linkage
  HCdata.average <- hclust(distance, method = "average")
  HCdataaverage.id <- cutree(HCdata.average, 3)
  I4 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I4[g, 1] <- g
  }
  I4[, 2] <- HCdataaverage.id
  I4[, 3] <- data0[, p + 1]
  CPP4 <- IndexCPP(I4)

  # Centroid linkage
  HCdata.centroid <- hclust(distance, method = "centroid")
  HCdatacentroid.id <- cutree(HCdata.centroid, 3)
  I5 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I5[g, 1] <- g
  }
  I5[, 2] <- HCdatacentroid.id
  I5[, 3] <- data0[, p + 1]
  CPP5 <- IndexCPP(I5)

  # Median linkage
  HCdata.median <- hclust(distance, method = "median")
  HCdatamedian.id <- cutree(HCdata.median, 3)
  I6 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I6[g, 1] <- g
  }
  I6[, 2] <- HCdatamedian.id
  I6[, 3] <- data0[, p + 1]
  CPP6 <- IndexCPP(I6)

  # Ward's method
  HCdata.ward <- hclust(distance, method = "ward.D")
  HCdataward.id <- cutree(HCdata.ward, 3)
  I7 <- matrix(0, nrow = n, ncol = 3)
  for (g in 1:n) {
    I7[g, 1] <- g
  }
  I7[, 2] <- HCdataward.id
  I7[, 3] <- data0[, p + 1]
  CPP7 <- IndexCPP(I7)
 })
  return(list(Xnew = XNIPALS, RMSE = RMSE, MMAE = MMAE, RRE = RRE, CPP1 = CPP1, CPP2 = CPP2, CPP3 = CPP3, CPP4 = CPP4, CPP5 = CPP5, CPP6 = CPP6, CPP7 = CPP7,timeNIPALS = timeNIPALS))
}