R/RPCA.R

In DTSR: Distributed Trimmed Scores Regression for Handling Missing Data

#' Robust Principal Component Analysis with Missing Data
#'
#' This function performs Robust Principal Component Analysis (RPCA) to handle missing data by imputing the missing values based on the correlation structure within the data. It also calculates various evaluation metrics including RMSE, MMAE, RRE, 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{MMAE}{The Mean Absolute Error.}
#' \item{RRE}{The Relative Relative 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{timeRPCA}{The RPCA 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 RPCA imputation
#' result <- RPCA(data0, data.sample, data.copy, mr, km)
#' # Print the results
#' print(result$RMSE)
#' print(result$MMAE)
#' print(result$RRE)
#' print(result$CPP1)
#' print(result$Xnew)
#'
#' @seealso \code{\link{princomp}} and \code{\link{svd}} for more information on PCA and SVD.
#' @keywords imputation RPCA PCA SVD
#' @importFrom stats kmeans hclust princomp cutree dist
#' @importFrom cluster silhouette
#' @importFrom MASS ginv
RPCA <- 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
  timeRPCA <- 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

    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)))))
    }

    Xnew <- Znew + matrix(rep(1, n * p), ncol = p) %*% diag(cm0)
    predicteds <- Xnew[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 <- sum(abs(predicteds - actuals)) / sum(actuals)

    # K-means clustering
    s <- scale(Xnew)
    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 <- Xnew
    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 = Xnew, RMSE = RMSE, MMAE = MMAE, RRE = RRE, CPP1 = CPP1, CPP2 = CPP2, CPP3 = CPP3, CPP4 = CPP4, CPP5 = CPP5, CPP6 = CPP6, CPP7 = CPP7,timeRPCA = timeRPCA))
}