R/iscca.R

In iSFun: Integrative Dimension Reduction Analysis for Multi-Source Data

##' @title Integrative sparse canonical correlation analysis
##'
##' @description This function provides a penalty-based integrative sparse canonical correlation analysis method to handle the multiple datasets with high dimensions generated under similar protocols, which consists of two built-in penalty items for selecting the important variables for users to choose, and two contrasted penalty functions for eliminating the diffierence (magnitude or sign) between estimators within each group.
##'
##' @param x list of data matrices, L datasets of explanatory variables.
##' @param y list of data matrices, L datasets of dependent variables.
##' @param L numeric, number of datasets.
##' @param mu1 numeric, sparsity penalty parameter for vector u.
##' @param mu2 numeric, contrasted penalty parameter for vector u.
##' @param mu3 numeric, sparsity penalty parameter for vector v.
##' @param mu4 numeric, contrasted penalty parameter for vector v.
##' @param eps numeric, the threshold at which the algorithm terminates.
##' @param pen1 character, "homogeneity" or "heterogeneity" type of the sparsity structure. If not specified, the default is homogeneity.
##' @param pen2 character, "magnitude" or "sign" based contrasted penalty. If not specified, the default is magnitude.
##' @param scale.x character, "TRUE" or "FALSE", whether or not to scale the variables x. The default is TRUE.
##' @param scale.y character, "TRUE" or "FALSE", whether or not to scale the variables y. The default is TRUE.
##' @param maxstep numeric, maximum iteration steps. The default value is 50.
##' @param submaxstep numeric, maximum iteration steps in the sub-iterations. The default value is 10.
##' @param trace character, "TRUE" or "FALSE". If TRUE, prints out its screening results of variables.
##' @param draw character, "TRUE" or "FALSE". If TRUE, plot the convergence path of loadings and the heatmap of coefficient beta.
##'
##' @return An 'iscca' object that contains the list of the following items.
##' \itemize{
##' \item{x:}{ list of data matrices, L datasets of explanatory variables with centered columns. If scale.x is TRUE, the columns of L datasets are standardized to have mean 0 and standard deviation 1.}
##' \item{y:}{ list of data matrices, L datasets of dependent variables with centered columns. If scale.y is TRUE, the columns of L datasets are standardized to have mean 0 and standard deviation 1.}
##' \item{loading.x:}{ the estimated canonical vector of variables x.}
##' \item{loading.y:}{ the estimated canonical vector of variables y.}
##' \item{variable.x:}{ the screening results of variables x. }
##' \item{variable.y:}{ the screening results of variables y.}
##' \item{meanx:}{ list of numeric vectors, column mean of the original datasets x.}
##' \item{normx:}{ list of numeric vectors, column standard deviation of the original datasets x.}
##' \item{meany:}{ list of numeric vectors, column mean of the original datasets y.}
##' \item{normy:}{ list of numeric vectors, column standard deviation of the original datasets y.}
##' }
##' @seealso See Also as \code{\link{preview.cca}}, \code{\link{iscca.cv}}, \code{\link{meta.scca}}, \code{\link{scca}}.
##'
##' @import caret
##' @import irlba
##' @import graphics
##' @import stats
##' @importFrom grDevices rainbow
##' @export
##' @examples
##' # Load a list with 3 data sets
##' library(iSFun)
##' data("simData.cca")
##' x <- simData.cca$x
##' y <- simData.cca$y
##' L <- length(x)
##' mu1 <- mu3 <- 0.4
##' mu2 <- mu4 <- 2.5
##'
##' prev_cca <- preview.cca(x = x, y = y, L = L, scale.x = TRUE, scale.y = TRUE)
##' res_homo_m <- iscca(x = x, y = y, L = L, mu1 = mu1, mu2 = mu2, mu3 = mu3, mu4 = mu4,
##'                     eps = 5e-2, maxstep = 50, submaxstep = 10, trace = TRUE, draw = TRUE)
##'
##' \donttest{
##' res_homo_s <- iscca(x = x, y = y, L = L, mu1 = mu1, mu2 = mu2, mu3 = mu3, mu4 = mu4,
##'                     eps = 5e-2, pen1 = "homogeneity", pen2 = "sign", scale.x = TRUE,
##'                     scale.y = TRUE, maxstep = 50, submaxstep = 10, trace = FALSE, draw = FALSE)
##'
##' mu1 <- mu3 <- 0.3
##' mu2 <- mu4 <- 2
##' res_hete_m <- iscca(x = x, y = y, L = L, mu1 = mu1, mu2 = mu2, mu3 = mu3, mu4 = mu4,
##'                     eps = 5e-2, pen1 = "heterogeneity", pen2 = "magnitude", scale.x = TRUE,
##'                     scale.y = TRUE, maxstep = 50, submaxstep = 10, trace = FALSE, draw = FALSE)
##'
##' res_hete_s <- iscca(x = x, y = y, L = L, mu1 = mu1, mu2 = mu2, mu3 = mu3, mu4 = mu4,
##'                     eps = 5e-2, pen1 = "heterogeneity", pen2 = "sign", scale.x = TRUE,
##'                     scale.y = TRUE, maxstep = 50, submaxstep = 10, trace = FALSE, draw = FALSE)
##' }

iscca <- function(x, y, L, mu1, mu2, mu3, mu4, eps = 1e-4, pen1 = "homogeneity", pen2 = "magnitude", scale.x = TRUE, scale.y = TRUE, maxstep = 50, submaxstep = 10, trace = FALSE, draw = FALSE) {

  if (class(x) != "list") { stop("x should be of list type.") }
  if (class(y) != "list") { stop("y should be of list type.") }

  # initialization
  x  <- lapply(x, as.matrix)
  y  <- lapply(y, as.matrix)
  nl <- as.numeric(lapply(x, nrow))
  pl <- as.numeric(lapply(x, ncol))
  ql <- as.numeric(lapply(y, ncol))
  p  <- unique(pl)
  q  <- unique(ql)
  if(length(p) > 1){ stop("The dimension of data x should be consistent among different datasets.")}
  if(length(q) > 1){ stop("The dimension of data y should be consistent among different datasets.")}
  ip <- c(1:p)
  iq <- c(1:q)

  # center & scale x & y
  meanx <- lapply(1:L, function(l) drop( matrix(1, 1, nl[l]) %*% x[[l]] / nl[l] ) )
  meany <- lapply(1:L, function(l) drop( matrix(1, 1, nl[l]) %*% y[[l]] / nl[l] ) )
  x <- lapply(1:L, function(l) scale(x[[l]], meanx[[l]], FALSE) )
  y <- lapply(1:L, function(l) scale(y[[l]], meany[[l]], FALSE) )

  x.scale <- function(l){
    one <- matrix(1, 1, nl[l])
    normx <- sqrt(drop(one %*% (x[[l]]^2)) / (nl[l] - 1))
    if (any(normx < .Machine$double.eps)) {
      stop("Some of the columns of the predictor matrix have zero variance.")
    }
    return(normx)
  }
  y.scale <- function(l){
    one <- matrix(1, 1, nl[l])
    normy <- sqrt(drop(one %*% (y[[l]]^2)) / (nl[l] - 1))
    if (any(normy < .Machine$double.eps)) {
      stop("Some of the columns of the response matrix have zero variance.")
    }
    return(normy)
  }

  if (scale.x) { normx <- lapply(1:L, x.scale ) } else { normx <- rep(list(rep(1,p)), L) }
  if (scale.y) { normy <- lapply(1:L, y.scale ) } else { normy <- rep(list(rep(1,q)), L) }
  if (scale.x) { x <- lapply(1:L, function(l) scale(x[[l]], FALSE, normx[[l]]) ) }
  if (scale.y) { y <- lapply(1:L, function(l) scale(y[[l]], FALSE, normy[[l]]) ) }

  ro <- function(x, mu, alpha) {
    f <- function(x) mu * (1 > x / (mu * alpha)) * (1 - x / (mu * alpha))
    r <- integrate(f, 0, x)
    return(r)
  }

  ro_d1st <- function(x, mu, alpha) {
    r <- mu * (1 > x / (mu * alpha)) * (1 - x / (mu * alpha))
    return(r)
  }

  u_value_homo <- function(u, v, p, q, L, mu1, mu2, pen2) {
    fun.s <- function(j, l) {
      s1 <- t(v[, l]) %*% t(y[[l]]) %*% t(matrix(x[[l]][, j], nrow = 1)) / (nl[l])
      if (pen2 == "magnitude") s2 <- mu2 * sum(u[j, -l])
      if (pen2 == "sign") {
        s2 <- mu2 * sum(mapply(function(x) {
          r <- x / sqrt(x^2 + 0.5)
          return(r)
        }, u[j, -l])) / sqrt(u[j, l]^2 + 0.5)
      }
      s <- s1 + s2
      return(s)
    }
    result.s <- mapply(fun.s, rep(c(1:p), times = L), rep(c(1:L), each = p))
    s <- matrix(result.s, nrow = p, ncol = L)

    norm_u_j <- apply(u, 1, function(x) {
      return(sqrt(sum(x^2)))
    })
    ro_d <- ro_d1st(norm_u_j, mu1, 6)

    fun.c <- function(j, l) {
      s_norm <- sqrt(sum(s[j, ]^2))
      if (pen2 == "magnitude") c <- (s_norm > ro_d[j]) * s[j, l] * (s_norm - ro_d[j]) / ( ( mu2 * (L - 1) ) * s_norm )

      if (pen2 == "sign") c <- (s_norm > ro_d[j]) * s[j, l] * (s_norm - ro_d[j]) / ( ( mu2 * (L - 1) / (u[j, l]^2 + 0.5) ) * s_norm)

      return(c)
    }
    c <- matrix(mapply(fun.c, rep(c(1:p), times = L), rep(c(1:L), each = p)), nrow = p, ncol = L)
    return(c)
  }

  u_value_hetero <- function(u, v, p, q, L, mu1, mu2, pen2) {
    fun.mu <- function(j, l) {
      u_j <- u[j, ]
      ro_j <- mapply(ro, abs(u_j), mu1, 6)
      s_ro <- sum(as.data.frame(ro_j[1, ]))
      mu_jl <- ro_d1st(s_ro, 1, 1 / 2 * L * 6 * mu1^2) * ro_d1st(abs(u_j[l]), mu1, 6)
      return(mu_jl)
    }
    result.mu <- mapply(fun.mu, rep(c(1:p), times = L), rep(c(1:L), each = p))
    mu <- matrix(result.mu, nrow = p, ncol = L)

    fun.s <- function(j, l) {
      s1 <- t(v[, l]) %*% t(y[[l]]) %*% t(matrix(x[[l]][, j], nrow = 1)) / (nl[l])
      if (pen2 == "magnitude") s2 <- mu2 * sum(u[j, -l])
      if (pen2 == "sign") {
        s2 <- mu2 * sum(mapply(function(x) {
          r <- x / sqrt(x^2 + 0.5)
          return(r)
        }, u[j, -l])) / sqrt(u[j, l]^2 + 0.5)
      }
      s <- s1 + s2
      return(s)
    }
    result.s <- mapply(fun.s, rep(c(1:p), times = L), rep(c(1:L), each = p))
    s <- matrix(result.s, nrow = p, ncol = L)

    fun.c <- function(j, l) {
      if (pen2 == "magnitude") c <- sign(s[j, l]) * (abs(s[j, l]) > mu[j, l]) * (abs(s[j, l]) - mu[j, l]) / ( mu2 * (L - 1) )

      if (pen2 == "sign") c <- sign(s[j, l]) * (abs(s[j, l]) > mu[j, l]) * (abs(s[j, l]) - mu[j, l]) / ( mu2 * (L - 1) / (u[j, l]^2 + 0.5) )

      return(c)
    }
    c <- matrix(mapply(fun.c, rep(c(1:p), times = L), rep(c(1:L), each = p)), nrow = p, ncol = L)
    return(c)
  }

  v_value_homo <- function(u, v, p, q, L, mu1, mu2, pen2) {
    fun.s <- function(j, l) {
      s1 <- t(u[, l]) %*% t(x[[l]]) %*% t(matrix(y[[l]][, j], nrow = 1)) / (nl[l])
      if (pen2 == "magnitude") s2 <- mu2 * sum(v[j, -l])
      if (pen2 == "sign") {
        s2 <- mu2 * sum(mapply(function(x) {
          r <- x / sqrt(x^2 + 0.5)
          return(r)
        }, v[j, -l])) / sqrt(v[j, l]^2 + 0.5)
      }
      s <- s1 + s2
      return(s)
    }
    result.s <- mapply(fun.s, rep(c(1:q), times = L), rep(c(1:L), each = q))
    s <- matrix(result.s, nrow = q, ncol = L)

    norm_v_j <- apply(v, 1, function(x) {
      return(sqrt(sum(x^2)))
    })
    ro_d <- ro_d1st(norm_v_j, mu1, 6)

    fun.c <- function(j, l) {
      s_norm <- sqrt(sum(s[j, ]^2))
      if (pen2 == "magnitude") c <- (s_norm > ro_d[j]) * s[j, l] * (s_norm - ro_d[j]) / ( ( mu2 * (L - 1) ) * s_norm )

      if (pen2 == "sign") c <- (s_norm > ro_d[j]) * s[j, l] * (s_norm - ro_d[j]) / ( ( mu2 * (L - 1) / (v[j, l]^2 + 0.5) ) * s_norm)

      return(c)
    }
    c <- matrix(mapply(fun.c, rep(c(1:q), times = L), rep(c(1:L), each = q)), nrow = q, ncol = L)
    return(c)
  }

  v_value_hetero <- function(u, v, p, q, L, mu1, mu2, pen2) {
    fun.mu <- function(j, l) {
      v_j <- v[j, ]
      ro_j <- mapply(ro, abs(v_j), mu1, 6)
      s_ro <- sum(as.data.frame(ro_j[1, ]))
      mu_jl <- ro_d1st(s_ro, 1, 1 / 2 * L * 6 * mu1^2) * ro_d1st(abs(v_j[l]), mu1, 6)
      return(mu_jl)
    }
    result.mu <- mapply(fun.mu, rep(c(1:q), times = L), rep(c(1:L), each = q))
    mu <- matrix(result.mu, nrow = q, ncol = L)

    fun.s <- function(j, l) {
      s1 <- t(u[, l]) %*% t(x[[l]]) %*% t(matrix(y[[l]][, j], nrow = 1)) / (nl[l])
      if (pen2 == "magnitude") s2 <- mu2 * sum(v[j, -l])
      if (pen2 == "sign") {
        s2 <- mu2 * sum(mapply(function(x) {
          r <- x / sqrt(x^2 + 0.5)
          return(r)
        }, v[j, -l])) / sqrt(v[j, l]^2 + 0.5)
      }
      s <- s1 + s2
      return(s)
    }
    result.s <- mapply(fun.s, rep(c(1:q), times = L), rep(c(1:L), each = q))
    s <- matrix(result.s, nrow = q, ncol = L)

    fun.c <- function(j, l) {
      if (pen2 == "magnitude") c <- sign(s[j, l]) * (abs(s[j, l]) > mu[j, l]) * (abs(s[j, l]) - mu[j, l]) / ( mu2 * (L - 1) )

      if (pen2 == "sign") c <- sign(s[j, l]) * (abs(s[j, l]) > mu[j, l]) * (abs(s[j, l]) - mu[j, l]) / ( mu2 * (L - 1) / (u[j, l]^2 + 0.5) )

      return(c)
    }
    c <- matrix(mapply(fun.c, rep(c(1:q), times = L), rep(c(1:L), each = q)), nrow = q, ncol = L)
    return(c)
  }

  # initilize objects
  what <- matrix(0, p, L)
  fun.1 <- function(l) {
    Z_l <- irlba( t(x[[l]]) %*% y[[l]] , nu =1, nv = 1)
    u_l <- Z_l$u
    return(u_l)
  }
  U <- matrix(mapply(fun.1, 1:L), nrow = p)

  fun.2 <- function(l) {
    Z_l <- irlba( t(x[[l]]) %*% y[[l]] , nu =1, nv = 1)
    v_l <- Z_l$v
    return(v_l)
  }
  V <- mapply(fun.2, 1:L)

  sgn1 <- mapply(function(l) sign(U[, l]%*% t(x[[l]]) %*% y[[l]] %*% V[, l]), 1:L )
  for (l in 1:L) { V[, l] <- sgn1[l]*V[, l] }

  sgn2 <- mapply(function(l) sign((U[,1]%*%U[,l])/(sqrt(sum(U[,1]^2))*sqrt(sum(U[,l]^2)))), 2:L )
  for (l in 2:L) {
    U[, l] <- sgn2[l-1]*U[, l]
    V[, l] <- sgn2[l-1]*V[, l]
  }

  u <- U
  v <- V
  iter <- 1
  dis.u.iter <- 10
  dis.v.iter <- 10
  loading_trace_u <- matrix(0, nrow = p * L, ncol = maxstep)
  loading_trace_v <- matrix(0, nrow = q * L, ncol = maxstep)

  if (trace) {
    cat("The variables that join the set of selected variables at each step:\n")
  }

  while (dis.u.iter > eps & dis.v.iter > eps & iter <= maxstep){
    u.iter <- u
    v.iter <- v
    subiter_u <- 1
    dis.u <- 10
    while (dis.u > eps & subiter_u <= submaxstep) {
      u.old <- u
      if (pen1 == "homogeneity") u <- u_value_homo(u, v, p, q, L, mu1, mu2, pen2)
      if (pen1 == "heterogeneity") u <- u_value_hetero(u, v, p, q, L, mu1, mu2, pen2)
      u_norm <- sqrt(colSums(u^2))
      u_norm <- ifelse(u_norm == 0, 0.0001, u_norm)
      u <- t(t(u) / u_norm)
      dis.u <- max( sqrt(colSums((u - u.old)^2)) / sqrt(colSums(u.old^2)) )

      what <- u
      what_cut <- ifelse(abs(what) > 1e-4, what, 0)
      what_cut_norm <- sqrt(colSums(what_cut^2))
      what_cut_norm <- ifelse(what_cut_norm == 0, 0.0001, what_cut_norm)
      what_dir <- t(t(what_cut) / what_cut_norm)
      what_cut <- ifelse(abs(what_dir) > 1e-4, what_dir, 0)

      if (sum(apply(u, 2, function(x) sum(abs(x) <= 1e-4) == p)) > 0) {
        cat("Stop! The value of mu1 is too large");
        break}
      subiter_u <- subiter_u + 1
    }

    loading_trace_u[, iter] <- as.numeric(what_cut)

    subiter_v <- 1
    dis.v <- 10
    while (dis.v > eps & subiter_v <= submaxstep) {
      v.old <- v

      if (pen1 == "homogeneity") v <- v_value_homo(u, v, p, q, L, mu1 = mu3, mu2 = mu4, pen2)
      if (pen1 == "heterogeneity") v <- v_value_hetero(u, v, p, q, L, mu1 = mu3, mu2 = mu4, pen2)
      v_norm <- sqrt(colSums(v^2))
      v_norm <- ifelse(v_norm == 0, 0.0001, v_norm)
      v <- t(t(v) / v_norm)
      dis.v <- max( sqrt(colSums((v - v.old)^2)) / sqrt(colSums(v.old^2)) )

      what_v <- v
      what_cut_v <- ifelse(abs(what_v) > 1e-4, what_v, 0)
      what_cut_norm <- sqrt(colSums(what_cut_v^2))
      what_cut_norm <- ifelse(what_cut_norm == 0, 0.0001, what_cut_norm)
      what_dir <- t(t(what_cut_v) / what_cut_norm)
      what_cut_v <- ifelse(abs(what_dir) > 1e-4, what_dir, 0)

      if (sum(apply(v, 2, function(x) sum(abs(x) <= 1e-4) == q)) > 0) {
        cat("Stop! The value of mu1 is too large");
        break}
      subiter_v <- subiter_v + 1
    }
    loading_trace_v[, iter] <- as.numeric(what_cut_v)

    dis.u.iter <- max( sqrt(colSums((u - u.iter)^2)) / sqrt(colSums(u.iter^2)) )
    dis.v.iter <- max( sqrt(colSums((v - v.iter)^2)) / sqrt(colSums(v.iter^2)) )
    if (sum(apply(u, 2, function(x) sum(abs(x) <= 1e-4) == p)) > 0 | sum(apply(v, 2, function(x) sum(abs(x) <= 1e-4) == q)) > 0) { break }

    if (trace) {
      # selected variables of x
      new2A <- list()
      new2A <- mapply(function(l) {
        A <- ip[what_cut[, l] != 0]
        return(A)
      }, c(1:L), SIMPLIFY = FALSE)
      cat("\n")
      cat(paste("--------------------", "\n"))
      cat(paste("----- Step", iter, " -----\n", sep = " "))
      cat(paste("--------------------", "\n"))
      for (l in 1:L)
      {
        new2A_l <- new2A[[l]]
        if (length(new2A_l) <= 10) {
          cat(paste("DataSet ", l, ":\n", sep = ""))
          cat(paste("X", new2A_l, ", ", sep = " "))
          cat("\n")
        } else {
          cat(paste("DataSet ", l, ":\n", sep = ""))
          nlines <- ceiling(length(new2A_l) / 10)
          for (i in 0:(nlines - 2))
          {
            cat(paste("X", new2A_l[(10 * i + 1):(10 * (i + 1))], ", ", sep = " "))
            cat("\n")
          }
          cat(paste("X", new2A_l[(10 * (nlines - 1) + 1):length(new2A_l)], ", ", sep = " "))
          cat("\n")
        }
      }

      # selected variables of y
      new2A_v <- list()
      new2A_v <- mapply(function(l) {
        A <- iq[what_cut_v[, l] != 0]
        return(A)
      }, c(1:L), SIMPLIFY = FALSE)

      for (l in 1:L)
      {
        new2A_l <- new2A_v[[l]]
        if (length(new2A_l) <= 10) {
          cat(paste("DataSet ", l, ":\n", sep = ""))
          cat(paste("Y", new2A_l, ", ", sep = " "))
          cat("\n")
        } else {
          cat(paste("DataSet ", l, ":\n", sep = ""))
          nlines <- ceiling(length(new2A_l) / 10)
          for (i in 0:(nlines - 2))
          {
            cat(paste("Y", new2A_l[(10 * i + 1):(10 * (i + 1))], ", ", sep = " "))
            cat("\n")
          }
          cat(paste("Y", new2A_l[(10 * (nlines - 1) + 1):length(new2A_l)], ", ", sep = " "))
          cat("\n")
        }
      }
    }
    iter <- iter + 1
  }

  if(iter > 1){
    loading_trace_u <- loading_trace_u[,1:(iter-1)]
    loading_trace_v <- loading_trace_v[,1:(iter-1)]
  }

  # normalization
  what.u <- what_cut
  what.v <- what_cut_v

  # selected variables
  new2A <- list()
  new2A <- mapply(function(l) {
    A <- ip[what.u[, l] != 0]
    return(A)
  }, c(1:L), SIMPLIFY = FALSE)

  new2A_v <- list()
  new2A_v <- mapply(function(l) {
    A <- iq[what.v[, l] != 0]
    return(A)
  }, c(1:L), SIMPLIFY = FALSE)

  # print out variables that join the active set
  plot_draw <- function(draw){
    opar <- par(mfrow = c(1,2))
    on.exit(par(opar))
    if(draw){
      for (l in 1:L) {
        matplot(t(loading_trace_u[((l-1)*p+1):(l*p),]), type = 'l',
                main = paste("Dataset ", l, "\n", "Convergence path of elements in vector u"),
                xlab = "Number of iterations", ylab = "Weight")
        matplot(t(loading_trace_v[((l-1)*q+1):(l*q),]), type = 'l',
                main = paste("Dataset ", l, "\n", "Convergence path of elements in vector v"),
                xlab = "Number of iterations", ylab = "Weight")
      }
    }
  }
  plot_draw(draw = draw)

  listname <- mapply(function(l) paste("Dataset ", l), 1:L)
  names(new2A) <- listname
  names(new2A_v) <- listname
  names(meanx) <- listname
  names(meany) <- listname
  names(normx) <- listname
  names(normy) <- listname
  names(x) <- listname
  names(y) <- listname
  colnames(what) <- listname
  rownames(what) <- c(1 : p)
  colnames(what.v) <- listname
  rownames(what.v) <- c(1 : q)

  # return objects
  object <- list(
    x = x, y = y, loading.x = what, loading.y = what.v,
    variable.x = new2A, variable.y = new2A_v, meanx = meanx,
    normx = normx, meany = meany, normy = normy,pen1 = pen1, pen2 = pen2,
    mu1 = mu1, mu2 = mu2, mu3 = mu3, mu4 = mu4,
    loading_trace_u = loading_trace_u, loading_trace_v = loading_trace_v
  )
  class(object) <- "iscca"
  return(object)
}