R/sobolowen.R

In sensitivity: Global Sensitivity Analysis of Model Outputs and Importance Measures

# Sobol' indices estimation by the method of Owen
#Authors: B. Ramos and T. Touati (2014)

sobolowen <- function(model = NULL, X1, X2, X3, nboot = 0, conf = 0.95, varest = 2, ...) {
  if ((ncol(X1) != ncol(X2)) | (nrow(X1) != nrow(X2)) | (ncol(X2) != ncol(X3)) | (nrow(X2) != nrow(X3)))
    stop("The samples X1, X2 and X3 must have the same dimensions")
  p <- ncol(X1)
  
  X <- rbind(X1, X2)
  for (i in 1:p) {
    Xb <- X1
    Xb[,i] <- X3[,i]
    X <- rbind(X, Xb)
  }
  for (i in 1:p) {
    Xb <- X2
    Xb[,i] <- X1[,i]
    X <- rbind(X, Xb)
  }
  for (i in 1:p) {
    Xb <- X3
    Xb[,i] <- X2[,i]
    X <- rbind(X, Xb)
  }
  
  x <- list(model = model, X1 = X1, X2 = X2, X3 = X3, nboot = nboot, conf = conf, X = X,
            call = match.call())
  class(x) <- "sobolowen"
  
  if (!is.null(x$model)) {
    response(x, ...)
    tell(x,varest=varest)
  }
  
  return(x)
}

estim.sobolowen <- function(data, i=1:nrow(data), varest=2) {
  d <- as.matrix(data[i, ]) # as.matrix for colSums
  n <- nrow(d)
  p <- (ncol(d)-2)/3
  if (varest==1) {
    V <- var(d[, 1])    
  } else {
    V <- numeric(0)    
    for (k in 1:p) {
      V[k] <- mean(apply(d[,c(1,2,2+k,2+k+p)]^2,1,mean))-(mean(apply(d[,c(1,2,2+k,2+k+p)],1,mean)))^2
    }
  }
  VCE <- (colSums((d[,1] - d[, 3:(2+p)])*(d[, (3+p):(2+2*p)] - d[,2])) / n)
  VCE.compl <- V - (colSums((d[,2] - d[, (3+2*p):(2+3*p)])*(d[, (3+p):(2+2*p)] - d[,1])) / n)
  # COMM: VCE correspond a underline{tau_u}=\int (f(Y)-f(Y_U:Z_{-U}))(f(X_U:Y_{-U})-f(X))
  # COMM: VCE.compl CORRESPOND A overline{tau_u}=sigma^2-underline{tau_{-u}}=sigma^2 - \int (f(Y)-f(Y_U:Z_{-U}))(f(X_U:Y_{-U})-f(X)) qui est obtenu en echangeant x et y de l'estimateur d'en haut et en remplacant u par -u
  c(V, VCE, VCE.compl)
}

tell.sobolowen <- function(x, y = NULL, return.var = NULL, varest=2, ...) {
  id <- deparse(substitute(x))
  
  if (! is.null(y)) {
    x$y <- y
  } else if (is.null(x$y)) {
    stop("y not found")
  }
  
  p <- ncol(x$X1)
  n <- nrow(x$X1)
  
  data <- matrix(x$y, nrow = n)
  
  # estimation of the partial variances (V, D1 and Dt)
  if (x$nboot == 0){
    V <- data.frame(original = estim.sobolowen(data,varest=varest))
  }
  else{
    #    eval(parse(text=paste("func <- function(data,i) {estim.sobolowen(data,i,varest=",varest,") }",sep="")))
    func <- function(data,i) {estim.sobolowen(data,i,varest)}
    V.boot <- boot(data=data, statistic=func, R = x$nboot)
    V <- bootstats(V.boot, x$conf, "basic")
  }
  rownames(V) <- c(paste("global",colnames(x$X1)), colnames(x$X1), paste("-", colnames(x$X1), sep = ""))
  
  # estimation of the Sobol' indices (S1 and St)
  if (varest==1) {
    k <- 1
  } else {
    k <- p
  }
  if (x$nboot == 0) {
    S <- V[(k + 1):(p + k), 1, drop = FALSE] / V[1:k,1]
    T <- V[(p + k + 1):(2 * p + k), 1, drop = FALSE] / V[1:k,1]
  } else {
    S.boot <- V.boot
    S.boot$t0 <- V.boot$t0[(k+1):(p + k)] / V.boot$t0[1:k]
    S.boot$t <- V.boot$t[,(k+1):(p + k)] / V.boot$t[,(1:k)]
    S <- bootstats(S.boot, x$conf, "basic")
    
    T.boot <- V.boot
    T.boot$t0 <- V.boot$t0[(p + k + 1):(2 * p + k)] / V.boot$t0[1:k]
    T.boot$t <- V.boot$t[,(p + k + 1):(2 * p + k)] / V.boot$t[,(1:k)]
    T <- bootstats(T.boot, x$conf, "basic")
  }
  rownames(S) <- colnames(x$X1)
  rownames(T) <- colnames(x$X1)
  
  # return
  x$V <- V
  x$S <- S
  x$T <- T
  
  for (i in return.var) {
    x[[i]] <- get(i)
  }
  
  assign(id, x, parent.frame())
}

print.sobolowen <- function(x, ...) {
  cat("\nCall:\n", deparse(x$call), "\n", sep = "")
  if (!is.null(x$y)) {
    cat("\nModel runs:", length(x$y), "\n")
    cat("\nFirst order indices:\n")
    print(x$S)
    cat("\nTotal indices:\n")
    print(x$T)
  }
}

plot.sobolowen <- function(x, ylim = c(0, 1), ...) {
  if (!is.null(x$y)) {
    p <- ncol(x$X1)
    pch = c(21, 24)
    nodeplot(x$S, xlim = c(1, p + 1), ylim = ylim, pch = pch[1])
    nodeplot(x$T, xlim = c(1, p + 1), ylim = ylim, labels = FALSE,
             pch = pch[2], at = (1:p)+.3, add = TRUE)
    legend(x = "topright", legend = c("main effect", "total effect"), pch = pch)
  }
}

ggplot.sobolowen <- function(data, mapping = aes(), ylim = c(0,1), ..., environment = parent.frame()) {
  x <- data
  if (!is.null(x$y)) {
    p <- ncol(x$X1)
    pch = c(21, 24)
    nodeggplot(listx = list(x$S,x$T), xname = c("Main effet","Total effect"), ylim = ylim, pch = pch)
  }
}