Nothing
R/sobolGP.R
In sensitivity: Global Sensitivity Analysis of Model Outputs and Importance Measures
Defines functions
sobolT2007pickfreeze
sobol2007pickfreeze
sobolTjansenpickfreeze
sobolT2002pickfreeze
sobolEffpickfreeze
soboljansenpickfreeze
sobolpickfreeze
simulateGP.sobol
tell.sobolGP
ask.sobolGP
print.sobolGP
plot.sobolGP
Documented in
ask.sobolGP
plot.sobolGP
print.sobolGP
tell.sobolGP
# Kriging-based global sensitivity analysis taking into account both
# the meta-model and the Monte-Carlo errors
# Author : loic le Gratiet, 2014
sobolGP <- function (
model,
type="SK",
MCmethod="sobol",
X1,
X2,
nsim=100,
nboot=1,
conf = 0.95,
sequential = FALSE,
candidate = NULL,
sequential.tot=FALSE,
max_iter = 1000
)
{
if(sequential){
ncandidate <- dim(candidate)[1]
dcandidate <- dim(candidate)[2]
}
if ((ncol(X1) != ncol(X2)) | (nrow(X1) != nrow(X2)))
stop("The samples X1 and X2 must have the same dimensions")
if(sequential){
if(is.null(candidate)){
stop("The number of candidate points must be greater than zero")
}
if (ncol(X1) != ncol(candidate)){
stop("The candidate points, X1 and X2 must have the same dimensions")
}
}
if(MCmethod=="sobol"||MCmethod=="sobolEff"){
if(sequential.tot){
stop("Sequential design for total indices is only available for sobol2002, sobol2007 and soboljansen methods")
}
}
p <- ncol(X1)
S <- list()
if(sequential){
Svar <- matrix(nrow = ncandidate, ncol = p)
}
if(sequential.tot){
STotvar <- matrix(nrow = ncandidate, ncol = p)
}
output <- list()
output$call$X1 <- X1
output$call$X2 <- X2
output$call$conf <- conf
output$call$nboot <- nboot
output$call$candidate <- candidate
output$call$sequential <- sequential
output$call$max_iter <- max_iter
output$call$sequential.tot <- sequential.tot
output$call$model <- model
Tot=FALSE
if(MCmethod=="sobol2002"||MCmethod=="sobol2007"||MCmethod=="soboljansen") Tot=TRUE
if(MCmethod!="sobol2007"){
for (i in 1:p) {
Xb <- X2
Xb[, i] <- X1[, i]
X <- rbind(X1, Xb)
nX <- dim(X)[1]
if(sequential){
X <- rbind(X, data.frame(candidate))
}
rm(list=c("Xb"))
ysimu <- simulateGP.sobol(object = model, nsim = nsim, newdata=X,
cond=TRUE, checkNames=FALSE, max_iter=1000,type)
if(MCmethod=="sobol"||MCmethod=="sobol2002"){
S[[i]] <- sobolpickfreeze(ysimu[,1:(nX/2)] , ysimu[,(nX/2+1):nX],nboot)
}
if(MCmethod=="sobolEff"){
S[[i]] <- sobolEffpickfreeze(ysimu[,1:(nX/2)] , ysimu[,(nX/2+1):nX],nboot)
}
if(MCmethod=="soboljansen"){
S[[i]] <- soboljansenpickfreeze(ysimu[,1:(nX/2)] , ysimu[,(nX/2+1):nX],nboot)
}
if(sequential){
predCov <- predictGP.sobol(object = model, newdata1=data.frame(X), newdata2=data.frame(candidate), type=type, prednewdata1 = FALSE, prednewdata2 = TRUE)
for(k in 1:ncandidate){
ynew <- predCov$mean2[k]
zsimu <- t(as.matrix(predCov$cov[-c((nX+1):(nX+ncandidate)),k])%*%rep(1,nsim))*(ynew-ysimu[,(nX+k)])/predCov$cov[(nX+k),k]+ysimu[,-c((nX+1):(nX+ncandidate))]
if(MCmethod=="sobol"||MCmethod=="sobol2002"){
Scand <- sobolpickfreeze(zsimu[,1:(nX/2)] , zsimu[,(nX/2+1):nX],nboot=1)
}
if(MCmethod=="sobolEff"){
Scand <- sobolEffpickfreeze(zsimu[,1:(nX/2)] , zsimu[,(nX/2+1):nX],nboot=1)
}
if(MCmethod=="soboljansen"){
Scand <- soboljansenpickfreeze(zsimu[,1:(nX/2)] , zsimu[,(nX/2+1):nX],nboot=1)
}
rm(list=c("zsimu"))
Svar[k,i] <- var(Scand)
rm(list=c("Scand"))
}
rm(list=c("predCov", "ynew"))
}
rm(list=c("ysimu" ))
}
if(Tot){
STot <- list()
for (i in 1:p) {
Xb <- X1
Xb[, i] <- X2[, i]
X <- rbind(X1, Xb)
nX <- dim(X)[1]
if(sequential.tot){
X <- rbind(X, data.frame(candidate))
}
rm(list=c("Xb"))
ysimu <- simulateGP.sobol(object = model, nsim = nsim, newdata=X,
cond=TRUE, checkNames=FALSE, max_iter=1000,type)
if(MCmethod=="sobol2002"){
STot[[i]] <- sobolT2002pickfreeze(ysimu[,1:(nX/2)],ysimu[,(nX/2+1):nX],nboot)
}
if(MCmethod=="soboljansen"){
STot[[i]] <- sobolTjansenpickfreeze(ysimu[,1:(nX/2)],ysimu[,(nX/2+1):nX],nboot)
}
if(sequential.tot){
predCov <- predictGP.sobol(object = model, newdata1=data.frame(X), newdata2=data.frame(candidate), type=type)
for(k in 1:ncandidate){
ynew <- predCov$mean[k]
zsimu <- t(as.matrix(predCov$cov[-c((nX+1):(nX+ncandidate)),k])%*%rep(1,nsim))*(ynew-ysimu[,(nX+k)])/predCov$cov[(nX+k),k]+ysimu[,-c((nX+1):(nX+ncandidate))]
if(MCmethod=="sobol2002"){
STotcand <- sobolT2002pickfreeze(zsimu[,1:(nX/2)],zsimu[,(nX/2+1):nX],nboot=1)
}
if(MCmethod=="soboljansen"){
STotcand <- sobolTjansenpickfreeze(zsimu[,1:(nX/2)],zsimu[,(nX/2+1):nX],nboot=1)
}
rm(list=c("zsimu"))
STotvar[k,i] <- var(STotcand)
rm(list=c("STotcand"))
}
rm(list=c("predCov", "ynew"))
}
rm(list=c("ysimu" ))
}
}
rm(list=c("X", "X1","X2"))
}
if(MCmethod=="sobol2007"){
STot <- list()
for (i in 1:p) {
Xb <- X1
Xb[, i] <- X2[, i]
X <- rbind(X1, Xb, X2)
nX <- dim(X)[1]
if(sequential||sequential.tot){
X <- rbind(X, data.frame(candidate))
}
rm(list=c("Xb"))
ysimu <- simulateGP.sobol(object = model, nsim = nsim, newdata=X,
cond=TRUE, checkNames=FALSE, max_iter=1000,type)
S[[i]] <- sobol2007pickfreeze(ysimu[,1:(nX/3)] , ysimu[,(nX/3+1):(2*nX/3)] , ysimu[,(2*nX/3+1):nX],nboot)
STot[[i]] <- sobolT2007pickfreeze(ysimu[,1:(nX/3)] ,ysimu[,(nX/3+1):(2*nX/3)] ,nboot)
if(sequential||sequential.tot){
predCov <- predictGP.sobol(object = model, newdata1=data.frame(X), newdata2=data.frame(candidate), type=type)
for(k in 1:ncandidate){
ynew <- predCov$mean[k]
zsimu <- t(as.matrix(predCov$cov[-c((nX+1):(nX+ncandidate)),k])%*%rep(1,nsim))*(ynew-ysimu[,(nX+k)])/predCov$cov[(nX+k),k]+ysimu[,-c((nX+1):(nX+ncandidate))]
if(sequential){
Scand <- sobol2007pickfreeze(zsimu[,1:(nX/3)] , zsimu[,(nX/3+1):(2*nX/3)] , zsimu[,(2*nX/3+1):nX],nboot=1)
Svar[k,i] <- var(Scand)
rm(list=c("Scand"))
}
if(sequential.tot){
STotcand <- sobolT2007pickfreeze(zsimu[,1:(nX/3)] ,zsimu[,(nX/3+1):(2*nX/3)], nboot=1 )
STotvar[k,i] <- var(STotcand)
rm(list=c("STotcand"))
}
rm(list=c("zsimu"))
}
rm(list=c("predCov", "ynew"))
}
rm(list=c("ysimu"))
}
}
namesS <- c()
for (i in 1:p){
namesS <- c(namesS,paste("S",i,sep=""))
}
names(S) <- namesS
if(Tot){
namesStot <- c()
for (i in 1:p){
namesStot <- c(namesStot,paste("T",i,sep=""))
}
names(STot) <- namesStot
}
output$S <- S
rm(list=c("S"))
if(sequential){
SumVar <- apply(Svar,1,sum)
output$S$xnew <- candidate[which.min(SumVar),]
output$S$xnewi <- which.min(SumVar)
rm(list=c("Svar","SumVar"))
}
if(Tot){
output$T <- STot
rm(list=c("STot"))
if(sequential.tot){
SumVar <- apply(STotvar,1,sum)
output$T$xnew <- candidate[which.min(SumVar),]
output$T$xnewi <- which.min(SumVar)
rm(list=c("STotvar","SumVar"))
}
}
class(output) <- "sobolGP"
output$S$mean <- matrix(nrow = 1, ncol = p)
output$S$var <- matrix(nrow = 1, ncol = p)
output$S$ci <- matrix(nrow = 2, ncol = p)
output$S$varPG <- matrix(nrow = 1, ncol = p)
output$S$varMC <- matrix(nrow = 1, ncol = p)
if(Tot){
output$T$mean <- matrix(nrow = 1, ncol = p)
output$T$var <- matrix(nrow = 1, ncol = p)
output$T$ci <- matrix(nrow = 2, ncol = p)
output$T$varPG <- matrix(nrow = 1, ncol = p)
output$T$varMC <- matrix(nrow = 1, ncol = p)
}
for(i in 1:p){
output$S$mean[1,i] <- mean(as.numeric(output$S[[i]]))
output$S$var[1,i] <- var(as.numeric(output$S[[i]]))
output$S$ci[1,i] <- quantile(as.numeric(output$S[[i]]), (1-conf)/2)
output$S$ci[2,i] <- quantile(as.numeric(output$S[[i]]), (1+conf)/2)
if(nboot==1){
output$S$varPG[1,i] <- var(output$S[[i]])
} else {
output$S$varPG[1,i] <- mean(apply(output$S[[i]],1,var))
output$S$varMC[1,i] <- mean(apply(output$S[[i]],2,var))
}
if(Tot){
output$T$mean[1,i] <- mean(as.numeric(output$T[[i]]))
output$T$var[1,i] <- var(as.numeric(output$T[[i]]))
output$T$ci[1,i] <- quantile(as.numeric(output$T[[i]]), (1-conf)/2)
output$T$ci[2,i] <- quantile(as.numeric(output$T[[i]]), (1+conf)/2)
if(nboot==1){
output$T$varPG[1,i] <- var(output$T[[i]])
} else {
output$T$varPG[1,i] <- mean(apply(output$T[[i]],1,var))
output$T$varMC[1,i] <- mean(apply(output$T[[i]],2,var))
}
}
}
output$call$tot <- Tot
output$call$method <- MCmethod
output$call$type <- type
output$call$nsim <- nsim
return(output)
}
###############################################################################
plot.sobolGP <- function(x, ...){
sGP <- x
dev.new(width = 12, height = 5)
if(sGP$call$tot){
par(mfrow=c(1,3))
} else {
par(mfrow=c(1,2))
}
d <- length(sGP$S$mean)
ylim=c(-0.05,1.05)
xlim=c(1,d+0.1)
plot(1:d,sGP$S$mean, xlim = xlim, ylim = ylim, axes = FALSE,
xlab = "", ylab = "", pch = 21)
axis(side = 1, at = 1:d, labels = 1:d)
axis(side = 2, at = seq(0,1,0.1), labels = seq(0,1,0.1))
segments(1:d,sGP$S$ci[1,],1:d,sGP$S$ci[2,])
box()
if(sGP$call$tot){
points((1:d)+0.1,sGP$T$mean, pch = 24)
legend("topright", legend = c("main effect", "total effect"), pch = c(21,24))
segments(1:d+0.1,sGP$T$ci[1,],1:d+0.1,sGP$T$ci[2,],lty=2)
} else {
legend("topright", legend = c("main effect"), pch = 21)
}
abline(0,0)
nboot <- sGP$call$nboot
if(nboot!=1){
data <- data.frame(rbind(sGP$S$varPG,sGP$S$varMC))
names(data) <- names(sGP$S)[1:d]
barplot(as.matrix(data), angle = c(45,-45), density = 20, col = "black",
legend = c("GP variance","MC variance"),ylim=c(0,1.2*max(sGP$S$varPG+sGP$S$varMC)))
title(main = list("Variance decomposition of the main effects", font = 4))
if(sGP$call$tot){
data <- data.frame(rbind(sGP$T$varPG,sGP$T$varMC))
names(data) <- names(sGP$T)[1:d]
barplot(as.matrix(data), angle = c(45,-45), density = 20, col = "black",
legend = c("GP variance","MC variance"),ylim=c(0,1.2*max(sGP$T$varPG+sGP$T$varMC)))
title(main = list("Variance decomposition of the total effects", font = 4))
}
} else {
data <- data.frame(sGP$S$varPG)
names(data) <- names(sGP$S)[1:d]
barplot(as.matrix(data), angle = 45, density = 20, col = "black",
legend = c("GP variance"),ylim=c(0,1.2*max(sGP$S$varPG)))
title(main = list("Variance of the main effects", font = 4))
if(sGP$call$tot){
data <- data.frame(sGP$T$varPG)
names(data) <- names(sGP$T)[1:d]
barplot(as.matrix(data), angle = 45, density = 20, col = "black",
legend = c("GP variance"),ylim=c(0,1.2*max(sGP$T$varPG)))
title(main = list("Variance of the total effects", font = 4))
}
}
}
######################################
print.sobolGP <- function(x, ...){
sGP <- x
cat("\nMethod: ", sGP$call$method, "\n", sep = "")
cat("\nModel runs:", sGP$call$model@n, "\n")
cat("\nNumber of GP realizations:", sGP$call$nsim, "\n")
cat("\nKriging type:", sGP$call$type, "\n")
d <- length(sGP$S$mean)
df=data.frame(pointEstimate=t(sGP$S$mean), stdErr=t(sqrt(sGP$S$var)), minCI=as.matrix(sGP$S$ci[1,]), maxCI=as.matrix(sGP$S$ci[2,]))
colnames(df)=c("estimate","std. error","min. c.i.","max. c.i.")
rownames(df)=names(sGP$S)[1:d]
cat("\n")
print(df)
cat("\n")
if(sGP$call$tot){
df=data.frame(pointEstimate=t(sGP$T$mean), stdErr=t(sqrt(sGP$T$var)), minCI=as.matrix(sGP$T$ci[1,]), maxCI=as.matrix(sGP$T$ci[2,]))
colnames(df)=c("estimate","std. error","min. c.i.","max. c.i.")
rownames(df)=names(sGP$T)[1:d]
print(df)
cat("\n")
}
}
######################################
ask.sobolGP <- function(x, tot=FALSE, ...){
sGP <- x
output <- c()
if(is.null(sGP$S$xnew)){
stop("The function sobolGP must be run with the argument sequential = TRUE")
}
if(!tot){
output <- sGP$S$xnew
} else {
if(is.null(sGP$T$xnew)){
stop("The function sobolGP must be run with the argument sequential.tot = TRUE")
}
output <- sGP$T$xnew
}
d <- dim(sGP$call$candidate)[2]
return(output)
}
######################################
tell.sobolGP <- function(x, y=NULL, xpoint=NULL, newcandidate=NULL, ...){
sGP <- x
xx <- xpoint
if(is.null(newcandidate)){
d <- dim(sGP$call$candidate)[2]
test = c(sGP$call$candidate[,1] == xx[,1])
for(i in 2:d){
test <- test*(sGP$call$candidate[,i] == xx[,i])
}
newcandidate <- sGP$call$candidate[-which(test == 1),]
}
xx <- as.matrix(xx)
if(dim(xx)[2]==1){
xx <- t(xx)
}
rownames(xx) = NULL
m <- update(object = sGP$call$model, newX = xx, newy = y, cov.reestim = FALSE, trend.reestim = FALSE)
res <- sobolGP(
model = m,
type = sGP$call$type,
MCmethod = sGP$call$method,
X1 = sGP$call$X1,
X2 = sGP$call$X2,
nsim = sGP$call$nsim,
conf = sGP$call$conf,
nboot = sGP$call$nboot,
sequential = sGP$call$sequential,
candidate = newcandidate,
sequential.tot = sGP$call$sequential.tot,
max_iter = sGP$call$max_iter)
return(res)
}
######################################
predictGP.sobol <- function (
object,
newdata1 ,
newdata2,
type,
bias.correct = FALSE,
checkNames = FALSE,
prednewdata1 = FALSE,
prednewdata2 = TRUE)
{
nugget.flag <- object@covariance@nugget.flag
X <- object@X
y <- object@y
if (checkNames) {
newdata1 <- checkNames(X1 = X, X2 = newdata1, X1.name = "the design",
X2.name = "newdata")
newdata2 <- checkNames(X1 = X, X2 = newdata2, X1.name = "the design",
X2.name = "newdata")
} else {
newdata1 <- as.matrix(newdata1)
newdata2 <- as.matrix(newdata2)
d.newdata <- ncol(newdata1)
if (!identical(d.newdata, object@d)) {
stop("newdata must have the same numbers of columns than the experimental design")
}
if (!identical(colnames(newdata1), colnames(X))) {
colnames(newdata1) <- colnames(X)
}
if (!identical(colnames(newdata2), colnames(X))) {
colnames(newdata2) <- colnames(X)
}
}
T <- object@T
z <- object@z
M <- object@M
beta <- object@trend.coef
F.newdata1 <- model.matrix(object@trend.formula, data = data.frame(newdata1))
F.newdata2 <- model.matrix(object@trend.formula, data = data.frame(newdata2))
if(prednewdata1){
y.predict.trend1 <- F.newdata1 %*% beta
}
if(prednewdata2){
y.predict.trend2 <- F.newdata2 %*% beta
}
if (requireNamespace("DiceKriging", quietly = TRUE)){
c.newdata1 <- DiceKriging::covMat1Mat2(object@covariance, X1 = X, X2 = newdata1,
nugget.flag = object@covariance@nugget.flag)
c.newdata2 <- DiceKriging::covMat1Mat2(object@covariance, X1 = X, X2 = newdata2,
nugget.flag = object@covariance@nugget.flag)
}
Tinv.c.newdata1 <- backsolve(t(T), c.newdata1, upper.tri = FALSE)
Tinv.c.newdata2 <- backsolve(t(T), c.newdata2, upper.tri = FALSE)
output.list <- list()
if(prednewdata1){
y.predict.complement1 <- t(Tinv.c.newdata1) %*% z
y.predict1 <- y.predict.trend1 + y.predict.complement1
rm(list=c("y.predict.complement1","y.predict.trend1"))
y.predict1 <- as.numeric(y.predict1)
output.list$mean1 <- y.predict1
rm(list=c("y.predict1"))
}
if(prednewdata2){
y.predict.complement2 <- t(Tinv.c.newdata2) %*% z
y.predict2 <- y.predict.trend2 + y.predict.complement2
rm(list=c("y.predict.complement2","y.predict.trend2"))
y.predict2 <- as.numeric(y.predict2)
output.list$mean2 <- y.predict2
rm(list=c("y.predict2"))
}
if (requireNamespace("DiceKriging", quietly = TRUE)){
C.newdata <- DiceKriging::covMat1Mat2(object@covariance, X1 = newdata1, X2 = newdata2,
nugget.flag = object@covariance@nugget.flag)
}
rm(list=c("newdata1","newdata2"))
cond.cov <- C.newdata - crossprod(Tinv.c.newdata1,Tinv.c.newdata2 )
if (type == "UK") {
T.M <- chol(t(M) %*% M)
s2.predict.mat1 <- backsolve(t(T.M), t(F.newdata1 -
t(Tinv.c.newdata1) %*% M), upper.tri = FALSE)
s2.predict.mat2 <- backsolve(t(T.M), t(F.newdata2 -
t(Tinv.c.newdata2) %*% M), upper.tri = FALSE)
cond.cov <- cond.cov + crossprod(s2.predict.mat1,s2.predict.mat2)
if (bias.correct)
cond.cov <- cond.cov * object@n/(object@n - object@p)
}
output.list$cov <- cond.cov
return(output.list)
}
######################################
simulateGP.sobol <- function(
object,
nsim = 100,
newdata = NULL,
cond = TRUE,
checkNames = FALSE,
max_iter = 1000,
type
) {
if (!is.logical(cond)) stop("'cond' must be TRUE/FALSE")
if ((!is.null(newdata)) && (checkNames)) {
newdata <- checkNames(X1 = object@X, X2 = newdata, X1.name = "the design", X2.name = "newdata")
}
if (is.null(newdata)) {
newdata <- object@X
F.newdata <- object@F
} else {
newdata <- as.matrix(newdata)
m <- nrow(newdata)
if (!identical(ncol(newdata), object@d))
stop("newdata must have the same numbers of columns than the experimental design")
if (!identical(colnames(newdata), colnames(object@X))) {
colnames(newdata) <- colnames(object@X)
}
newdata <- rbind(newdata, object@X)
row.names(newdata) <- NULL
F.newdata <- model.matrix(object@trend.formula, data = data.frame(newdata))
}
n <- object@n
# non conditional simulations
sample_test <- sample.int(m,min(100,m))
iter <- 1
test <- FALSE
yc <- matrix(0,nsim,m+n)
#RMSE_save <- c() #-!!-#
while((!test&iter<1000)){
a <- sample.int(m+n, size = nsim, replace = FALSE, prob = NULL)
ya <- rnorm(nsim,0,1)
if (requireNamespace("DiceKriging", quietly = TRUE)){
Cab <- DiceKriging::covMat1Mat2(object@covariance, X1 = as.matrix(newdata[a,]), X2 = as.matrix(newdata),
nugget.flag = object@covariance@nugget.flag)/object@covariance@sd2
}
yc <- yc + Cab* (as.matrix(ya-diag(yc[,a]))%*%t(as.matrix(rep(1,m+n))))
diag(yc[1:nsim,a]) <- ya
if(iter%%20==0){
Cemp <- var(yc[,sample_test])
Ctheo <- DiceKriging::covMatrix(object@covariance, X = as.matrix(newdata[sample_test,]))[[1]]/
object@covariance@sd2
RMSE_C <- sqrt(mean((Cemp-Ctheo)^2))
#RMSE_save <- c(RMSE_save,RMSE_C) #-!!-#
if(RMSE_C < 0.1){test <- TRUE}
}
iter <- iter+1
}
rm(list=c("Cemp","Ctheo","ya" ,"Cab","a" ,"sample_test"))
yc <- yc*sqrt(object@covariance@sd2*object@n/(object@n-object@p))
if (cond){
if(type=="UK"){
y.predict.trend <- F.newdata %*% object@trend.coef
if (requireNamespace("DiceKriging", quietly = TRUE)){
c.newdata <- DiceKriging::covMat1Mat2(object@covariance, X1 = object@X, X2 = newdata,
nugget.flag = object@covariance@nugget.flag)
}
rm(list=c("newdata"))
Tinv.c.newdata <- backsolve(t(object@T), c.newdata, upper.tri = FALSE)
rm(list=c("c.newdata"))
y.predict.complement <- t(Tinv.c.newdata) %*% object@z
y.predict <- y.predict.trend + y.predict.complement
rm(list=c("y.predict.complement"))
y.predict <- as.numeric(y.predict)
ZD <- t(yc[,(m+1):(m+n)])
if(!identical(object@noise.var,numeric(0))){
ZD <- ZD + matrix(rnorm(nsim*n,0,1),nrow=n)*matrix(rep(sqrt(object@noise.var),nsim),ncol=nsim)
}
xtilde <- backsolve(t(object@T), ZD, upper.tri = FALSE)
Betatilde <- solve(t(object@M)%*%object@M )%*%t(object@M)%*%xtilde
rm(list=c("xtilde"))
ztilde <- backsolve(t(object@T), ZD - object@F%*%Betatilde ,
upper.tri = FALSE)
rm(list=c("ZD"))
ytilde.predict.complement <- t(Tinv.c.newdata) %*% ztilde
rm(list=c("ztilde","Tinv.c.newdata"))
ytilde.predict.trend <- F.newdata %*% Betatilde
rm(list=c("F.newdata","Betatilde"))
ytilde.predict <- ytilde.predict.trend + ytilde.predict.complement
rm(list=c("ytilde.predict.trend","ytilde.predict.complement","y.predict.trend"))
yc <- t(matrix(rep(y.predict,nsim),ncol=nsim)) - t(ytilde.predict) +yc
rm(list=c("y.predict","ytilde.predict"))
}
if(type=="SK"){
y.predict.trend <- F.newdata %*% object@trend.coef
if (requireNamespace("DiceKriging", quietly = TRUE)){
c.newdata <- DiceKriging::covMat1Mat2(object@covariance, X1 = object@X, X2 = newdata,
nugget.flag = object@covariance@nugget.flag)
}
Tinv.c.newdata <- backsolve(t(object@T), c.newdata, upper.tri = FALSE)
y.predict.complement <- t(Tinv.c.newdata) %*% object@z
y.predict <- y.predict.trend + y.predict.complement
y.predict <- as.numeric(y.predict)
ZD <- t(yc[,(m+1):(m+n)])
if(!identical(object@noise.var,numeric(0))){
ZD <- ZD + matrix(rnorm(nsim*n,0,1),nrow=n)*matrix(rep(sqrt(object@noise.var),nsim),ncol=nsim)
}
ztilde <- backsolve(t(object@T), ZD , upper.tri = FALSE)
ytilde.predict <- t(Tinv.c.newdata) %*% ztilde
rm(list=c("ztilde","c.newdata","Tinv.c.newdata","newdata","y.predict.trend","F.newdata","ZD"))
yc <- t(matrix(rep(y.predict,nsim),ncol=nsim)) - t(ytilde.predict) +yc
rm(list=c("y.predict","ytilde.predict"))
}
}
return(yc[,-c((m+1):(m+n))])
}
##############################################################################
sobolpickfreeze <- function(y1,y2,nboot){
output <- (apply(y1*y2,1,mean) - apply(y1,1,mean)*apply(y2,1,mean))/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(apply(y1[,b]*y2[,b],1,mean) - apply(y1[,b],1,mean)*apply(y2[,b],1,mean))/apply(y1[,b],1,var))
}
}
return(output)
}
soboljansenpickfreeze <- function(y1,y2,nboot){
varY <- apply(cbind(y1,y2),1,var)
output <- (varY-apply((y1-y2)^2,1,mean)/2)/varY
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
varY <- apply(cbind(y1[,b],y2[,b]),1,var)
output <- rbind(output,(varY-apply((y1[,b]-y2[,b])^2,1,mean)/2)/varY)
}
}
return(output)
}
sobolEffpickfreeze <- function(y1,y2,nboot){
output <- (apply(y1*y2,1,mean) - apply(cbind(y1,y2),1,mean)^2)/apply(cbind(y1,y2),1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(apply(y1[,b]*y2[,b],1,mean) - apply(cbind(y1[,b],y2[,b]),1,mean)^2)/apply(cbind(y1[,b],y2[,b]),1,var))
}
}
return(output)
}
sobolT2002pickfreeze <- function(y1,y2,nboot){
output <- (1-(apply(y1*y2,1,mean) - apply(y1,1,mean)^2)/apply(y1,1,var))
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(1-(apply(y1[,b]*y2[,b],1,mean) - apply(y1[,b],1,mean)^2)/apply(y1[,b],1,var)))
}
}
return(output)
}
sobolTjansenpickfreeze <- function(y1,y2,nboot){
output <- (apply((y1-y2)^2,1,mean)/2)/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(apply((y1[,b]-y2[,b])^2,1,mean)/2)/apply(y1[,b],1,var))
}
}
return(output)
}
sobol2007pickfreeze <- function(y1,y2,y3,nboot){
output <- apply(y3*(y2-y1),1,mean)/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,apply(y3[,b]*(y2[,b]-y1[,b]),1,mean)/apply(y1[,b],1,var))
}
}
return(output)
}
sobolT2007pickfreeze <- function(y1,y2,nboot){
output <- apply(y1*(y1-y2),1,mean)/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,apply(y1[,b]*(y1[,b]-y2[,b]),1,mean)/apply(y1[,b],1,var))
}
}
return(output)
}
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Defines functions sobolT2007pickfreeze sobol2007pickfreeze sobolTjansenpickfreeze sobolT2002pickfreeze sobolEffpickfreeze soboljansenpickfreeze sobolpickfreeze simulateGP.sobol tell.sobolGP ask.sobolGP print.sobolGP plot.sobolGP
Documented in ask.sobolGP plot.sobolGP print.sobolGP tell.sobolGP
# Kriging-based global sensitivity analysis taking into account both
# the meta-model and the Monte-Carlo errors
# Author : loic le Gratiet, 2014
sobolGP <- function (
model,
type="SK",
MCmethod="sobol",
X1,
X2,
nsim=100,
nboot=1,
conf = 0.95,
sequential = FALSE,
candidate = NULL,
sequential.tot=FALSE,
max_iter = 1000
)
{
if(sequential){
ncandidate <- dim(candidate)[1]
dcandidate <- dim(candidate)[2]
}
if ((ncol(X1) != ncol(X2)) | (nrow(X1) != nrow(X2)))
stop("The samples X1 and X2 must have the same dimensions")
if(sequential){
if(is.null(candidate)){
stop("The number of candidate points must be greater than zero")
}
if (ncol(X1) != ncol(candidate)){
stop("The candidate points, X1 and X2 must have the same dimensions")
}
}
if(MCmethod=="sobol"||MCmethod=="sobolEff"){
if(sequential.tot){
stop("Sequential design for total indices is only available for sobol2002, sobol2007 and soboljansen methods")
}
}
p <- ncol(X1)
S <- list()
if(sequential){
Svar <- matrix(nrow = ncandidate, ncol = p)
}
if(sequential.tot){
STotvar <- matrix(nrow = ncandidate, ncol = p)
}
output <- list()
output$call$X1 <- X1
output$call$X2 <- X2
output$call$conf <- conf
output$call$nboot <- nboot
output$call$candidate <- candidate
output$call$sequential <- sequential
output$call$max_iter <- max_iter
output$call$sequential.tot <- sequential.tot
output$call$model <- model
Tot=FALSE
if(MCmethod=="sobol2002"||MCmethod=="sobol2007"||MCmethod=="soboljansen") Tot=TRUE
if(MCmethod!="sobol2007"){
for (i in 1:p) {
Xb <- X2
Xb[, i] <- X1[, i]
X <- rbind(X1, Xb)
nX <- dim(X)[1]
if(sequential){
X <- rbind(X, data.frame(candidate))
}
rm(list=c("Xb"))
ysimu <- simulateGP.sobol(object = model, nsim = nsim, newdata=X,
cond=TRUE, checkNames=FALSE, max_iter=1000,type)
if(MCmethod=="sobol"||MCmethod=="sobol2002"){
S[[i]] <- sobolpickfreeze(ysimu[,1:(nX/2)] , ysimu[,(nX/2+1):nX],nboot)
}
if(MCmethod=="sobolEff"){
S[[i]] <- sobolEffpickfreeze(ysimu[,1:(nX/2)] , ysimu[,(nX/2+1):nX],nboot)
}
if(MCmethod=="soboljansen"){
S[[i]] <- soboljansenpickfreeze(ysimu[,1:(nX/2)] , ysimu[,(nX/2+1):nX],nboot)
}
if(sequential){
predCov <- predictGP.sobol(object = model, newdata1=data.frame(X), newdata2=data.frame(candidate), type=type, prednewdata1 = FALSE, prednewdata2 = TRUE)
for(k in 1:ncandidate){
ynew <- predCov$mean2[k]
zsimu <- t(as.matrix(predCov$cov[-c((nX+1):(nX+ncandidate)),k])%*%rep(1,nsim))*(ynew-ysimu[,(nX+k)])/predCov$cov[(nX+k),k]+ysimu[,-c((nX+1):(nX+ncandidate))]
if(MCmethod=="sobol"||MCmethod=="sobol2002"){
Scand <- sobolpickfreeze(zsimu[,1:(nX/2)] , zsimu[,(nX/2+1):nX],nboot=1)
}
if(MCmethod=="sobolEff"){
Scand <- sobolEffpickfreeze(zsimu[,1:(nX/2)] , zsimu[,(nX/2+1):nX],nboot=1)
}
if(MCmethod=="soboljansen"){
Scand <- soboljansenpickfreeze(zsimu[,1:(nX/2)] , zsimu[,(nX/2+1):nX],nboot=1)
}
rm(list=c("zsimu"))
Svar[k,i] <- var(Scand)
rm(list=c("Scand"))
}
rm(list=c("predCov", "ynew"))
}
rm(list=c("ysimu" ))
}
if(Tot){
STot <- list()
for (i in 1:p) {
Xb <- X1
Xb[, i] <- X2[, i]
X <- rbind(X1, Xb)
nX <- dim(X)[1]
if(sequential.tot){
X <- rbind(X, data.frame(candidate))
}
rm(list=c("Xb"))
ysimu <- simulateGP.sobol(object = model, nsim = nsim, newdata=X,
cond=TRUE, checkNames=FALSE, max_iter=1000,type)
if(MCmethod=="sobol2002"){
STot[[i]] <- sobolT2002pickfreeze(ysimu[,1:(nX/2)],ysimu[,(nX/2+1):nX],nboot)
}
if(MCmethod=="soboljansen"){
STot[[i]] <- sobolTjansenpickfreeze(ysimu[,1:(nX/2)],ysimu[,(nX/2+1):nX],nboot)
}
if(sequential.tot){
predCov <- predictGP.sobol(object = model, newdata1=data.frame(X), newdata2=data.frame(candidate), type=type)
for(k in 1:ncandidate){
ynew <- predCov$mean[k]
zsimu <- t(as.matrix(predCov$cov[-c((nX+1):(nX+ncandidate)),k])%*%rep(1,nsim))*(ynew-ysimu[,(nX+k)])/predCov$cov[(nX+k),k]+ysimu[,-c((nX+1):(nX+ncandidate))]
if(MCmethod=="sobol2002"){
STotcand <- sobolT2002pickfreeze(zsimu[,1:(nX/2)],zsimu[,(nX/2+1):nX],nboot=1)
}
if(MCmethod=="soboljansen"){
STotcand <- sobolTjansenpickfreeze(zsimu[,1:(nX/2)],zsimu[,(nX/2+1):nX],nboot=1)
}
rm(list=c("zsimu"))
STotvar[k,i] <- var(STotcand)
rm(list=c("STotcand"))
}
rm(list=c("predCov", "ynew"))
}
rm(list=c("ysimu" ))
}
}
rm(list=c("X", "X1","X2"))
}
if(MCmethod=="sobol2007"){
STot <- list()
for (i in 1:p) {
Xb <- X1
Xb[, i] <- X2[, i]
X <- rbind(X1, Xb, X2)
nX <- dim(X)[1]
if(sequential||sequential.tot){
X <- rbind(X, data.frame(candidate))
}
rm(list=c("Xb"))
ysimu <- simulateGP.sobol(object = model, nsim = nsim, newdata=X,
cond=TRUE, checkNames=FALSE, max_iter=1000,type)
S[[i]] <- sobol2007pickfreeze(ysimu[,1:(nX/3)] , ysimu[,(nX/3+1):(2*nX/3)] , ysimu[,(2*nX/3+1):nX],nboot)
STot[[i]] <- sobolT2007pickfreeze(ysimu[,1:(nX/3)] ,ysimu[,(nX/3+1):(2*nX/3)] ,nboot)
if(sequential||sequential.tot){
predCov <- predictGP.sobol(object = model, newdata1=data.frame(X), newdata2=data.frame(candidate), type=type)
for(k in 1:ncandidate){
ynew <- predCov$mean[k]
zsimu <- t(as.matrix(predCov$cov[-c((nX+1):(nX+ncandidate)),k])%*%rep(1,nsim))*(ynew-ysimu[,(nX+k)])/predCov$cov[(nX+k),k]+ysimu[,-c((nX+1):(nX+ncandidate))]
if(sequential){
Scand <- sobol2007pickfreeze(zsimu[,1:(nX/3)] , zsimu[,(nX/3+1):(2*nX/3)] , zsimu[,(2*nX/3+1):nX],nboot=1)
Svar[k,i] <- var(Scand)
rm(list=c("Scand"))
}
if(sequential.tot){
STotcand <- sobolT2007pickfreeze(zsimu[,1:(nX/3)] ,zsimu[,(nX/3+1):(2*nX/3)], nboot=1 )
STotvar[k,i] <- var(STotcand)
rm(list=c("STotcand"))
}
rm(list=c("zsimu"))
}
rm(list=c("predCov", "ynew"))
}
rm(list=c("ysimu"))
}
}
namesS <- c()
for (i in 1:p){
namesS <- c(namesS,paste("S",i,sep=""))
}
names(S) <- namesS
if(Tot){
namesStot <- c()
for (i in 1:p){
namesStot <- c(namesStot,paste("T",i,sep=""))
}
names(STot) <- namesStot
}
output$S <- S
rm(list=c("S"))
if(sequential){
SumVar <- apply(Svar,1,sum)
output$S$xnew <- candidate[which.min(SumVar),]
output$S$xnewi <- which.min(SumVar)
rm(list=c("Svar","SumVar"))
}
if(Tot){
output$T <- STot
rm(list=c("STot"))
if(sequential.tot){
SumVar <- apply(STotvar,1,sum)
output$T$xnew <- candidate[which.min(SumVar),]
output$T$xnewi <- which.min(SumVar)
rm(list=c("STotvar","SumVar"))
}
}
class(output) <- "sobolGP"
output$S$mean <- matrix(nrow = 1, ncol = p)
output$S$var <- matrix(nrow = 1, ncol = p)
output$S$ci <- matrix(nrow = 2, ncol = p)
output$S$varPG <- matrix(nrow = 1, ncol = p)
output$S$varMC <- matrix(nrow = 1, ncol = p)
if(Tot){
output$T$mean <- matrix(nrow = 1, ncol = p)
output$T$var <- matrix(nrow = 1, ncol = p)
output$T$ci <- matrix(nrow = 2, ncol = p)
output$T$varPG <- matrix(nrow = 1, ncol = p)
output$T$varMC <- matrix(nrow = 1, ncol = p)
}
for(i in 1:p){
output$S$mean[1,i] <- mean(as.numeric(output$S[[i]]))
output$S$var[1,i] <- var(as.numeric(output$S[[i]]))
output$S$ci[1,i] <- quantile(as.numeric(output$S[[i]]), (1-conf)/2)
output$S$ci[2,i] <- quantile(as.numeric(output$S[[i]]), (1+conf)/2)
if(nboot==1){
output$S$varPG[1,i] <- var(output$S[[i]])
} else {
output$S$varPG[1,i] <- mean(apply(output$S[[i]],1,var))
output$S$varMC[1,i] <- mean(apply(output$S[[i]],2,var))
}
if(Tot){
output$T$mean[1,i] <- mean(as.numeric(output$T[[i]]))
output$T$var[1,i] <- var(as.numeric(output$T[[i]]))
output$T$ci[1,i] <- quantile(as.numeric(output$T[[i]]), (1-conf)/2)
output$T$ci[2,i] <- quantile(as.numeric(output$T[[i]]), (1+conf)/2)
if(nboot==1){
output$T$varPG[1,i] <- var(output$T[[i]])
} else {
output$T$varPG[1,i] <- mean(apply(output$T[[i]],1,var))
output$T$varMC[1,i] <- mean(apply(output$T[[i]],2,var))
}
}
}
output$call$tot <- Tot
output$call$method <- MCmethod
output$call$type <- type
output$call$nsim <- nsim
return(output)
}
###############################################################################
plot.sobolGP <- function(x, ...){
sGP <- x
dev.new(width = 12, height = 5)
if(sGP$call$tot){
par(mfrow=c(1,3))
} else {
par(mfrow=c(1,2))
}
d <- length(sGP$S$mean)
ylim=c(-0.05,1.05)
xlim=c(1,d+0.1)
plot(1:d,sGP$S$mean, xlim = xlim, ylim = ylim, axes = FALSE,
xlab = "", ylab = "", pch = 21)
axis(side = 1, at = 1:d, labels = 1:d)
axis(side = 2, at = seq(0,1,0.1), labels = seq(0,1,0.1))
segments(1:d,sGP$S$ci[1,],1:d,sGP$S$ci[2,])
box()
if(sGP$call$tot){
points((1:d)+0.1,sGP$T$mean, pch = 24)
legend("topright", legend = c("main effect", "total effect"), pch = c(21,24))
segments(1:d+0.1,sGP$T$ci[1,],1:d+0.1,sGP$T$ci[2,],lty=2)
} else {
legend("topright", legend = c("main effect"), pch = 21)
}
abline(0,0)
nboot <- sGP$call$nboot
if(nboot!=1){
data <- data.frame(rbind(sGP$S$varPG,sGP$S$varMC))
names(data) <- names(sGP$S)[1:d]
barplot(as.matrix(data), angle = c(45,-45), density = 20, col = "black",
legend = c("GP variance","MC variance"),ylim=c(0,1.2*max(sGP$S$varPG+sGP$S$varMC)))
title(main = list("Variance decomposition of the main effects", font = 4))
if(sGP$call$tot){
data <- data.frame(rbind(sGP$T$varPG,sGP$T$varMC))
names(data) <- names(sGP$T)[1:d]
barplot(as.matrix(data), angle = c(45,-45), density = 20, col = "black",
legend = c("GP variance","MC variance"),ylim=c(0,1.2*max(sGP$T$varPG+sGP$T$varMC)))
title(main = list("Variance decomposition of the total effects", font = 4))
}
} else {
data <- data.frame(sGP$S$varPG)
names(data) <- names(sGP$S)[1:d]
barplot(as.matrix(data), angle = 45, density = 20, col = "black",
legend = c("GP variance"),ylim=c(0,1.2*max(sGP$S$varPG)))
title(main = list("Variance of the main effects", font = 4))
if(sGP$call$tot){
data <- data.frame(sGP$T$varPG)
names(data) <- names(sGP$T)[1:d]
barplot(as.matrix(data), angle = 45, density = 20, col = "black",
legend = c("GP variance"),ylim=c(0,1.2*max(sGP$T$varPG)))
title(main = list("Variance of the total effects", font = 4))
}
}
}
######################################
print.sobolGP <- function(x, ...){
sGP <- x
cat("\nMethod: ", sGP$call$method, "\n", sep = "")
cat("\nModel runs:", sGP$call$model@n, "\n")
cat("\nNumber of GP realizations:", sGP$call$nsim, "\n")
cat("\nKriging type:", sGP$call$type, "\n")
d <- length(sGP$S$mean)
df=data.frame(pointEstimate=t(sGP$S$mean), stdErr=t(sqrt(sGP$S$var)), minCI=as.matrix(sGP$S$ci[1,]), maxCI=as.matrix(sGP$S$ci[2,]))
colnames(df)=c("estimate","std. error","min. c.i.","max. c.i.")
rownames(df)=names(sGP$S)[1:d]
cat("\n")
print(df)
cat("\n")
if(sGP$call$tot){
df=data.frame(pointEstimate=t(sGP$T$mean), stdErr=t(sqrt(sGP$T$var)), minCI=as.matrix(sGP$T$ci[1,]), maxCI=as.matrix(sGP$T$ci[2,]))
colnames(df)=c("estimate","std. error","min. c.i.","max. c.i.")
rownames(df)=names(sGP$T)[1:d]
print(df)
cat("\n")
}
}
######################################
ask.sobolGP <- function(x, tot=FALSE, ...){
sGP <- x
output <- c()
if(is.null(sGP$S$xnew)){
stop("The function sobolGP must be run with the argument sequential = TRUE")
}
if(!tot){
output <- sGP$S$xnew
} else {
if(is.null(sGP$T$xnew)){
stop("The function sobolGP must be run with the argument sequential.tot = TRUE")
}
output <- sGP$T$xnew
}
d <- dim(sGP$call$candidate)[2]
return(output)
}
######################################
tell.sobolGP <- function(x, y=NULL, xpoint=NULL, newcandidate=NULL, ...){
sGP <- x
xx <- xpoint
if(is.null(newcandidate)){
d <- dim(sGP$call$candidate)[2]
test = c(sGP$call$candidate[,1] == xx[,1])
for(i in 2:d){
test <- test*(sGP$call$candidate[,i] == xx[,i])
}
newcandidate <- sGP$call$candidate[-which(test == 1),]
}
xx <- as.matrix(xx)
if(dim(xx)[2]==1){
xx <- t(xx)
}
rownames(xx) = NULL
m <- update(object = sGP$call$model, newX = xx, newy = y, cov.reestim = FALSE, trend.reestim = FALSE)
res <- sobolGP(
model = m,
type = sGP$call$type,
MCmethod = sGP$call$method,
X1 = sGP$call$X1,
X2 = sGP$call$X2,
nsim = sGP$call$nsim,
conf = sGP$call$conf,
nboot = sGP$call$nboot,
sequential = sGP$call$sequential,
candidate = newcandidate,
sequential.tot = sGP$call$sequential.tot,
max_iter = sGP$call$max_iter)
return(res)
}
######################################
predictGP.sobol <- function (
object,
newdata1 ,
newdata2,
type,
bias.correct = FALSE,
checkNames = FALSE,
prednewdata1 = FALSE,
prednewdata2 = TRUE)
{
nugget.flag <- object@covariance@nugget.flag
X <- object@X
y <- object@y
if (checkNames) {
newdata1 <- checkNames(X1 = X, X2 = newdata1, X1.name = "the design",
X2.name = "newdata")
newdata2 <- checkNames(X1 = X, X2 = newdata2, X1.name = "the design",
X2.name = "newdata")
} else {
newdata1 <- as.matrix(newdata1)
newdata2 <- as.matrix(newdata2)
d.newdata <- ncol(newdata1)
if (!identical(d.newdata, object@d)) {
stop("newdata must have the same numbers of columns than the experimental design")
}
if (!identical(colnames(newdata1), colnames(X))) {
colnames(newdata1) <- colnames(X)
}
if (!identical(colnames(newdata2), colnames(X))) {
colnames(newdata2) <- colnames(X)
}
}
T <- object@T
z <- object@z
M <- object@M
beta <- object@trend.coef
F.newdata1 <- model.matrix(object@trend.formula, data = data.frame(newdata1))
F.newdata2 <- model.matrix(object@trend.formula, data = data.frame(newdata2))
if(prednewdata1){
y.predict.trend1 <- F.newdata1 %*% beta
}
if(prednewdata2){
y.predict.trend2 <- F.newdata2 %*% beta
}
if (requireNamespace("DiceKriging", quietly = TRUE)){
c.newdata1 <- DiceKriging::covMat1Mat2(object@covariance, X1 = X, X2 = newdata1,
nugget.flag = object@covariance@nugget.flag)
c.newdata2 <- DiceKriging::covMat1Mat2(object@covariance, X1 = X, X2 = newdata2,
nugget.flag = object@covariance@nugget.flag)
}
Tinv.c.newdata1 <- backsolve(t(T), c.newdata1, upper.tri = FALSE)
Tinv.c.newdata2 <- backsolve(t(T), c.newdata2, upper.tri = FALSE)
output.list <- list()
if(prednewdata1){
y.predict.complement1 <- t(Tinv.c.newdata1) %*% z
y.predict1 <- y.predict.trend1 + y.predict.complement1
rm(list=c("y.predict.complement1","y.predict.trend1"))
y.predict1 <- as.numeric(y.predict1)
output.list$mean1 <- y.predict1
rm(list=c("y.predict1"))
}
if(prednewdata2){
y.predict.complement2 <- t(Tinv.c.newdata2) %*% z
y.predict2 <- y.predict.trend2 + y.predict.complement2
rm(list=c("y.predict.complement2","y.predict.trend2"))
y.predict2 <- as.numeric(y.predict2)
output.list$mean2 <- y.predict2
rm(list=c("y.predict2"))
}
if (requireNamespace("DiceKriging", quietly = TRUE)){
C.newdata <- DiceKriging::covMat1Mat2(object@covariance, X1 = newdata1, X2 = newdata2,
nugget.flag = object@covariance@nugget.flag)
}
rm(list=c("newdata1","newdata2"))
cond.cov <- C.newdata - crossprod(Tinv.c.newdata1,Tinv.c.newdata2 )
if (type == "UK") {
T.M <- chol(t(M) %*% M)
s2.predict.mat1 <- backsolve(t(T.M), t(F.newdata1 -
t(Tinv.c.newdata1) %*% M), upper.tri = FALSE)
s2.predict.mat2 <- backsolve(t(T.M), t(F.newdata2 -
t(Tinv.c.newdata2) %*% M), upper.tri = FALSE)
cond.cov <- cond.cov + crossprod(s2.predict.mat1,s2.predict.mat2)
if (bias.correct)
cond.cov <- cond.cov * object@n/(object@n - object@p)
}
output.list$cov <- cond.cov
return(output.list)
}
######################################
simulateGP.sobol <- function(
object,
nsim = 100,
newdata = NULL,
cond = TRUE,
checkNames = FALSE,
max_iter = 1000,
type
) {
if (!is.logical(cond)) stop("'cond' must be TRUE/FALSE")
if ((!is.null(newdata)) && (checkNames)) {
newdata <- checkNames(X1 = object@X, X2 = newdata, X1.name = "the design", X2.name = "newdata")
}
if (is.null(newdata)) {
newdata <- object@X
F.newdata <- object@F
} else {
newdata <- as.matrix(newdata)
m <- nrow(newdata)
if (!identical(ncol(newdata), object@d))
stop("newdata must have the same numbers of columns than the experimental design")
if (!identical(colnames(newdata), colnames(object@X))) {
colnames(newdata) <- colnames(object@X)
}
newdata <- rbind(newdata, object@X)
row.names(newdata) <- NULL
F.newdata <- model.matrix(object@trend.formula, data = data.frame(newdata))
}
n <- object@n
# non conditional simulations
sample_test <- sample.int(m,min(100,m))
iter <- 1
test <- FALSE
yc <- matrix(0,nsim,m+n)
#RMSE_save <- c() #-!!-#
while((!test&iter<1000)){
a <- sample.int(m+n, size = nsim, replace = FALSE, prob = NULL)
ya <- rnorm(nsim,0,1)
if (requireNamespace("DiceKriging", quietly = TRUE)){
Cab <- DiceKriging::covMat1Mat2(object@covariance, X1 = as.matrix(newdata[a,]), X2 = as.matrix(newdata),
nugget.flag = object@covariance@nugget.flag)/object@covariance@sd2
}
yc <- yc + Cab* (as.matrix(ya-diag(yc[,a]))%*%t(as.matrix(rep(1,m+n))))
diag(yc[1:nsim,a]) <- ya
if(iter%%20==0){
Cemp <- var(yc[,sample_test])
Ctheo <- DiceKriging::covMatrix(object@covariance, X = as.matrix(newdata[sample_test,]))[[1]]/
object@covariance@sd2
RMSE_C <- sqrt(mean((Cemp-Ctheo)^2))
#RMSE_save <- c(RMSE_save,RMSE_C) #-!!-#
if(RMSE_C < 0.1){test <- TRUE}
}
iter <- iter+1
}
rm(list=c("Cemp","Ctheo","ya" ,"Cab","a" ,"sample_test"))
yc <- yc*sqrt(object@covariance@sd2*object@n/(object@n-object@p))
if (cond){
if(type=="UK"){
y.predict.trend <- F.newdata %*% object@trend.coef
if (requireNamespace("DiceKriging", quietly = TRUE)){
c.newdata <- DiceKriging::covMat1Mat2(object@covariance, X1 = object@X, X2 = newdata,
nugget.flag = object@covariance@nugget.flag)
}
rm(list=c("newdata"))
Tinv.c.newdata <- backsolve(t(object@T), c.newdata, upper.tri = FALSE)
rm(list=c("c.newdata"))
y.predict.complement <- t(Tinv.c.newdata) %*% object@z
y.predict <- y.predict.trend + y.predict.complement
rm(list=c("y.predict.complement"))
y.predict <- as.numeric(y.predict)
ZD <- t(yc[,(m+1):(m+n)])
if(!identical(object@noise.var,numeric(0))){
ZD <- ZD + matrix(rnorm(nsim*n,0,1),nrow=n)*matrix(rep(sqrt(object@noise.var),nsim),ncol=nsim)
}
xtilde <- backsolve(t(object@T), ZD, upper.tri = FALSE)
Betatilde <- solve(t(object@M)%*%object@M )%*%t(object@M)%*%xtilde
rm(list=c("xtilde"))
ztilde <- backsolve(t(object@T), ZD - object@F%*%Betatilde ,
upper.tri = FALSE)
rm(list=c("ZD"))
ytilde.predict.complement <- t(Tinv.c.newdata) %*% ztilde
rm(list=c("ztilde","Tinv.c.newdata"))
ytilde.predict.trend <- F.newdata %*% Betatilde
rm(list=c("F.newdata","Betatilde"))
ytilde.predict <- ytilde.predict.trend + ytilde.predict.complement
rm(list=c("ytilde.predict.trend","ytilde.predict.complement","y.predict.trend"))
yc <- t(matrix(rep(y.predict,nsim),ncol=nsim)) - t(ytilde.predict) +yc
rm(list=c("y.predict","ytilde.predict"))
}
if(type=="SK"){
y.predict.trend <- F.newdata %*% object@trend.coef
if (requireNamespace("DiceKriging", quietly = TRUE)){
c.newdata <- DiceKriging::covMat1Mat2(object@covariance, X1 = object@X, X2 = newdata,
nugget.flag = object@covariance@nugget.flag)
}
Tinv.c.newdata <- backsolve(t(object@T), c.newdata, upper.tri = FALSE)
y.predict.complement <- t(Tinv.c.newdata) %*% object@z
y.predict <- y.predict.trend + y.predict.complement
y.predict <- as.numeric(y.predict)
ZD <- t(yc[,(m+1):(m+n)])
if(!identical(object@noise.var,numeric(0))){
ZD <- ZD + matrix(rnorm(nsim*n,0,1),nrow=n)*matrix(rep(sqrt(object@noise.var),nsim),ncol=nsim)
}
ztilde <- backsolve(t(object@T), ZD , upper.tri = FALSE)
ytilde.predict <- t(Tinv.c.newdata) %*% ztilde
rm(list=c("ztilde","c.newdata","Tinv.c.newdata","newdata","y.predict.trend","F.newdata","ZD"))
yc <- t(matrix(rep(y.predict,nsim),ncol=nsim)) - t(ytilde.predict) +yc
rm(list=c("y.predict","ytilde.predict"))
}
}
return(yc[,-c((m+1):(m+n))])
}
##############################################################################
sobolpickfreeze <- function(y1,y2,nboot){
output <- (apply(y1*y2,1,mean) - apply(y1,1,mean)*apply(y2,1,mean))/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(apply(y1[,b]*y2[,b],1,mean) - apply(y1[,b],1,mean)*apply(y2[,b],1,mean))/apply(y1[,b],1,var))
}
}
return(output)
}
soboljansenpickfreeze <- function(y1,y2,nboot){
varY <- apply(cbind(y1,y2),1,var)
output <- (varY-apply((y1-y2)^2,1,mean)/2)/varY
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
varY <- apply(cbind(y1[,b],y2[,b]),1,var)
output <- rbind(output,(varY-apply((y1[,b]-y2[,b])^2,1,mean)/2)/varY)
}
}
return(output)
}
sobolEffpickfreeze <- function(y1,y2,nboot){
output <- (apply(y1*y2,1,mean) - apply(cbind(y1,y2),1,mean)^2)/apply(cbind(y1,y2),1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(apply(y1[,b]*y2[,b],1,mean) - apply(cbind(y1[,b],y2[,b]),1,mean)^2)/apply(cbind(y1[,b],y2[,b]),1,var))
}
}
return(output)
}
sobolT2002pickfreeze <- function(y1,y2,nboot){
output <- (1-(apply(y1*y2,1,mean) - apply(y1,1,mean)^2)/apply(y1,1,var))
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(1-(apply(y1[,b]*y2[,b],1,mean) - apply(y1[,b],1,mean)^2)/apply(y1[,b],1,var)))
}
}
return(output)
}
sobolTjansenpickfreeze <- function(y1,y2,nboot){
output <- (apply((y1-y2)^2,1,mean)/2)/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,(apply((y1[,b]-y2[,b])^2,1,mean)/2)/apply(y1[,b],1,var))
}
}
return(output)
}
sobol2007pickfreeze <- function(y1,y2,y3,nboot){
output <- apply(y3*(y2-y1),1,mean)/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,apply(y3[,b]*(y2[,b]-y1[,b]),1,mean)/apply(y1[,b],1,var))
}
}
return(output)
}
sobolT2007pickfreeze <- function(y1,y2,nboot){
output <- apply(y1*(y1-y2),1,mean)/apply(y1,1,var)
if(nboot > 1){
n <- dim(y1)[2]
for (i in 1:nboot){
b <- sample(n,replace = TRUE)
output <- rbind(output,apply(y1[,b]*(y1[,b]-y2[,b]),1,mean)/apply(y1[,b],1,var))
}
}
return(output)
}
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