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##' Analytical gradient of the Expected Improvement criterion
##'
##' Computes the gradient of the Expected Improvement at the current location.
##' The current minimum of the observations can be replaced by an arbitrary
##' value (plugin), which is usefull in particular in noisy frameworks.
##'
##'
##' @param x a vector representing the input for which one wishes to calculate
##' \code{\link{EI}}.
##' @param model an object of class \code{\link[DiceKriging]{km}}.
##' @param plugin optional scalar: if provided, it replaces the minimum of the
##' current observations,
##' @param type Kriging type: "SK" or "UK"
##' @param minimization logical specifying if EI is used in minimiziation or in
##' maximization,
##' @param envir an optional environment specifying where to get intermediate
##' values calculated in \code{\link{EI}}.
##' @param proxy an optional Boolean, if TRUE EI is replaced by the kriging mean (to minimize)
##' @return The gradient of the expected improvement criterion with respect to
##' x. Returns 0 at design points (where the gradient does not exist).
##' @author David Ginsbourger
##'
##' Olivier Roustant
##'
##' Victor Picheny
##' @seealso \code{\link{EI}}
##' @references
##'
##' D. Ginsbourger (2009), \emph{Multiples metamodeles pour l'approximation et
##' l'optimisation de fonctions numeriques multivariables}, Ph.D. thesis, Ecole
##' Nationale Superieure des Mines de Saint-Etienne, 2009.
##'
##' J. Mockus (1988), \emph{Bayesian Approach to Global Optimization}. Kluwer
##' academic publishers.
##'
##' T.J. Santner, B.J. Williams, and W.J. Notz (2003), \emph{The design and
##' analysis of computer experiments}, Springer.
##'
##' M. Schonlau (1997), \emph{Computer experiments and global optimization},
##' Ph.D. thesis, University of Waterloo.
##' @keywords models optimize
##' @examples
##'
##' set.seed(123)
##' # a 9-points factorial design, and the corresponding response
##' d <- 2; n <- 9
##' design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3))
##' names(design.fact)<-c("x1", "x2")
##' design.fact <- data.frame(design.fact)
##' names(design.fact)<-c("x1", "x2")
##' response.branin <- apply(design.fact, 1, branin)
##' response.branin <- data.frame(response.branin)
##' names(response.branin) <- "y"
##'
##' # model identification
##' fitted.model1 <- km(~1, design=design.fact, response=response.branin,
##' covtype="gauss", control=list(pop.size=50,trace=FALSE), parinit=c(0.5, 0.5))
##'
##' # graphics
##' n.grid <- 9 # Increase to 50 for a nicer picture
##' x.grid <- y.grid <- seq(0,1,length=n.grid)
##' design.grid <- expand.grid(x.grid, y.grid)
##' #response.grid <- apply(design.grid, 1, branin)
##' EI.grid <- apply(design.grid, 1, EI,fitted.model1)
##' #EI.grid <- apply(design.grid, 1, EI.plot,fitted.model1, gr=TRUE)
##'
##' z.grid <- matrix(EI.grid, n.grid, n.grid)
##'
##' contour(x.grid,y.grid,z.grid,20)
##' title("Expected Improvement for the Branin function known at 9 points")
##' points(design.fact[,1], design.fact[,2], pch=17, col="blue")
##'
##' # graphics
##' n.gridx <- 5 # increase to 15 for nicer picture
##' n.gridy <- 5 # increase to 15 for nicer picture
##' x.grid2 <- seq(0,1,length=n.gridx)
##' y.grid2 <- seq(0,1,length=n.gridy)
##' design.grid2 <- expand.grid(x.grid2, y.grid2)
##'
##' EI.envir <- new.env()
##' environment(EI) <- environment(EI.grad) <- EI.envir
##'
##' for(i in seq(1, nrow(design.grid2)) )
##' {
##' x <- design.grid2[i,]
##' ei <- EI(x, model=fitted.model1, envir=EI.envir)
##' eigrad <- EI.grad(x , model=fitted.model1, envir=EI.envir)
##' if(!(is.null(ei)))
##' {
##' suppressWarnings(arrows(x$Var1,x$Var2,
##' x$Var1 + eigrad[1]*2.2*10e-5, x$Var2 + eigrad[2]*2.2*10e-5,
##' length = 0.04, code=2, col="orange", lwd=2))
##' }
##' }
##'
##' @export EI.grad
EI.grad <- function(x, model, plugin=NULL, type="UK", minimization = TRUE, envir=NULL, proxy=FALSE){
########################################################################################
if (is.null(plugin)){
if (minimization) {
plugin <- min(model@y)
} else {
plugin <- -max(model@y)
}
}
m <- plugin
########################################################################################
# Convert x in proper format(s)
d <- length(x)
if (d != model@d){ stop("x does not have the right size") }
newdata.num <- as.numeric(x)
newdata <- data.frame(t(newdata.num))
colnames(newdata) = colnames(model@X)
########################################################################################
# Get quantities related to the model
T <- model@T
X <- model@X
z <- model@z
u <- model@M
covStruct <- model@covariance
# Get quantities related to the prediction
if (is.null(envir))
{
predx <- predict(object=model, newdata=newdata, type=type, checkNames = FALSE,se.compute=TRUE,cov.compute=FALSE)
kriging.mean <- predx$mean
if(!minimization) kriging.mean <- -kriging.mean
kriging.sd <- predx$sd
v <- predx$Tinv.c
c <- predx$c
xcr <- (m - kriging.mean)/kriging.sd
xcr.prob <- pnorm(xcr)
xcr.dens <- dnorm(xcr)
} else
{ # If uploaded through "envir", no prediction computation is necessary
toget <- matrix(c("xcr", "xcr.prob", "xcr.dens", "kriging.sd", "c", "Tinv.c"), 1, 6)
apply(toget, 2, get, envir=envir)
xcr <- envir$xcr
xcr.prob <- envir$xcr.prob
xcr.dens <- envir$xcr.dens
kriging.sd <- envir$kriging.sd
c <- envir$c
v <- envir$Tinv.c
}
F.newdata <- model.matrix(model@trend.formula, data=newdata)
########################################################################################
# Pursue calculation only if standard deviation is non-zero
if ( kriging.sd/sqrt(model@covariance@sd2) < 1e-06)
{ ei.grad <- rep(0,d)
} else
{ # Compute derivatives of the covariance and trend functions
dc <- covVector.dx(x=newdata.num, X=X, object=covStruct, c=c)
f.deltax <- trend.deltax(x=newdata.num, model=model)
# Compute gradients of the kriging mean and variance
W <- backsolve(t(T), dc, upper.tri=FALSE)
kriging.mean.grad <- t(W)%*%z + t(model@trend.coef%*%f.deltax)
if (!minimization) kriging.mean.grad <- -kriging.mean.grad
if (proxy) {
ei.grad <- - kriging.mean.grad
} else {
if (type=="UK")
{ tuuinv <- solve(t(u)%*%u)
kriging.sd2.grad <- t( -2*t(v)%*%W +
2*(F.newdata - t(v)%*%u )%*% tuuinv %*%
(f.deltax - t(t(W)%*%u) ))
} else
{ kriging.sd2.grad <- t( -2*t(v)%*%W) }
kriging.sd.grad <- kriging.sd2.grad / (2*kriging.sd)
# Compute gradient of EI
ei.grad <- - kriging.mean.grad * xcr.prob + kriging.sd.grad * xcr.dens
}
}
########################################################################################
return(ei.grad)
}
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