Analytical gradient of the Expected Improvement criterion

Description

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.

Usage

EI.grad(
x,
model,
plugin = NULL,
type = "UK",
minimization = TRUE,
envir = NULL,
proxy = FALSE
)


Arguments

 x a vector representing the input for which one wishes to calculate EI. model an object of class km. plugin optional scalar: if provided, it replaces the minimum of the current observations, type Kriging type: "SK" or "UK" minimization logical specifying if EI is used in minimiziation or in maximization, envir an optional environment specifying where to get intermediate values calculated in EI. proxy an optional Boolean, if TRUE EI is replaced by the kriging mean (to minimize)

Value

The gradient of the expected improvement criterion with respect to x. Returns 0 at design points (where the gradient does not exist).

Author(s)

David Ginsbourger

Olivier Roustant

Victor Picheny

References

D. Ginsbourger (2009), 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), Bayesian Approach to Global Optimization. Kluwer academic publishers.

T.J. Santner, B.J. Williams, and W.J. Notz (2003), The design and analysis of computer experiments, Springer.

M. Schonlau (1997), Computer experiments and global optimization, Ph.D. thesis, University of Waterloo.

EI

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()

for(i in seq(1, nrow(design.grid2)) )
{
x <- design.grid2[i,]
ei <- EI(x, model=fitted.model1, envir=EI.envir)
suppressWarnings(arrows(x$Var1,x$Var2,
x$Var1 + eigrad[1]*2.2*10e-5, x$Var2 + eigrad[2]*2.2*10e-5,