AEI.grad {DiceOptim}

AEI's Gradient

Description

Analytical gradient of the Augmented Expected Improvement (AEI) criterion.

Usage

AEI.grad(x, model, new.noise.var = 0, y.min = NULL, type = "UK", envir = NULL)

Arguments

x

the input vector at which one wants to evaluate the criterion

model

a Kriging model of "km" class

new.noise.var

the (scalar) noise variance of the new observation.

y.min

The kriging predictor at the current best point (point with smallest kriging quantile). If not provided, this quantity is evaluated.

type

Kriging type: "SK" or "UK"

envir

environment for inheriting intermediate calculations from AEI

Value

Gradient of the Augmented Expected Improvement

Author(s)

Victor Picheny

David Ginsbourger

Examples

set.seed(421)

# Set test problem parameters
doe.size <- 12
dim <- 2
test.function <- get("branin2")
lower <- rep(0,1,dim)
upper <- rep(1,1,dim)
noise.var <- 0.2

# Generate DOE and response
doe <- as.data.frame(matrix(runif(doe.size*dim),doe.size))
y.tilde <- rep(0, 1, doe.size)
for (i in 1:doe.size)  {
y.tilde[i] <- test.function(doe[i,]) + sqrt(noise.var)*rnorm(n=1)
}
y.tilde <- as.numeric(y.tilde)

# Create kriging model
model <- km(y~1, design=doe, response=data.frame(y=y.tilde),
        covtype="gauss", noise.var=rep(noise.var,1,doe.size), 
	lower=rep(.1,dim), upper=rep(1,dim), control=list(trace=FALSE))

# Compute actual function and criterion on a grid
n.grid <- 8 # Change to 21 for a nicer picture
x.grid <- y.grid <- seq(0,1,length=n.grid)
design.grid <- expand.grid(x.grid, y.grid)
nt <- nrow(design.grid)

crit.grid <- apply(design.grid, 1, AEI, model=model, new.noise.var=noise.var)
crit.grad <- t(apply(design.grid, 1, AEI.grad, model=model, new.noise.var=noise.var))

z.grid <- matrix(crit.grid, n.grid, n.grid)
contour(x.grid,y.grid, z.grid, 30)
title("AEI and its gradient")
points(model@X[,1],model@X[,2],pch=17,col="blue")

for (i in 1:nt)
{
 x <- design.grid[i,]
 suppressWarnings(arrows(x$Var1,x$Var2, x$Var1+crit.grad[i,1]*.6,x$Var2+crit.grad[i,2]*.6,
length=0.04,code=2,col="orange",lwd=2))
}