EQI.grad {DiceOptim}

EQI's Gradient

Description

Analytical gradient of the Expected Quantile Improvement (EQI) criterion.

Usage

EQI.grad(
  x,
  model,
  new.noise.var = 0,
  beta = 0.9,
  q.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

(scalar) noise variance of the future observation. Default value is 0 (noise-free observation).

beta

Quantile level (default value is 0.9)

q.min

Best kriging quantile. If not provided, this quantity is evaluated.

type

Kriging type: "SK" or "UK"

envir

environment for inheriting intermediate calculations from EQI

Value

Gradient of the Expected Quantile 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 <- 9  # change to 21 for nicer visuals
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, EQI, model=model, new.noise.var=noise.var, beta=.9)
crit.grad <- t(apply(design.grid, 1, EQI.grad, model=model, new.noise.var=noise.var, beta=.9))

z.grid <- matrix(crit.grid, n.grid, n.grid)
contour(x.grid,y.grid, z.grid, 30)
title("EQI 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$Var1+crit.grad[i,1]*.2,x$Var2+crit.grad[i,2]*.2,
length=0.04,code=2,col="orange",lwd=2))
}