AKG {DiceOptim}

Approximate Knowledge Gradient (AKG)

Description

Evaluation of the Approximate Knowledge Gradient (AKG) criterion.

Usage

AKG(x, model, new.noise.var = 0, 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).

type

Kriging type: "SK" or "UK"

envir

environment for saving intermediate calculations and reusing them within AKG.grad

Value

Approximate Knowledge Gradient

Author(s)

Victor Picheny

David Ginsbourger

References

Scott, W., Frazier, P., Powell, W. (2011). The correlated knowledge gradient for simulation optimization of continuous parameters using gaussian process regression. SIAM Journal on Optimization, 21(3), 996-1026.

Examples

##########################################################################
###    AKG SURFACE ASSOCIATED WITH AN ORDINARY KRIGING MODEL          ####
### OF THE BRANIN FUNCTION KNOWN AT A 12-POINT LATIN HYPERCUBE DESIGN ####
##########################################################################
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 <- 12 # 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, AKG, model=model, new.noise.var=noise.var)
func.grid <- apply(design.grid, 1, test.function)

# Compute kriging mean and variance on a grid
names(design.grid) <- c("V1","V2")
pred <- predict.km(model, newdata=design.grid, type="UK")
mk.grid <- pred$m
sk.grid <- pred$sd

# Plot actual function
z.grid <- matrix(func.grid, n.grid, n.grid)
filled.contour(x.grid,y.grid, z.grid, nlevels=50, color = topo.colors,
               plot.axes = {title("Actual function");
                            points(model@X[,1],model@X[,2],pch=17,col="blue"); 
                            axis(1); axis(2)})

# Plot Kriging mean
z.grid <- matrix(mk.grid, n.grid, n.grid)
filled.contour(x.grid,y.grid, z.grid, nlevels=50, color = topo.colors,
               plot.axes = {title("Kriging mean");
                            points(model@X[,1],model@X[,2],pch=17,col="blue"); 
                            axis(1); axis(2)})

# Plot Kriging variance
z.grid <- matrix(sk.grid^2, n.grid, n.grid)
filled.contour(x.grid,y.grid, z.grid, nlevels=50, color = topo.colors,
               plot.axes = {title("Kriging variance");
                            points(model@X[,1],model@X[,2],pch=17,col="blue"); 
                            axis(1); axis(2)})

# Plot AKG criterion
z.grid <- matrix(crit.grid, n.grid, n.grid)
filled.contour(x.grid,y.grid, z.grid, nlevels=50, color = topo.colors,
               plot.axes = {title("AKG");
                            points(model@X[,1],model@X[,2],pch=17,col="blue"); 
                            axis(1); axis(2)})