bichon_optim {KrigInv}

Bichon et al.'s Expected Feasibility criterion

Description

Evaluation of Bichon's Expected Feasibility criterion. To be used in optimization routines, like in max_infill_criterion.

Usage

bichon_optim(x, model, T, method.param = 1)

Arguments

x

Input vector at which one wants to evaluate the criterion. This argument can be either a vector of size d (for an evaluation at a single point) or a p*d matrix (for p simultaneous evaluations of the criterion at p different points).

model

An object of class km (Kriging model).

T

Target value (scalar).

method.param

Scalar tolerance around the target T. Default value is 1.

Value

Bichon EF criterion. When the argument x is a vector, the function returns a scalar. When the argument x is a p*d matrix, the function returns a vector of size p.

Author(s)

Victor Picheny (INRA, Toulouse, France)

David Ginsbourger (IDIAP Martigny and University of Bern, Switzerland)

Clement Chevalier (University of Neuchatel, Switzerland)

References

Bichon B.J., Eldred M.S., Swiler L.P., Mahadevan S., McFarland J.M. (2008) Efficient global reliability analysis for nonlinear implicit performance functions, AIAA Journal 46(10), pp 2459-2468

See Also

EGI, max_infill_criterion

Examples

#bichon_optim

set.seed(9)
N <- 20 #number of observations
T <- 80 #threshold
testfun <- branin

#a 20 points initial design
design <- data.frame( matrix(runif(2*N),ncol=2) )
response <- testfun(design)

#km object with matern3_2 covariance 
#params estimated by ML from the observations
model <- km(formula=~., design = design, 
	response = response,covtype="matern3_2")

x <- c(0.5,0.4) #one evaluation of the bichon criterion
bichon_optim(x=x,T=T,model=model)

n.grid <- 20 # resolution. You may use a larger value.
x.grid <- y.grid <- seq(0,1,length=n.grid)
x <- expand.grid(x.grid, y.grid)
bichon.grid <- bichon_optim(x=x,T=T,model=model)
z.grid <- matrix(bichon.grid, n.grid, n.grid)

#plots: contour of the criterion, DOE points and new point
image(x=x.grid,y=y.grid,z=z.grid,col=grey.colors(10))
contour(x=x.grid,y=y.grid,z=z.grid,25,add=TRUE)
points(design, col="black", pch=17, lwd=4,cex=2)

i.best <- which.max(bichon.grid)
points(x[i.best,], col="blue", pch=17, lwd=4,cex=3)

#plots the real (unknown in practice) curve f(x)=T
testfun.grid <- apply(x,1,testfun)
z.grid.2 <- matrix(testfun.grid, n.grid, n.grid)
contour(x.grid,y.grid,z.grid.2,levels=T,col="blue",add=TRUE,lwd=5)
title("Contour lines of Bichon criterion (black) and of f(x)=T (blue)")