bichon_optim: 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 = NULL)

Arguments

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).

Target value (scalar). The sampling algorithm and the underlying kriging model aim to find the points below (resp. over) T.

method.param

Scalar tolerance around the target T. If not provided, default value used 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.

References

Bect J., Ginsbourger D., Li L., Picheny V., Vazquez E. (2010), Sequential design of computer experiments for the estimation of a probability of failure, Statistics and Computing, pp.1-21, 2011, http://arxiv.org/abs/1009.5177

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

Examples

Run this code

# NOT RUN {
#bichon_optim

set.seed(8)
N <- 9 #number of observations
T <- 80 #threshold
testfun <- branin

#a 9 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 #you can run it with 100
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)")
# }