sur_optim: SUR criterion

Description

Evaluation of the SUR criterion for a candidate point. To be used in optimization routines, like in max.sur. The new point is added to the design of experiments only if it is not too close to an existing observation, or if there is some observation noise. The criterion is the integral of the posterior probability of misclassification.

Usage

sur_optim(x, y.integration=NULL, integration.points, model, T, 
new.noise.var=0, type="UK")

Arguments

the input vector at which one wants to evaluate the criterion

the targeted (scalar) output value

y.integration

the vector of integration points in the y space; if not provided, 10 points using importance sampling are used

model

An object of class km (Kriging model)

type

Kriging type (string): "SK" or "UK" (default)

integration.points

Matrix of points for numerical integration in the X space

new.noise.var

optional scalar value of the noise variance for the new observation (default value is 0)

Value

SUR value

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, accepted with minor revisions to the Journal of Statistics and Computing, http://arxiv.org/abs/1009.5177 Vazquez, E., Bect, J.: A sequential Bayesian algorithm to estimate a probability of failure. In: Proceedings of the 15th IFAC Symposium on System Identification, (SYSID 2009), Saint-Malo, France (2009)

Examples

Run this code

#####################################################################
#a 9-point full factorial initial design
design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3))

design.fact <- data.frame(design.fact)
names(design.fact) <- c ( "x1","x2")
testfun <- camelback2			#our test function

#the response
response <- testfun(design.fact)

#the initial km model
model <- km(formula=~., design = design.fact, response = response, 
covtype="matern5_2")

#the integration points
n.grid <- 20
x.grid <- y.grid <- seq(0,1,length=n.grid)
design.grid <- expand.grid(x.grid, y.grid)

#evaluate criterion on the grid
T <- 0
		
sur.grid <- apply(design.grid, 1, sur_optim, T=T, model=model, 
integration.points=design.grid)
z.grid <- matrix(sur.grid, n.grid, n.grid)

#plots: contour of the criterion, doe points and new point
contour(x.grid,y.grid,z.grid,25)
points(design.fact, col="black", pch=20, lwd=4)

i.best <- which.min(sur.grid)
points(design.grid[i.best,], col="blue", pch=20, lwd=4)

#plots: contour of the actual function at threshold
testfun.grid <- testfun(design.grid)
z.grid.2 <- matrix(testfun.grid, n.grid, n.grid)
contour(x.grid,y.grid,z.grid.2,levels=T,col="blue",add=TRUE)
title("Contour lines of sur criterion (black) and of f(x)=T (blue)")
#####################################################################