sur_optim: sur criterion

Description

Evaluation of the "sur" criterion for a candidate point. To be used in optimization routines, like in max_sur. To avoid numerical instabilities, the new point is evaluated 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 sur uncertainty.

Usage

sur_optim(x, integration.points, integration.weights = NULL, 
intpoints.oldmean, intpoints.oldsd, 
precalc.data, model, T, 
new.noise.var = NULL,current.sur=1e6)

Arguments

Input vector of size d at which one wants to evaluate the criterion.

integration.points

p*d matrix of points for numerical integration in the X space.

integration.weights

Vector of size p corresponding to the weights of these integration points.

intpoints.oldmean

Vector of size p corresponding to the kriging mean at the integration points before adding x to the design of experiments.

intpoints.oldsd

Vector of size p corresponding to the kriging standard deviation at the integration points before adding x to the design of experiments.

precalc.data

List containing useful data to compute quickly the updated kriging variance. This list can be generated using the precomputeUpdateData function.

model

Object of class km (Kriging model).

Target value (scalar)

new.noise.var

Optional scalar value of the noise variance for the new observations.

current.sur

Arbitrary value given to the "sur" criterion for candidate points that are too close to existing observations. This argument applies only if the noise variance is zero.

Value

sur value

References

Chevalier C., Bect J., Ginsbourger D., Vazquez E., Picheny V., Richet Y. (2011), Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set ,http://hal.archives-ouvertes.fr/hal-00641108/

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

Examples

Run this code

# NOT RUN {
#sur_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")

###we need to compute some additional arguments:
#integration points, and current kriging means and variances at these points
integcontrol <- list(n.points=50,distrib="sur",init.distrib="MC")
obj <- integration_design(integcontrol=integcontrol,lower=c(0,0),upper=c(1,1),
model=model,T=T)

integration.points <- obj$integration.points
integration.weights <- obj$integration.weights
pred <- predict_nobias_km(object=model,newdata=integration.points,
type="UK",se.compute=TRUE)

intpoints.oldmean <- pred$mean ; intpoints.oldsd<-pred$sd

#another precomputation
precalc.data <- precomputeUpdateData(model,integration.points)


x <- c(0.5,0.4)#one evaluation of the sur criterion
sur_optim(x=x,integration.points=integration.points,
         integration.weights=integration.weights,
         intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
         precalc.data=precalc.data,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)
sur.grid <- apply(X=x,FUN=sur_optim,MARGIN=1,integration.points=integration.points,
         integration.weights=integration.weights,
         intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
         precalc.data=precalc.data,T=T,model=model)#takes ~15seconds to run
z.grid <- matrix(sur.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,15,add=TRUE)
points(design, col="black", pch=17, lwd=4,cex=2)

i.best <- which.min(sur.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 sur criterion (black) and of f(x)=T (blue)")
# }