vorobVol_optim_parallel2 {KrigInv}

Compute volume criterion

Description

Compute the volume criterion. Useful for optimization routines.

Usage

vorobVol_optim_parallel2(x, other.points, integration.points,
  integration.weights = NULL, intpoints.oldmean, intpoints.oldsd,
  precalc.data, model, T, new.noise.var = NULL, batchsize, alpha,
  current.crit, typeEx = ">")

Arguments

x

One point where to evaluate the criterion

other.points

remaining points of the batch

integration.points

p*d matrix with the integration points for evaluating numerically the integral of the criterion

integration.weights

vector of size p containting the integration weights for the integral numerical evaluation

intpoints.oldmean

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

intpoints.oldsd

Vector of size p corresponding to the kriging standard deviation at the integration points before adding the batchsize points 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

a km Model

T

threshold

new.noise.var

Optional scalar with the noise variance at the new observation

batchsize

size of the batch of new points

alpha

threshold on pn obtained with conservative estimates

current.crit

starting value for criterion

typeEx

a character (">" or "<") identifying the type of excursion

Value

The value of the criterion at x.

Author(s)

Dario Azzimonti (IDSIA, Switzerland)

References

Azzimonti, D. and Ginsbourger, D. (2018). Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation. Journal of Computational and Graphical Statistics, 27(2), 255-267.

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

Azzimonti, D., Ginsbourger, D., Chevalier, C., Bect, J., and Richet, Y. (2018). Adaptive design of experiments for conservative estimation of excursion sets. Under revision. Preprint at hal-01379642

Chevalier, C., Bect, J., Ginsbourger, D., Vazquez, E., Picheny, V., and Richet, Y. (2014). Fast kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. Technometrics, 56(4):455-465.

See Also

EGIparallel, max_futureVol_parallel

Examples

#vorobVol_optim_parallel2

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")

###we need to compute some additional arguments:
#integration points, and current kriging means and variances at these points
integcontrol <- list(n.points=50,distrib="vorob",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

# alpha, the pn threshold should be computed with conservativeEstimate
# Here it is fixed at 0.992364
alpha <- 0.992364

## Not run: 
# You can compute it with the following code
CE_design=as.matrix (randtoolbox::sobol (n = 500*model@d,
                                      dim = model@d))
colnames(CE_design) <- colnames(model@X)

CE_pred = predict.km(object = model, newdata = CE_design,
                          type = "UK",cov.compute = TRUE)
CE_pred$cov <- CE_pred$cov +1e-7*diag(nrow = nrow(CE_pred$cov),ncol = ncol(CE_pred$cov))

Cestimate <- anMC::conservativeEstimate(alpha = 0.95, pred=CE_pred,
                           design=CE_design, threshold=T, pn = NULL,
                           type = ">", verb = 1,
                           lightReturn = TRUE, algo = "GANMC")
alpha <- Cestimate$lvs

## End(Not run)


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)

batchsize <- 4
other.points <- c(0.7,0.5,0.5,0.9,0.9,0.8)
x <- c(0.1,0.2)
#one evaluation of the vorobVol_optim_parallel criterion2
#we calculate the expectation of the future volume vorobev uncertainty when
#1+3 points are added to the doe
#the 1+3 points are (0.1,0.2) and (0.7,0.5), (0.5,0.9), (0.9,0.8)
vorobVol_optim_parallel2(x=x,other.points=other.points,integration.points=integration.points,
                      integration.weights=integration.weights,
                      intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
                      precalc.data=precalc.data,T=T,model=model,
                      batchsize=batchsize,alpha=alpha)


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)
vorobVol_parallel.grid <- apply(X=x,FUN=vorobVol_optim_parallel2,MARGIN=1,other.points=other.points,
                             integration.points=integration.points,
                             integration.weights=integration.weights,
                             intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
                             precalc.data=precalc.data,T=T,model=model,
                             batchsize=batchsize,alpha=alpha)
z.grid <- matrix(vorobVol_parallel.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)
points(matrix(other.points,ncol=2,byrow=TRUE), col="red", pch=17, lwd=4,cex=2)

# Note that we want to maximize this criterion.
i.best <- which.max(vorobVol_parallel.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 vorobVol_parallel criterion (black) and of f(x)=T (blue)")