predict_update_km_parallel {KrigInv}

Quick update of kriging means and variances when one or many new points are added to the DOE.

Description

The functions uses the kriging update formulas to quickly compute kriging mean and variances at points newdata, when r new points newX are added.

Usage

predict_update_km_parallel(newXmean, newXvar, newXvalue, 
Sigma.r, newdata.oldmean, newdata.oldsd, kn)

Arguments

newXmean

Vector of size r: old kriging mean at points x_(n+1),...,x_(n+r).

newXvar

Vector of size r: kriging variance at points x_(n+1),...,x_(n+r).

newXvalue

Vector of size r: value of the objective function at x_(n+1),...,x_(n+r).

Sigma.r

An r*r matrix: kriging covariances between the points x_(n+1),...,x_(n+r).

newdata.oldmean

Vector: old kriging mean at the points newdata (before adding x_(n+1),...,x_(n+r))

newdata.oldsd

Vector: old kriging standard deviations at the points newdata (before adding x_(n+1),...,x_(n+r))

kn

Kriging covariances between the points newdata and the r points newX. These covariances can be computed using the function computeQuickKrigcov

Value

A list with the following fields:

mean

Updated kriging mean at points newdata

sd

Updated kriging standard deviation at points newdata

lambda

New kriging weight of x_(n+1),...,x_(n+r) for the prediction at points newdata

Author(s)

Clement Chevalier (University of Neuchatel, Switzerland)

References

Chevalier C., Ginsbourger D. (2014), Corrected Kriging update formulae for batch-sequential data assimilation, in Pardo-Iguzquiza, E., et al. (Eds.) Mathematics of Planet Earth, pp 119-122

See Also

computeQuickKrigcov, precomputeUpdateData

Examples

#predict_update_km_parallel

set.seed(9)
N <- 20 #number of observations
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")

#points where we want to compute prediction (if a point new.x is added to the doe)
n.grid <- 20 #you can run it with 100
x.grid <- y.grid <- seq(0,1,length=n.grid)
newdata <- expand.grid(x.grid,y.grid)
newdata <- as.matrix(newdata)
precalc.data <- precomputeUpdateData(model=model,integration.points=newdata)
pred2 <- predict_nobias_km(object=model,newdata=newdata,type="UK",se.compute=TRUE)
newdata.oldmean <- pred2$mean; newdata.oldsd <- pred2$sd

#the point that we are going to add
new.x <- matrix(c(0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8),ncol=2,byrow=TRUE)   
pred1 <- predict_nobias_km(object=model,newdata=new.x,type="UK",
se.compute=TRUE,cov.compute=TRUE)
newXmean <- pred1$mean; newXvar <- pred1$sd^2; newXvalue <- pred1$mean + 2*pred1$sd
Sigma.r <- pred1$cov

kn <- computeQuickKrigcov(model=model,integration.points=newdata,X.new=new.x,
                    precalc.data=precalc.data,F.newdata=pred1$F.newdata,
                    c.newdata=pred1$c)

updated.predictions <- predict_update_km_parallel(newXmean=newXmean,newXvar=newXvar,
                                         newXvalue=newXvalue,Sigma.r=Sigma.r,
                                         newdata.oldmean=newdata.oldmean,
                                         newdata.oldsd=newdata.oldsd,kn=kn)

#the new kriging variance is usually lower than the old one
updated.predictions$sd - newdata.oldsd 

z.grid1 <- matrix(newdata.oldsd, n.grid, n.grid)
z.grid2 <- matrix(updated.predictions$sd, n.grid, n.grid)

par(mfrow=c(1,2))

image(x=x.grid,y=y.grid,z=z.grid1,col=grey.colors(10))
contour(x=x.grid,y=y.grid,z=z.grid1,15,add=TRUE)
points(design, col="black", pch=17, lwd=4,cex=2)
title("Kriging standard deviation")

image(x=x.grid,y=y.grid,z=z.grid2,col=grey.colors(10))
contour(x=x.grid,y=y.grid,z=z.grid2,15,add=TRUE)
points(design, col="black", pch=17, lwd=4,cex=2)
points(new.x, col="red", pch=17, lwd=4,cex=2)
title("updated Kriging standard deviation")