predict.MuFicokm: Predictions and confidence intervals of Multi-Kriging Cokriging models at new data

Description

Provide predictive mean, variance and covariance of a multi-fidelity cokriging model. 95 % confidence intervals are given based on Gaussian process assumption. This might be abusive in particular in the case where the number of observations is small.

Usage

"predict"(	object, newdata,  type,  
		se.compute = TRUE, cov.compute = FALSE, checkNames = FALSE, ...)

Arguments

object

an object of class S3 ("MuFicokm") provided by the function "MuFicokm" corresponding to the multi-fidelity cokriging model.

newdata

a vector, matrix or data frame containing the points where to perform predictions.

type

a list of character strings corresponding to the kriging family of the Gaussian processes $\delta_k(x)$ with $k=1,...,nlevel$ (we use the convention $\delta_1(x) = Z_1(x)$), to be chosen between simple kriging ("SK"), or universal kriging ("UK"). (see "MuFicokm")

se.compute

a list of optional booleans for each level. If FALSE, only the kriging mean is computed. If TRUE, the kriging variance and confidence intervals are computed too. The length of the list must be either the number of levels or one. If the length is one, the same argument is repeated for all levels.

cov.compute

a list of optional booleans for each level. If TRUE, the conditional covariance matrix is computed. The length of the list must be either the number of levels or one. If the length is one, the same argument is repeated for all levels.

checkNames

a list of optional booleans for each level. If TRUE, a consistency test is performed between the names of newdata and the names of the experimental design (contained in object@X). The length of the list must be either the number of levels or one. If the length is one, the same argument is repeated for all levels.

...

no other argument for this method.

Value

mean: multi-fidelity co-kriging predictive mean computed at newdata.
sig2: multi-fidelity co-kriging predictive variance computed at newdata.
C: multi-fidelity co-kriging predictive conditional covariance matrix. Not computed if cov.compute=FALSE (default).
mux: multi-fidelity co-kriging predictive means at each level.
varx: multi-fidelity co-kriging predictive variances at each level.
CovMat: multi-fidelity co-kriging predictive conditional covariance matrices at each level.

References

KENNEDY, M.C. & O'HAGAN, A. (2000), Predicting the output from a complex computer code when fast approximations are available. Biometrika 87, 1-13

FORRESTER, A.I.J, SOBESTER A. & KEAN, A.J. (2007), Multi-Fidelity optimization via surrogate modelling. Proc. R. Soc. A 463, 3251-3269.

LE GRATIET, L. & GARNIER, J. (2012), Recursive co-kriging model for Design of Computer Experiments with multiple levels of fidelity, arXiv:1210.0686

LE GRATIET, L. (2012), Bayesian analysis of hierarchical multi-fidelity codes, arXiv:1112.5389

Examples

Run this code

#--- test functions (see [Le GRATIET, L. 2012])
	Funcf <- function(x){return(0.5*(6*x-2)^2*sin(12*x-4)+sin(10*cos(5*x)))}
	Funcc <- function(x){return((6*x-2)^2*sin(12*x-4)+10*(x-0.5)-5)}
#--- Data
	Dc <- seq(0,1,0.1)
	indDf <- c(1,3,7,11)
	DNest <- NestedDesign(Dc, nlevel=2 , indices = list(indDf) )
	zc <- Funcc(DNest$PX)
	Df <- ExtractNestDesign(DNest,2)
	zf <- Funcf(Df)
#--- Multi-fidelity cokriging creation without parameter estimations
#--- "SK" : Simple CoKriging, i.e. when parameters are known
#--- "UK" : Universal CoKriging, i.e. when parameters are estimated
	#-- model creation
		mymodelSK <- MuFicokm(
				formula = list(~1,~1), 
				MuFidesign = DNest, 
				response = list(zc,zf), 
				nlevel = 2,
				coef.trend=list(0,2),
				coef.rho=list(0.5),
				coef.var=list(2,2),
				coef.cov=list(0.1,0.2))
	#-- predictions with "SK"
		predictionsSK <- predict(
				object = mymodelSK, 
				newdata = seq(0,1,le=100),
				type = "SK")

#--- Multi-fidelity co-kriging building with parameter estimations
	#-- model creation
		mymodelUK <- MuFicokm(
				formula = list(~1,~1), 
				MuFidesign = DNest, 
				response = list(zc,zf), 
				nlevel = 2)
	#-- predictions with "UK" 
		predictionsUK <- predict(
				object = mymodelUK, 
				newdata = seq(0,1,le=100),
				type = "UK")

#--- Multi-fidelity co-kriging building with known and unknown parameters
	#-- model creation
		mymodelSK_UK <- MuFicokm(
				formula = list(~1,~1),
				MuFidesign = DNest,
				response = list(zc,zf),
				nlevel = 2,
				coef.trend=list(-5,NULL),
				coef.rho=list(NULL),
				coef.var=list(2,NULL),
				coef.cov=list(0.1,NULL))
	#-- predictions with "UK" 
		predictionsSK_UK <- predict(
				object = mymodelSK_UK, 
				newdata = seq(0,1,le=100),
				type = list("SK","UK"))

#--- plot
	op <- par(mfrow=c(3,1))
		x <- seq(0,1,le=100)
		curve(Funcf,ylim=c(-5,10),main="SK")
		polygon(c(x,rev(x)), c(predictionsSK$mean+2*sqrt(predictionsSK$sig2), 
						rev(predictionsSK$mean-2*sqrt(predictionsSK$sig2))),
		col="gray", border = "red",density=20 )
		lines(seq(0,1,le=100), predictionsSK$mean,col=2,lty=2,lwd=2)

		curve(Funcf,ylim=c(-5,10),main="UK")
		polygon(c(x,rev(x)), c(predictionsUK$mean+2*sqrt(predictionsUK$sig2), 
						rev(predictionsUK$mean-2*sqrt(predictionsUK$sig2))),
		col="gray", border = "red",density=20 )
		lines(seq(0,1,le=100), predictionsUK$mean,col=2,lty=2,lwd=2)

		curve(Funcf,ylim=c(-5,10),main="mix SK & UK")
		polygon(c(x,rev(x)), c(predictionsSK_UK$mean+2*sqrt(predictionsSK_UK$sig2), 
						rev(predictionsSK_UK$mean-2*sqrt(predictionsSK_UK$sig2))),
		col="gray", border = "red",density=20 )
		lines(seq(0,1,le=100), predictionsSK_UK$mean,col=2,lty=2,lwd=2)
	par(op)