CrossValidationMuFicokmAll: Cross Validation Procedure for Multi-Fidelity Cokriging...

In MuFiCokriging: Multi-Fidelity Cokriging models

Description

Provide the predictive errors and variances of the cross validation procedure when observations are removed from all code levels.

Usage

CrossValidationMuFicokmAll(model, indice)

Arguments

model	an object of class S3 ("MuFicokm") provided by the function "MuFicokm" corresponding to the multi-fidelity cokriging model.
indice	a vector containing the indices of the observations removed from the highest code level for the cross-validation procedure.

Details

This function performs all the possible cross-validation procedures. Indeed, due to the nested property of the experimental design sets, we can choose to remove observations only from the highest code level or the two highest code levels and so on.

Value

CVerr	a list of vectors indexed by q containing the predictive errors of the cross-validation procedure when the observations are removed from the q highest code levels.
CVvar	a list of vectors indexed by q containing the predictive variances of the cross-validation procedure when the observations are removed from the q highest code levels.
CVCov	a list indexed by q of the predictive covariance matrices of the cross-validation procedure when the observations are removed from the q highest code levels.
CVerrall	a vector containing the predictive errors of the cross-validation procedure when the observations are removed from all code levels.
CVvarall	a vector containing the predictive variances of the cross-validation procedure when the observations are removed from all code levels.
CVCovall	the predictive covariance matrix of the cross-validation procedure when the observations are removed from all code levels.

Author(s)

Loic Le Gratiet

References

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

See Also

MuFicokm, CrossValidationMuFicokm

Examples

#--- 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)
	zf <- Funcf(ExtractNestDesign(DNest,2))

#--- Model creation with parameter estimations
		mymodel <- MuFicokm(
				formula = list(~1,~1+X1), 
				MuFidesign = DNest, 
				response = list(zc,zf), 
				nlevel = 2,
				covtype = "matern5_2")
#--- Cross Validation
		indice <- c(1,3)
		CVAll <- CrossValidationMuFicokmAll(mymodel,indice)
	#-- predictive errors when we remove the observations from Funcf and Funcc
		CVAll$CVerrall
	#-- predictive variances when we remove the observations from Funcf and Funcc
		CVAll$CVvarall
	#-- predictive covariance matrix when we remove the observations from Funcf and Funcc
		CVAll$CVCovall
	#-- predictive errors when we remove the observations from Funcf
		CVAll$CVerr[[1]]
	#-- predictive variances when we remove the observations from Funcf
		CVAll$CVvar[[1]]
	#-- predictive covariance matrix when we remove the observations from Funcf
		CVAll$CVCov[[1]]

#--- Leave-One-Out Cross Validation
	#-- LOO CV predictive errors
		apply(matrix(1:DNest$n),1,function(x) CrossValidationMuFicokmAll(mymodel,x)$CVerrall)

Example output

Loading required package: DiceKriging

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  matern5_2 
  - nugget : NO
  - parameters lower bounds :  1e-10 
  - parameters upper bounds :  2 
  - best initial criterion value(s) :  -30.0395 

N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       30.039  |proj g|=       1.3554
At iterate     1  f =       30.033  |proj g|=     0.0054075
At iterate     2  f =       30.033  |proj g|=    0.00021824
At iterate     3  f =       30.033  |proj g|=    3.7897e-08

iterations 3
function evaluations 6
segments explored during Cauchy searches 3
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 3.78972e-08
final function value 30.0335

F = 30.0335
final  value 30.033468 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~X1
* covariance model : 
  - type :  matern5_2 
  - nugget : NO
  - parameters lower bounds :  1e-10 
  - parameters upper bounds :  2 
  - best initial criterion value(s) :  -0.3681792 

N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=      0.36818  |proj g|=     0.068155
At iterate     1  f =      0.32578  |proj g|=             0

iterations 1
function evaluations 2
segments explored during Cauchy searches 1
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0
final function value 0.32578

F = 0.32578
final  value 0.325780 
converged
[1] 1.1975188 0.1558947
[1] 3.3408142 0.5482324
           [,1]       [,2]
[1,]  3.3408142 -0.7254001
[2,] -0.7254001  0.5482324
[1]  2.0530830 -0.1630859
[1] 14.777187  2.164815
          [,1]      [,2]
[1,] 14.777187 -3.276186
[2,] -3.276186  2.164815
[1]  1.081382305 -0.834986538  0.370079124 -0.002980427

MuFiCokriging documentation built on May 2, 2019, 3:33 p.m.