summary.MuFicokm: Function summary for Multi-Fidelity Cokriging models

In MuFiCokriging: Multi-Fidelity Cokriging models

Description

Provide a summary of a multi-fidelity cokriging model. In particular, it provides the parameter estimations and the results of the cross-validation procedure.

Usage

## S3 method for class 'MuFicokm'
summary(object, CrossValidation = FALSE, ...)

Arguments

object	an object of class S3 ("MuFicokm") provided by the function MuFicokm corresponding to the multi-fidelity cokriging model.
CrossValidation	a Boolean. If TRUE, a Leave-One-Out cross validation procedure is performed. For the LOO procedure, the responses are removed from all code levels and the trend, adjustment and variance parameters are re-estimated after each removed observation.
...	no other argument for this method.

Details

"summary.MuFicokm" return the parameter estimations for each level and the result of the Leave-One-Out Cross-Validation (RMSE=Root Mean Squared Error ; Std RMSE=Standardized RMSE ; Q2=explained variance).

Value

A list with following items (see "MuFicokm"):

CovNames	a list of character strings giving the covariance structures used for the cokriging model. The element i of the list corresponds to the covariance structure of the Gaussian process δ_i(x) with δ_1(x) = Z_1(x). (see "MuFicokm")
Cov.val	a list of vectors giving the values of the hyper-parameters of the cokriging model. The element i of the list corresponds to the hyper-parameters of the Gaussian process δ_i(x) with δ_1(x) = Z_1(x). (see "MuFicokm")
Var.val	a list of numerics giving the values of the variance parameters of the cokriging model. The element i of the list corresponds to the variance of the Gaussian process δ_i(x) with δ_1(x) = Z_1(x). (see "MuFicokm")
Rho.val	a list of vectors giving the values of the trends γ_i of the adjustment parameters ρ_i of the cokriging model. The element i of the list corresponds to the adjustment parameter between Z_i and δ_i(x). (see "MuFicokm")
Trend.val	a list of vectors giving the values of the trend parameters of the Gaussian processes δ_i(x) and Z_1(x).

Author(s)

Loic Le Gratiet

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)
	Df <- ExtractNestDesign(DNest,2)
	zf <- Funcf(Df)
#--- Multi-fidelity cokriging creation without parameter estimations
		mymodel <- MuFicokm(
				formula = list(~1,~1), 
				MuFidesign = DNest, 
				response = list(zc,zf), 
				nlevel = 2)

		sum <- summary(object = mymodel, CrossValidation = TRUE)
		names(sum)
#--- Saving parameters
	#--covariance parameters
		sum$Cov.Val
	#--variance parameters
		sum$Var.Val
	#--trend parameters
		sum$Trend.Val
	#-- adjustment parameters
		sum$Rho.Val

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) :  -30.15302 

N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       30.153  |proj g|=      0.34507
At iterate     1  f =       30.096  |proj g|=       0.33244
At iterate     2  f =       30.039  |proj g|=        1.2483
At iterate     3  f =       30.034  |proj g|=       0.18578
At iterate     4  f =       30.033  |proj g|=      0.007017
At iterate     5  f =       30.033  |proj g|=    4.2326e-05
At iterate     6  f =       30.033  |proj g|=    9.5387e-09

iterations 6
function evaluations 8
segments explored during Cauchy searches 6
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 9.5387e-09
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 : ~1
* covariance model : 
  - type :  matern5_2 
  - nugget : NO
  - parameters lower bounds :  1e-10 
  - parameters upper bounds :  2 
  - best initial criterion value(s) :  -3.813998 

N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=        3.814  |proj g|=   1.7011e-09
At iterate     1  f =        3.814  |proj g|=     1.701e-09

iterations 1
function evaluations 2
segments explored during Cauchy searches 1
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 1.70105e-09
final function value 3.814

F = 3.814
final  value 3.813998 
converged
Level 1: parameter estimation 
 
 Covariance type: matern5_2 
 Variance estimation: 142.5592 
 Trend estimation: 1.856447 
 Correlation length estimation: 0.3014264 
 
Level  2 : parameter estimation 
 
 Covariance type: matern5_2 
 Variance estimation: 0.3942074 
 Trend estimation: 2.473242 
 Adjustment estimation: 0.3601547 
 Correlation length estimation: 0.01307602 
 
Leave One Cross Validation 

RMSE:0.9018081
Std RMSE:0.7447739
Q2:0.9299403

[1] "CovNames"  "Cov.Val"   "Var.Val"   "Rho.Val"   "Trend.Val"
[[1]]
[1] 0.3014264

[[2]]
[1] 0.01307602

[[1]]
[1] 142.5592

[[2]]
[1] 0.3942074

[[1]]
[1] 1.856447

[[2]]
[1] 2.473242

[[1]]
[1] 0.3601547

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