Evaluation of Ranjan's Expected Feasibility criterion. To be used in optimization routines, like in max_infill_criterion.
Usage
ranjan_optim(x, model, T, method.param = NULL)
Arguments
x
Input vector at which one wants to evaluate the criterion.
This argument can be either a vector of size d (for an evaluation at a single point) or a p*d matrix (for p simultaneous evaluations of the criterion at p different points).
model
An object of class km (Kriging model).
T
Target value (scalar).
method.param
Scalar tolerance around the target T (default value is 1).
Value
Ranjan EI criterion.
When the argument x is a vector the function returns a scalar.
When the argument x is a p*d matrix the function returns a vector of size p.
Author(s)
V. Picheny (CERFACS, Toulouse, France)
D. Ginsbourger (IMSV, University of Bern, Switzerland)
Clement Chevalier (IMSV, Switzerland, and IRSN, France)
References
Ranjan, P., Bingham, D., Michailidis, G.: Sequential experiment design for contour estimation from complex computer codes. Technometrics 50(4), 527-541 (2008)
Bect J., Ginsbourger D., Li L., Picheny V., Vazquez E. (2010), Sequential design of computer experiments for the estimation of a probability of failure, Statistics and Computing, pp.1-21, 2011, http://arxiv.org/abs/1009.5177
See Also
EGI, max_infill_criterion
Examples
########################################################################
#ranjan_optim
set.seed(8)
N <- 9 #number of observations
T <- 80 #threshold
testfun <- branin
#a 9 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")
x <- c(0.5,0.4)#one evaluation of the ranjan criterion
ranjan_optim(x=x,T=T,model=model)
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)
ranjan.grid <- ranjan_optim(x=x,T=T,model=model)
z.grid <- matrix(ranjan.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,25,add=TRUE)
points(design, col="black", pch=17, lwd=4,cex=2)
i.best <- which.max(ranjan.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 Ranjan criterion (black) and of f(x)=T (blue)")
Results
R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
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> library(KrigInv)
Loading required package: DiceKriging
Loading required package: pbivnorm
Loading required package: rgenoud
## rgenoud (Version 5.7-12.4, Build Date: 2015-07-19)
## See http://sekhon.berkeley.edu/rgenoud for additional documentation.
## Please cite software as:
## Walter Mebane, Jr. and Jasjeet S. Sekhon. 2011.
## ``Genetic Optimization Using Derivatives: The rgenoud package for R.''
## Journal of Statistical Software, 42(11): 1-26.
##
Loading required package: randtoolbox
Loading required package: rngWELL
This is randtoolbox. For overview, type 'help("randtoolbox")'.
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/KrigInv/ranjan_optim.Rd_%03d_medium.png", width=480, height=480)
> ### Name: ranjan_optim
> ### Title: Ranjan et al.'s Expected Improvement criterion
> ### Aliases: ranjan_optim
>
> ### ** Examples
>
> ########################################################################
> #ranjan_optim
>
> set.seed(8)
> N <- 9 #number of observations
> T <- 80 #threshold
> testfun <- branin
>
> #a 9 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")
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~X1 + X2
* covariance model :
- type : matern3_2
- nugget : NO
- parameters lower bounds : 1e-10 1e-10
- parameters upper bounds : 1.448893 1.853021
- best initial criterion value(s) : -25.38168
N = 2, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 25.382 |proj g|= 0.19431
At iterate 1 f = 25.027 |proj g|= 0.13259
At iterate 2 f = 25.014 |proj g|= 1.6725
At iterate 3 f = 25.002 |proj g|= 0.15969
At iterate 4 f = 25.001 |proj g|= 0.17792
At iterate 5 f = 24.999 |proj g|= 0.31318
At iterate 6 f = 24.998 |proj g|= 0.14968
At iterate 7 f = 24.998 |proj g|= 0.03446
At iterate 8 f = 24.998 |proj g|= 0.03458
At iterate 9 f = 24.998 |proj g|= 0.0084816
At iterate 10 f = 24.998 |proj g|= 0.038393
At iterate 11 f = 24.997 |proj g|= 1.3196
At iterate 12 f = 24.997 |proj g|= 1.3327
At iterate 13 f = 24.994 |proj g|= 1.8077
At iterate 14 f = 24.991 |proj g|= 1.8106
At iterate 15 f = 24.975 |proj g|= 1.8136
At iterate 16 f = 24.937 |proj g|= 1.8202
At iterate 17 f = 24.816 |proj g|= 1.8136
At iterate 18 f = 24.652 |proj g|= 0.81261
At iterate 19 f = 24.652 |proj g|= 0.25743
At iterate 20 f = 24.651 |proj g|= 0.0033442
At iterate 21 f = 24.651 |proj g|= 1.4045e-05
iterations 21
function evaluations 30
segments explored during Cauchy searches 22
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 1.40447e-05
final function value 24.6515
F = 24.6515
final value 24.651471
converged
>
> x <- c(0.5,0.4)#one evaluation of the ranjan criterion
> ranjan_optim(x=x,T=T,model=model)
[1] 2.121465e-59
>
> 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)
> ranjan.grid <- ranjan_optim(x=x,T=T,model=model)
> z.grid <- matrix(ranjan.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,25,add=TRUE)
> points(design, col="black", pch=17, lwd=4,cex=2)
>
> i.best <- which.max(ranjan.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 Ranjan criterion (black) and of f(x)=T (blue)")
>
>
>
>
>
> dev.off()
null device
1
>