Bichon et al.'s Expected Feasibility criterion

Description

Evaluation of Bichon's Expected Feasibility criterion. To be used in optimization routines, like in max_infill_criterion.

Usage

bichon_optim(x, model, T, method.param = NULL)

Arguments

Value

Bichon EF 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

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

Bichon, B.J., Eldred, M.S., Swiler, L.P., Mahadevan, S., McFarland, J.M.: Efficient global reliability analysis for nonlinear implicit performance functions. AIAA Journal 46 (10), 2459-2468 (2008)

See Also

EGI, max_infill_criterion

Examples

#bichon_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 bichon criterion
bichon_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)
bichon.grid <- bichon_optim(x=x,T=T,model=model)
z.grid <- matrix(bichon.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(bichon.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 Bichon 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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Platform: x86_64-pc-linux-gnu (64-bit)

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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/bichon_optim.Rd_%03d_medium.png", width=480, height=480)
> ### Name: bichon_optim
> ### Title: Bichon et al.'s Expected Feasibility criterion
> ### Aliases: bichon_optim
> 
> ### ** Examples
> 
> #bichon_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 bichon criterion
> bichon_optim(x=x,T=T,model=model)
[1] 3.574618e-60
> 
> 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)
> bichon.grid <- bichon_optim(x=x,T=T,model=model)
> z.grid <- matrix(bichon.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(bichon.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 Bichon criterion (black) and of f(x)=T (blue)")
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>