Targeted MSE criterion

Description

Evaluation of the Targeted MSE criterion. To be used in optimization routines, like in max_infill_criterion

Usage

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

Arguments

Value

targeted MSE value. 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 (Ecole Centrale Paris)

D. Ginsbourger (IMSV, University of Bern, Switzerland)

Clement Chevalier (IMSV, Switzerland, and IRSN, France)

References

Picheny, V., Ginsbourger, D., Roustant, O., Haftka, R.T., Adaptive designs of experiments for accurate approximation of a target region, J. Mech. Des. - July 2010 - Volume 132, Issue 7, http://dx.doi.org/10.1115/1.4001873

Picheny V., Improving accuracy and compensating for uncertainty in surrogate modeling, Ph.D. thesis, University of Florida and Ecole Nationale Superieure des Mines de Saint-Etienne

See Also

EGI, max_infill_criterion

Examples

#tmse_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 tmse criterion
tmse_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)
tmse.grid <- tmse_optim(x=x,T=T,model=model)
z.grid <- matrix(tmse.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(tmse.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 tmse 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/tmse_optim.Rd_%03d_medium.png", width=480, height=480)
> ### Name: tmse_optim
> ### Title: Targeted MSE criterion
> ### Aliases: tmse_optim
> 
> ### ** Examples
> 
> #tmse_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 tmse criterion
> tmse_optim(x=x,T=T,model=model)
[1] 5.213946e-65
> 
> 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)
> tmse.grid <- tmse_optim(x=x,T=T,model=model)
> z.grid <- matrix(tmse.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(tmse.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 tmse criterion (black) and of f(x)=T (blue)")
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>