Targeted IMSE criterion

Description

Evaluation of the "timse" criterion for a candidate point. To be used in optimization routines, like in max_timse. To avoid numerical instabilities, the new point is evaluated only if it is not too close to an existing observation, or if there is some observation noise. The criterion is the integral of the posterior timse uncertainty.

Usage

timse_optim(x, integration.points, integration.weights = NULL, 
intpoints.oldmean = NULL, intpoints.oldsd = NULL, 
precalc.data, model, T, new.noise.var = 0, weight = NULL)

Arguments

Value

targeted imse value

Author(s)

Victor Picheny (CERFACS, Toulouse, France)

David 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

Chevalier C., Bect J., Ginsbourger D., Vazquez E., Picheny V., Richet Y. (2011), Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set ,http://hal.archives-ouvertes.fr/hal-00641108/

See Also

EGI, max_timse

Examples

#timse_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")

###we need to compute some additional arguments:
#integration points, and current kriging means and variances at these points
integcontrol <- list(n.points=50,distrib="timse",init.distrib="MC")
obj <- integration_design(integcontrol=integcontrol,lower=c(0,0),
upper=c(1,1),model=model,T=T)

integration.points <- obj$integration.points
integration.weights <- obj$integration.weights
pred <- predict_nobias_km(object=model,newdata=integration.points,
type="UK",se.compute=TRUE)
intpoints.oldmean <- pred$mean ; intpoints.oldsd<-pred$sd

#another precomputation
precalc.data <- precomputeUpdateData(model,integration.points)

#we also need to compute weights. Otherwise the (more simple) 
#imse criterion will be evaluated
weight <- 1/sqrt(2*pi*intpoints.oldsd^2) * 
exp(-0.5*((intpoints.oldmean-T)/sqrt(intpoints.oldsd^2))^2)
weight[is.nan(weight)] <- 0

x <- c(0.5,0.4)#one evaluation of the timse criterion
timse_optim(x=x,integration.points=integration.points,
         integration.weights=integration.weights,
         intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
         precalc.data=precalc.data,T=T,model=model,weight=weight)

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)
timse.grid <- apply(X=x,FUN=timse_optim,MARGIN=1,integration.points=integration.points,
         integration.weights=integration.weights,
         intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
         precalc.data=precalc.data,T=T,model=model,weight=weight)

z.grid <- matrix(timse.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,15,add=TRUE)
points(design, col="black", pch=17, lwd=4,cex=2)

i.best <- which.min(timse.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 timse criterion (black) and of f(x)=T (blue)")

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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Type 'demo()' for some demos, 'help()' for on-line help, or
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Type 'q()' to quit R.

> 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/timse_optim.Rd_%03d_medium.png", width=480, height=480)
> ### Name: timse_optim
> ### Title: Targeted IMSE criterion
> ### Aliases: timse_optim
> 
> ### ** Examples
> 
> #timse_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
> 
> ###we need to compute some additional arguments:
> #integration points, and current kriging means and variances at these points
> integcontrol <- list(n.points=50,distrib="timse",init.distrib="MC")
> obj <- integration_design(integcontrol=integcontrol,lower=c(0,0),
+ upper=c(1,1),model=model,T=T)
> 
> integration.points <- obj$integration.points
> integration.weights <- obj$integration.weights
> pred <- predict_nobias_km(object=model,newdata=integration.points,
+ type="UK",se.compute=TRUE)
> intpoints.oldmean <- pred$mean ; intpoints.oldsd<-pred$sd
> 
> #another precomputation
> precalc.data <- precomputeUpdateData(model,integration.points)
> 
> #we also need to compute weights. Otherwise the (more simple) 
> #imse criterion will be evaluated
> weight <- 1/sqrt(2*pi*intpoints.oldsd^2) * 
+ exp(-0.5*((intpoints.oldmean-T)/sqrt(intpoints.oldsd^2))^2)
> weight[is.nan(weight)] <- 0
> 
> x <- c(0.5,0.4)#one evaluation of the timse criterion
> timse_optim(x=x,integration.points=integration.points,
+          integration.weights=integration.weights,
+          intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
+          precalc.data=precalc.data,T=T,model=model,weight=weight)
[1] 0.1199523
> 
> 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)
> timse.grid <- apply(X=x,FUN=timse_optim,MARGIN=1,integration.points=integration.points,
+          integration.weights=integration.weights,
+          intpoints.oldmean=intpoints.oldmean,intpoints.oldsd=intpoints.oldsd,
+          precalc.data=precalc.data,T=T,model=model,weight=weight)
> 
> z.grid <- matrix(timse.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,15,add=TRUE)
> points(design, col="black", pch=17, lwd=4,cex=2)
> 
> i.best <- which.min(timse.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 timse criterion (black) and of f(x)=T (blue)")
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>