max_timse: Minimizer of the targeted IMSE criterion

Description

Minimization, based on the package rgenoud (or on exhaustive search on a discrete set), of the targeted IMSE criterion. If not provided, the integration points (for the IMSE evaluation) are generated before optimization by latin hypercube sampling. The default number of integration points is 100 times the dimension of the problem.

Usage

max_timse(lower, upper, parinit=NULL, control=NULL, discrete.X=NULL, 
integration.points=NULL, T, epsilon=0, model, new.noise.var=0, type="UK")

Arguments

lower

vector containing the lower bounds of the variables to be optimized over

upper

vector containing the upper bounds of the variables to be optimized over

parinit

optional vector containing the initial values for the variables to be optimized over

control

optional list of control parameters for optimization. One can control "pop.size" (default : [4+3*log(nb of variables)]), "max.generations" (5), "wait.generations" (2) and "BFGSburnin" (0) of fun

target value (a real number)

epsilon

optional tolerance value (a real positive number); default value is 0.

model

An object of class km (Kriging model)

type

Kriging type (string): "SK" or "UK" (default)

new.noise.var

optional scalar value of the noise variance for the new observations

discrete.X

optional matrix of candidate points. If provided, the search for new observations is made on this discrete set instead of running the continuous optimisation, and it is also used as integration points

integration.points

optional scalar or matrix. If it is scalar, it defines the number of integration points, then generated as an LHS design; if it is a matrix, it defines the integration points directly. If discrete.X is provided, this input is not taken into accoun

Value

A list with components:
parthe best set of parameters found.
valuethe value tIMSE at par.

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

Examples

Run this code

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#a 9-point full factorial initial design
design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3))

design.fact <- data.frame(design.fact)
names(design.fact) <- c ( "x1","x2")
testfun <- camelback2			#our test function

#the response
response <- testfun(design.fact)

#the initial km model
model <- km(formula=~., design = design.fact, response = response, 
covtype="matern5_2")

#the integration points
n.grid <- 20
x.grid <- y.grid <- seq(0,1,length=n.grid)
design.grid <- expand.grid(x.grid, y.grid)

#compute weight on the grid
T <- 0
epsilon <- 0.01
krig  <- predict.km(model, newdata=as.data.frame(design.grid), "UK")
mk <- krig$mean
sk    <- krig$sd
weight <- 1/sqrt(2*pi*(sk^2+epsilon^2))*exp(-0.5*((mk-T)/sqrt(sk^2+epsilon^2))^2)
weight[is.nan(weight)] <- 0

#evaluate criterion on the grid		
timse.EI.grid <- apply(design.grid, 1, timse_optim, weight=weight, 
model=model, integration.points=design.grid)
z.grid <- matrix(timse.EI.grid, n.grid, n.grid)

#plots: contour of the criterion, doe points and new point
contour(x.grid,y.grid,z.grid,25)
points(design.fact, col="black", pch=20, lwd=4)

#plots: contour of the actual function at threshold
testfun.grid <- testfun(design.grid)
z.grid.2 <- matrix(testfun.grid, n.grid, n.grid)
contour(x.grid,y.grid,z.grid.2,levels=T,col="blue",add=TRUE)
title("Contour lines of timse criterion (black) and of f(x)=T (blue)")

#search best point with Genoud
opt <- max_timse(lower=c(0,0), upper=c(1,1), T=T, epsilon=epsilon, 
model=model, integration.points=design.grid)
points(opt$par, col="blue", pch=20, lwd=4)

                      
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