integration_design {KrigInv}

Construction of a sample of integration points and weights

Description

Generic function to build integration points for some sampling criterion. Available important sampling schemes are "sur", "jn", "timse", "vorob" and "imse". Each of them corresponds to a sampling criterion.

Usage

integration_design(integcontrol = NULL, d = NULL, 
lower, upper, model = NULL, T = NULL,min.prob=0.001)

Arguments

integcontrol

Optional list specifying the procedure to build the integration points and weights, relevant only for the sampling criteria based on numerical integration: ("imse", "timse", "sur", "vorob" or "jn"). Many options are possible. A) If nothing is specified, 100*d points are chosen using the Sobol sequence. B) One can directly set the field integration.points (a p * d matrix) for prespecified integration points. In this case these integration points and the corresponding vector integration.weights will be used for all the iterations of the algorithm. C) If the field integration.points is not set then the integration points are renewed at each iteration. In that case one can control the number of integration points n.points (default: 100*d) and a specific distribution distrib. Possible values for distrib are: "sobol", "MC", "timse", "imse", "sur", "vorob" and "jn" (default: "sobol"). C.1) The choice "sobol" corresponds to integration points chosen with the Sobol sequence in dimension d (equal weights). C.2) The choice "MC" corresponds to points chosen randomly, uniformly on the domain (equal weights). C.3) The choices "timse", "imse", "sur", "vorob" and "jn" correspond to importance sampling distributions (unequal weights). It is recommended to use the importance sampling distribution corresponding to the chosen sampling criterion. When important sampling procedures are chosen, n.points points are chosen using importance sampling among a discrete set of n.candidates points (default: n.points*10) which are distributed according to a distribution init.distrib (default: "sobol"). Possible values for init.distrib are "sobol" or "MC" (uniform random points) or an user defined distribution "spec". If the "spec" value is chosen the user must fill manually the field init.distrib.spec with a n.candidates*d matrix of points in dimension d.

d

The dimension of the input set. If not provided d is set equal to the length of lower.

lower

Vector containing the lower bounds of the design space.

upper

Vector containing the upper bounds of the design space.

model

A Kriging model of km class.

T

Array containing one or several thresholds.

min.prob

This argument applies only when importance sampling distributions are chosen. For numerical reasons we give a minimum probability for a point to belong to the importance sample. This avoids potential importance sampling weights equal to infinity. In an importance sample of M points, the maximum weight becomes 1/min.prob * 1/M.

Details

The important sampling aims at improving the accuracy of the computation of criteria which involve numerical integration, like "timse", "sur", etc.

Value

A list with components:

integration.points

p * d matrix of p points used for the numerical calculation of integrals

integration.weights

Vector of size p corresponding to the weights of each points. If all the points are equally weighted, integration.weights is set to NULL

alpha

If the "vorob" important sampling schemes is chosen, the function also returns a scalar, alpha, being the calculated Vorob'ev threshold

Author(s)

Clement Chevalier (University of Neuchatel, Switzerland)

References

Chevalier C., Bect J., Ginsbourger D., Vazquez E., Picheny V., Richet Y. (2014), Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set, Technometrics, vol. 56(4), pp 455-465

Chevalier C. (2013) Fast uncertainty reduction strategies relying on Gaussian process models Ph.D Thesis, University of Bern

See Also

max_timse_parallel, max_sur_parallel

Examples

#integration_design


#when nothing is specified: integration points 
#are chosen with the sobol sequence
integ.param <- integration_design(lower=c(0,0),upper=c(1,1))
plot(integ.param$integration.points)


#an example with pure random integration points
integcontrol <- list(distrib="MC",n.points=50)
integ.param <- integration_design(integcontrol=integcontrol,
		lower=c(0,0),upper=c(1,1))
plot(integ.param$integration.points)

#an example with important sampling distributions
#these distributions are used to compute integral criterion like
#"sur","timse" or "imse"

#for these, we need a kriging model
set.seed(9)
N <- 16;testfun <- branin
lower <- c(0,0);upper <- c(1,1)
design <- data.frame( matrix(runif(2*N),ncol=2) )
response <- testfun(design)
model <- km(formula=~., design = design, 
	response = response,covtype="matern3_2")
integcontrol <- list(distrib="sur",n.points=200,n.candidates=5000,
  init.distrib="MC")
  
T <- c(60,100) 
#we are interested in the set of points where the response is in [60,100]

integ.param <- integration_design(integcontrol=integcontrol,
		lower=c(0,0),upper=c(1,1), model=model,T=T)

print_uncertainty_2d(model=model,T=T,type="sur",
col.points.init="red",cex.points=2,
main="sur uncertainty and one sample of integration points")
points(integ.param$integration.points,pch=17,cex=1)