KrigInv (version 1.3.1)

integration_design: 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" 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" and "jn" (default: "sobol"). C.1) The choice "sobol" corresponds to integration points chosen with the Sobol sequence in dimension d (uniform weight). C.2) The choice "MC" corresponds to points chosen randomly, uniformly on the domain. C.3) The choices "timse", "imse", "sur" and "jn" correspond to importance sampling distributions (unequal weights). It is strongly 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 the space filling distributions "sobol" and "MC" or an user defined distribution "spec". The "sobol" and "MC" choices correspond to quasi random and random points in the domain. If the "spec" value is chosen the user must fill in manually the field init.distrib.spec to specify himself 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

Target value (scalar). The sampling algorithm and the underlying kriging model aim at finding the points for which the output is close to T.

min.prob

This argument applies only when importance sampling distributions are chosen (to compute integral criteria like "sur" or "timse"). For numerical reasons we give a minimum probability for a point to belong to the importance sample. This avoids probabilities equal to zero and importance sampling weights equal to infinity. In an importance sample of M points, the maximum weight becomes 1/min.prob * 1/M.

Value

A list with components:

integration.points

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

integration.weights

a vector of size p corresponding to the weight of each point. 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

Details

The important sampling aims at improving the accuracy of the calculation of numerical integrals present in these criteria.

References

Chevalier C., Picheny V., Ginsbourger D. (2012), The KrigInv package: An efficient and user-friendly R implementation of Kriging-based inversion algorithms , http://hal.archives-ouvertes.fr/hal-00713537/

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/

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

See Also

EGI, max_timse, max_sur

Examples

# NOT RUN {
#integration_design

# }
# NOT RUN {
set.seed(8)
#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$integration.points)
# }
# NOT RUN {
#an example with pure random integration points
integcontrol <- list(distrib="MC",n.points=50)
integ.param <- integration_design(integcontrol=integcontrol,
		lower=c(0,0),upper=c(1,1))
plot(integ.param$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
N <- 14;testfun <- branin; T <- 80
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="timse",n.points=200,n.candidates=5000,init.distrib="MC")
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="timse",
col.points.init="red",cex.points=2,
main="timse uncertainty and one sample of integration points")

points(integ.param$integration.points,pch=17,cex=1)

# }