PLIquantile: Perturbed-Law based sensitivity Indices (PLI) for quantile

Description

PLIquantile computes the Perturbed-Law based Indices (PLI) for quantile, which are sensitivity indices related to a quantile of a model output, estimated by a Monte Carlo method, See Sueur et al. (2016, 2017).

Usage

PLIquantile(order,x,y,quantilehat,deltasvector,
       InputDistributions,type="MOY",samedelta=TRUE)

Arguments

order

the order of the quantile to estimate.

the matrix of simulation points coordinates, one column per variable.

the vector of model outputs.

quantilehat

the estimation of quantile q.

deltasvector

a vector containing the values of delta for which the indices will be computed.

InputDistributions

a list of list. Each list contains, as a list, the name of the distribution to be used and the parameters. Implemented cases so far:

For a mean perturbation: Gaussian, Uniform, Triangle, Left Trucated Gaussian, Left Truncated Gumbel. Using Gumber requires the package evd.
For a variance perturbation: Gaussian, Uniform.

type

a character string in which the user will specify the type of perturbation wanted. The sense of "deltasvector" varies according to the type of perturbation:

type can take the value "MOY",in which case deltasvector is a vector of perturbated means.
type can take the value "VAR",in which case deltasvector is a vector of perturbated variances, therefore needs to be positive integers.

samedelta

a boolean used with the value "MOY" for type.

If it is set at TRUE, the mean perturbation will be the same for all the variables.
If not, the mean perturbation will be new_mean = mean+sigma*delta where mean, sigma are parameters defined in InputDistributions and delta is a value of deltasvector.

Value

PLIquantile returns a matrix where the PLI are stored. Each column corresponds to an input, each line corresponds to a twist of amplitude delta.

References

P. Lemaitre, E. Sergienko, A. Arnaud, N. Bousquet, F. Gamboa and B. Iooss, 2015, Density modification based reliability sensitivity analysis, Journal of Statistical Computation and Simulation, 85:1200-1223.

R. Sueur, N. Bousquet, B. Iooss and J. Bect, 2016, Perturbed-Law based sensitivity Indices for sensitivity analysis in structural reliability, Proceedings of the SAMO 2016 Conference, Reunion Island, France, December 2016.

R. Sueur, B. Iooss and T. Delage, 2017, Sensitivity analysis using perturbed-law based indices for quantiles and application to an industrial case, 10th International Conference on Mathematical Methods in Reliability (MMR 2017), Grenoble, France, July 2017.

Examples

Run this code

# NOT RUN {
# Model: 3D function 

  distribution = list()
	for (i in 1:3) distribution[[i]]=list("norm",c(0,1))
  
# Monte Carlo sampling to obtain failure points

  N = 10000
	X = matrix(0,ncol=3,nrow=N)
	for(i in 1:3) X[,i] = rnorm(N,0,1)
     
	Y = 2 * X[,1] + X[,2] + X[,3]/2
	q95 = quantile(Y,0.95)
	
# sensitivity indices with perturbation of the mean 
  
	v_delta = seq(-1,1,1/10) 
	toto = PLIquantile(0.95,X,Y,q95,deltasvector=v_delta,
	  InputDistributions=distribution,type="MOY",samedelta=TRUE)

	par(mar=c(4,5,1,1))
	plot(v_delta,toto[,2],ylim=c(-4.5,4.5),xlab=expression(delta),
		ylab=expression(hat(S[i*delta])),pch=19,cex=1.5)
	points(v_delta,toto[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,toto[,3],col="red",pch=17,cex=1.5)
	abline(h=0,lty=2)
	legend(0.8,4.4,legend=c("X1","X2","X3"),
		col=c("darkgreen","black","red"),pch=c(15,19,17),cex=1.5)
  
# }