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

Description

PLIquantile computes the Perturbed-Law based Indices (PLI) for quantile, which are robustness indices related to a quantile of a model output, estimated by a Monte Carlo method, See Sueur et al. (2017) and Iooss et al. (2020).

Usage

PLIquantile(order,x,y,deltasvector,InputDistributions,type="MOY",samedelta=TRUE,
            percentage=TRUE,nboot=0,conf=0.95,bootsample=TRUE)

Value

PLIquantile returns a list of matrices (each column corresponds to an input, each line corresponds to a twist of amplitude delta) containing the following components:

PLI: the PLI.
PLICIinf: the bootstrap lower confidence interval values of the PLI.
PLICIsup: the bootstrap upper confidence interval values of the PLI.
quantile: the perturbed quantile.
quantileCIinf: the bootstrap lower confidence interval values of the perturbed quantile.
quantileCIsup: the bootstrap upper confidence interval values of the perturbed quantile.

Arguments

order

the order of the quantile to estimate.

x

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

y

the vector of model outputs.

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 Gumbel 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.

percentage

a boolean that defines the formula used for the PLI.

If it is set at FALSE, the initially proposed formula is used (see Sueur et al., 2017).
If not (set as TRUE), the PLI is given in percentage of variation of the quantile (see Iooss et al., 2020).

nboot

the number of bootstrap replicates.

conf

the confidence level for bootstrap confidence intervals.

bootsample

If TRUE, the uncertainty about the original quantile estimation is taken into account in the PLI confidence intervals (see Iooss et al., 2021). If FALSE, standard confidence intervals are computed for the PLI. It mainly changes the CI at small delta values.

Author

Paul Lemaitre, Bertrand Iooss, Thibault Delage and Roman Sueur

References

T. Delage, R. Sueur and B. Iooss, 2018, Robustness analysis of epistemic uncertainties propagation studies in LOCA assessment thermal-hydraulic model, ANS Best Estimate Plus Uncertainty International Conference (BEPU 2018), Lucca, Italy, May 13-19, 2018.

C. Gauchy, J. Stenger, R. Sueur and B. Iooss, 2022, An information geometry approach for robustness analysis in uncertainty quantification of computer codes, Technometrics, 64:80-91.

B. Iooss, V. Verges and V. Larget, 2022, BEPU robustness analysis via perturbed law-based sensitivity indices, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 236:855-865.

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

# \donttest{

# Model: 3D function 

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

  N = 5000
	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
	alpha <- 0.95 # quantile order
	
	q95 = quantile(Y,alpha)
	
	nboot=20 # put nboot=200 for consistency
	
# sensitivity indices with perturbation of the mean 
  
	v_delta = seq(-1,1,1/10) 
	toto = PLIquantile(alpha,X,Y,deltasvector=v_delta,
	  InputDistributions=distribution,type="MOY",samedelta=TRUE,
	  percentage=FALSE,nboot=nboot)

# Plotting the PLI

  par(mar=c(4,5,1,1))
	plot(v_delta,toto$PLI[,2],ylim=c(-1.5,1.5),xlab=expression(delta),
		ylab=expression(hat(PLI[i*delta])),pch=19,cex=1.5)
	points(v_delta,toto$PLI[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,toto$PLI[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,toto$PLICIinf[,2],col="black")
	lines(v_delta,toto$PLICIsup[,2],col="black")
	lines(v_delta,toto$PLICIinf[,1],col="darkgreen")
	lines(v_delta,toto$PLICIsup[,1],col="darkgreen")
	lines(v_delta,toto$PLICIinf[,3],col="red")
	lines(v_delta,toto$PLICIsup[,3],col="red")
	abline(h=0,lty=2)
	legend(0.8,1.5,legend=c("X1","X2","X3"),
		col=c("darkgreen","black","red"),pch=c(15,19,17),cex=1.5)
  
# Plotting the perturbed quantiles

  par(mar=c(4,5,1,1))
	plot(v_delta,toto$quantile[,2],ylim=c(1.5,6.5),xlab=expression(delta),
		ylab=expression(hat(q[i*delta])),pch=19,cex=1.5)
	points(v_delta,toto$quantile[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,toto$quantile[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,toto$quantileCIinf[,2],col="black")
	lines(v_delta,toto$quantileCIsup[,2],col="black")
	lines(v_delta,toto$quantileCIinf[,1],col="darkgreen")
	lines(v_delta,toto$quantileCIsup[,1],col="darkgreen")
	lines(v_delta,toto$quantileCIinf[,3],col="red")
	lines(v_delta,toto$quantileCIsup[,3],col="red")
	abline(h=q95,lty=2)
	legend(0.5,2.4,legend=c("X1","X2","X3"),
		col=c("darkgreen","black","red"),pch=c(15,19,17),cex=1.5)
		
###########################################################		
# Plotting the PLI in percentage with refined confidence intervals

	toto = PLIquantile(alpha,X,Y,deltasvector=v_delta,
	  InputDistributions=distribution,type="MOY",samedelta=TRUE,
	  percentage=TRUE,nboot=nboot,bootsample=FALSE)
	  
  par(mar=c(4,5,1,1))
	plot(v_delta,toto$PLI[,2],ylim=c(-0.6,0.6),xlab=expression(delta),
		ylab=expression(hat(PLI[i*delta])),pch=19,cex=1.5)
	points(v_delta,toto$PLI[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,toto$PLI[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,toto$PLICIinf[,2],col="black")
	lines(v_delta,toto$PLICIsup[,2],col="black")
	lines(v_delta,toto$PLICIinf[,1],col="darkgreen")
	lines(v_delta,toto$PLICIsup[,1],col="darkgreen")
	lines(v_delta,toto$PLICIinf[,3],col="red")
	lines(v_delta,toto$PLICIsup[,3],col="red")
	abline(h=0,lty=2)
	legend(0,0.6,legend=c("X1","X2","X3"),
		col=c("darkgreen","black","red"),pch=c(15,19,17),cex=1.5)

###################################################		
# another visualization by using the plotCI() fct 
# (from plotrix package) for the CI plotting(from Vanessa Verges)

  library(plotrix)
  parameters = list(colors=c("darkgreen","black","red"),
                  symbols=c(15,19,17),overlay=c(FALSE,TRUE,TRUE))
  par(mar=c(4,5,1,1),xpd=TRUE)
  for (i in 1:3){
    plotCI(v_delta,toto$PLI[,i],ui=toto$PLICIsup[,i],li=toto$PLICIinf[,i],
         cex=1.5,col=parameters$colors[i],pch=parameters$symbols[i],
         add=parameters$overlay[i], xlab="", ylab="")
  }
  title(xlab=expression(delta),ylab=expression(hat(PLI[i*delta])),
       main=bquote("PLI-quantile (N ="~.(N) ~ ","~alpha~"="~.(alpha)~
       ") of Y="~2*X[1] + X[2] + X[3]/2))
  abline(h=0,lty=2)
  legend("topleft",legend=c("X1","X2","X3"),col=parameters$colors,
          pch=parameters$symbols,cex=1.5)

	
# }

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