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sensitivity (version 1.30.1)

PLI: Perturbed-Law based sensitivity Indices (PLI) for failure probability

Description

PLI computes the Perturbed-Law based Indices (PLI), also known as the Density Modification Based Reliability Sensitivity Indices (DMBRSI), which are robustness indices related to a probability of exceedence of a model output (i.e. a failure probability), estimated by a Monte Carlo method. See Lemaitre et al. (2015).

Usage

PLI(failurepoints,failureprobabilityhat,samplesize,deltasvector,
       InputDistributions,type="MOY",samedelta=TRUE)

Value

PLI returns a list of matrices, containing:

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

  • A matrix where their standard deviation are stored.

Arguments

failurepoints

a matrix of failure points coordinates, one column per variable.

failureprobabilityhat

the estimation of failure probability P through rough Monte Carlo method.

samplesize

the size of the sample used to estimate P. One must have Pchap=dim(failurepoints)[1]/samplesize

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.

Author

Paul Lemaitre and Bertrand Iooss

References

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

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

E. Borgonovo and B. Iooss, 2017, Moment independent importance measures and a common rationale, In: Springer Handbook on UQ, R. Ghanem, D. Higdon and H. Owhadi (Eds).

See Also

PLIquantile, PLIquantile_multivar, PLIsuperquantile, PLIsuperquantile_multivar

Examples

Run this code
# \donttest{

# Model: Ishigami function with a treshold at -7
# Failure points are those < -7

  distributionIshigami = list()
	for (i in 1:3){
		distributionIshigami[[i]]=list("unif",c(-pi,pi))
		distributionIshigami[[i]]$r=("runif")
	}
  
# Monte Carlo sampling to obtain failure points

  N = 100000
	X = matrix(0,ncol=3,nrow=N)
	for( i in 1:3) X[,i] = runif(N,-pi,pi)
	T = ishigami.fun(X)
	s = sum(as.numeric(T < -7)) # Number of failure
	pdefchap = s/N      # Failure probability
	ptsdef = X[T < -7,]	# Failure points
	
# sensitivity indices with perturbation of the mean 
  
	v_delta = seq(-3,3,1/20) 
	Toto = PLI(failurepoints=ptsdef,failureprobabilityhat=pdefchap,samplesize=N,
		deltasvector=v_delta,InputDistributions=distributionIshigami,type="MOY",
		samedelta=TRUE)
	BIshm = Toto[[1]]
	SIshm = Toto[[2]]

	par(mfrow=c(1,1),mar=c(4,5,1,1))
	plot(v_delta,BIshm[,2],ylim=c(-4,4),xlab=expression(delta),
		ylab=expression(hat(PLI[i*delta])),pch=19,cex=1.5)
	points(v_delta,BIshm[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,BIshm[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,BIshm[,2]+1.96*SIshm[,2],col="black")
	lines(v_delta,BIshm[,2]-1.96*SIshm[,2],col="black")
	lines(v_delta,BIshm[,1]+1.96*SIshm[,1],col="darkgreen")
	lines(v_delta,BIshm[,1]-1.96*SIshm[,1],col="darkgreen")
	lines(v_delta,BIshm[,3]+1.96*SIshm[,3],col="red")
	lines(v_delta,BIshm[,3]-1.96*SIshm[,3],col="red")
	abline(h=0,lty=2)
	legend(0,3,legend=c("X1","X2","X3"),
		col=c("darkgreen","black","red"),pch=c(15,19,17),cex=1.5)
  
# sensitivity indices with perturbation of the variance 

	v_delta = seq(1,5,1/4) # user parameter. (the true variance is 3.29)	
	Toto = PLI(failurepoints=ptsdef,failureprobabilityhat=pdefchap,samplesize=N,
		deltasvector=v_delta,InputDistributions=distributionIshigami,type="VAR",
		samedelta=TRUE)
	BIshv=Toto[[1]]
	SIshv=Toto[[2]]

	par(mfrow=c(2,1),mar=c(1,5,1,1)+0.1)
	plot(v_delta,BIshv[,2],ylim=c(-.5,.5),xlab=expression(V_f),
		ylab=expression(hat(PLI[i*delta])),pch=19,cex=1.5)
	points(v_delta,BIshv[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,BIshv[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,BIshv[,2]+1.96*SIshv[,2],col="black")
	lines(v_delta,BIshv[,2]-1.96*SIshv[,2],col="black")
	lines(v_delta,BIshv[,1]+1.96*SIshv[,1],col="darkgreen")
	lines(v_delta,BIshv[,1]-1.96*SIshv[,1],col="darkgreen")
	lines(v_delta,BIshv[,3]+1.96*SIshv[,3],col="red")
	lines(v_delta,BIshv[,3]-1.96*SIshv[,3],col="red")

	par(mar=c(4,5.1,1.1,1.1))
	plot(v_delta,BIshv[,2],ylim=c(-30,.7),xlab=expression(V[f]),
		ylab=expression(hat(PLI[i*delta])),pch=19,cex=1.5)
	points(v_delta,BIshv[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,BIshv[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,BIshv[,2]+1.96*SIshv[,2],col="black")
	lines(v_delta,BIshv[,2]-1.96*SIshv[,2],col="black")
	lines(v_delta,BIshv[,1]+1.96*SIshv[,1],col="darkgreen")
	lines(v_delta,BIshv[,1]-1.96*SIshv[,1],col="darkgreen")
	lines(v_delta,BIshv[,3]+1.96*SIshv[,3],col="red")
	lines(v_delta,BIshv[,3]-1.96*SIshv[,3],col="red")
	legend(2.5,-10,legend=c("X1","X2","X3"),col=c("darkgreen","black","red"),
		pch=c(15,19,17),cex=1.5)
  
##############################################################
# Example with an inverse probability transform 
# (to obtain Gaussian inputs from Uniform ones)

# Monte Carlo sampling (the inputs are Uniform)

  N = 100000
	X = matrix(0,ncol=3,nrow=N)
	for( i in 1:3) X[,i] = runif(N,-pi,pi)
	T = ishigami.fun(X)
	s = sum(as.numeric(T < -7)) # Number of failure
	pdefchap = s/N      # Failure probability
	
# Empirical transform (applied on the sample)

  Xn <- matrix(0,nrow=N,ncol=3)
  for (i in 1:3){
    ecdfx <- ecdf(X[,i])
    q <- ecdfx(X[,i])
    Xn[,i] <- qnorm(q) # Gaussian anamorphosis
    # infinite max values => putting the symetrical values of min values
    Xn[which(Xn[,i]==Inf),i] <- - Xn[which.min(Xn[,i]),i] 
    }
# Visualization of a perturbed density (the one of X1 perturbed on the mean)
  delta_mean_gauss <- 1 # perturbed value on the mean of the Gaussian transform
  Xtr <- quantile(ecdfx,pnorm(Xn[,1] + delta_mean_gauss)) # backtransform
	par(mfrow=c(1,1))
  plot(density(Xtr), col="red") ; lines(density(X[,1]))
  
# sensitivity indices with perturbation of the mean 
  
  distributionIshigami = list()
	for (i in 1:3){
		distributionIshigami[[i]]=list("norm",c(0,1))
		distributionIshigami[[i]]$r=("rnorm")
	}
	
	ptsdef = Xn[T < -7,]	# Failure points # failure points with Gaussian distrib.
	
	v_delta = seq(-1.5,1.5,1/20) 
	Toto = PLI(failurepoints=ptsdef,failureprobabilityhat=pdefchap,samplesize=N,
		deltasvector=v_delta,InputDistributions=distributionIshigami,type="MOY",
		samedelta=TRUE)
	BIshm = Toto[[1]]
	SIshm = Toto[[2]]

	par(mfrow=c(1,1),mar=c(4,5,1,1))
	plot(v_delta,BIshm[,2],ylim=c(-4,4),xlab=expression(delta),
		ylab=expression(hat(PLI[i*delta])),pch=19,cex=1.5)
	points(v_delta,BIshm[,1],col="darkgreen",pch=15,cex=1.5)
	points(v_delta,BIshm[,3],col="red",pch=17,cex=1.5)
	lines(v_delta,BIshm[,2]+1.96*SIshm[,2],col="black")
	lines(v_delta,BIshm[,2]-1.96*SIshm[,2],col="black")
	lines(v_delta,BIshm[,1]+1.96*SIshm[,1],col="darkgreen")
	lines(v_delta,BIshm[,1]-1.96*SIshm[,1],col="darkgreen")
	lines(v_delta,BIshm[,3]+1.96*SIshm[,3],col="red")
	lines(v_delta,BIshm[,3]-1.96*SIshm[,3],col="red")
	abline(h=0,lty=2)
	legend(0,3,legend=c("X1","X2","X3"),
		col=c("darkgreen","black","red"),pch=c(15,19,17),cex=1.5)
  
# }

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