PLI: Perturbed-Law based sensitivity Indices (PLI) for failure...
In sensitivity: Global Sensitivity Analysis of Model Outputs
PLI R Documentation
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)
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.
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.
Author(s)
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
# 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)
sensitivity documentation built on Aug. 31, 2023, 5:10 p.m.
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| PLI | R Documentation |
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)
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:
|
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:
|
samedelta |
a boolean used with the value "MOY" for type.
|
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.
Author(s)
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
# 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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