PoincareConstant: Poincare constants for Derivative-based Global Sensitivity Measures (DGSM)

Description

A DGSM is a sensitivity index relying on the integral (over the space domain of the input variables) of the squared derivatives of a model output with respect to one model input variable. The product between a DGSM and a Poincare Constant (Roustant et al., 2014: Roustant et al., 2017) gives an upper bound of the total Sobol' index corresponding to the same input (Lamboni et al., 2013; Kucherenko and Iooss, 2016).

This Poincare constant depends on the type of probability distribution of the input variable. In the particular case of log-concave distribution, analytical formulas are available for double-exponential transport by the way of the median value (Lamboni et al., 2013). For truncated log-concave distributions, different formulas are available (Roustant et al., 2014). For general distributions (truncated or not), some Poincare constants can be computed via a relatively simple optimization process using different formula coming from transport inequalities (Roustant et al., 2017).

Notice that the analytical formula based on the log-concave law cases is a subcase of the double-exponential transport. In all cases, with this function, the smallest constant is obtained using the logistic transport formula. PoincareOptimal allows to obtained the best (optimal) constant using another (spectral) method.

IMPORTANT: This program is useless for the two following input variable distributions:

uniform on \([min,max]\) interval: The optimal Poincare constant is \(\frac{(max-min)^2}{pi^2}\).
normal with a standard deviation \(sd\): The optimal Poincare constant is \(sd^2\).

Usage

PoincareConstant(dfct=dnorm, qfct=qnorm, pfct=pnorm,
                 logconcave=FALSE, transport="logistic", 
                 optimize.interval=c(-100, 100),
                 truncated=FALSE, min=0, max=1,  ...)

Value

PoincareConstant returns the value of the Poincare constant.

Arguments

dfct: the probability density function of the input variable
qfct: the quantile function of the input variable
pfct: the distribution function of the input variable
logconcave: logical value: TRUE for a log-concave distribution (analyical formula will be used). Requires argument 'dfct' and 'qfct'. FALSE (default value) means that the calculations will be performed using transport-based formulas (applicable for log-concave and non-log concave cases)
transport: If logconcave=FALSE, choice of the transport inequalities to be used: "double_exp" (default value) for double exponential transport and "logistic" for logistic transport". Requires argument 'dfct' and 'pfct'
optimize.interval: In the transport-based case (logconcave=FALSE), a vector containing the end-points of the interval to be searched for the maximum of the function to be optimized
truncated: logical value: TRUE for a truncated distribution. Default value is FALSE
min: the minimal bound in the case of a truncated distribution
max: the maximal bound in the case of a truncated distribution
...: additional arguments

Author

Jana Fruth, Bertrand Iooss and Olivier Roustant

Details

In the case of truncated distributions (truncated=TRUE), in addition to the min and max arguments: - the truncated distribution name has to be passed in the 'dfct' and 'pfct' arguments if logconcave=FALSE, - the non-truncated distribution name has to be passed in the 'dfct' and 'qfct' arguments if logconcave=TRUE. Moreover, if min and max are finite, optimize.interval is required to be defined as c(min,max).

References

S. Kucherenko and B. Iooss, Derivative-based global sensitivity measures, In: R. Ghanem, D. Higdon and H. Owhadi (eds.), Handbook of Uncertainty Quantification, 2016.

M. Lamboni, B. Iooss, A-L. Popelin and F. Gamboa, Derivative-based global sensitivity measures: General links with Sobol' indices and numerical tests, Mathematics and Computers in Simulation, 87:45-54, 2013.

O. Roustant, F. Barthe and B. Iooss, Poincare inequalities on intervals - application to sensitivity analysis, Electronic Journal of Statistics, Vol. 11, No. 2, 3081-3119, 2017.

O. Roustant, J. Fruth, B. Iooss and S. Kuhnt, Crossed-derivative-based sensitivity measures for interaction screening, Mathematics and Computers in Simulation, 105:105-118, 2014.

Examples

Run this code


# Exponential law (log-concave)
PoincareConstant(dfct=dexp,qfct=qexp,pfct=NULL,rate=1,
  logconcave=TRUE) # log-concave assumption
PoincareConstant(dfct=dexp,qfct=NULL,pfct=pexp,rate=1,
  optimize.interval=c(0, 15)) # logistic transport approach

# Weibull law (log-concave)
PoincareConstant(dfct=dweibull,qfct=NULL,pfct=pweibull,
  optimize.interval=c(0, 15),shape=1,scale=1) # logistic transport approach

# \donttest{
# Triangular law (log-concave)
library(triangle)
PoincareConstant(dfct=dtriangle, qfct=qtriangle, pfct=NULL, a=-1, b=1, c=0, 
  logconcave=TRUE) # log-concave assumption
PoincareConstant(dfct=dtriangle, qfct=NULL, pfct=ptriangle, a=-1, b=1, c=0, 
  transport="double_exp", optimize.interval=c(-1,1)) # Double-exp transport 
PoincareConstant(dfct=dtriangle, qfct=NULL, pfct=ptriangle, a=-1, b=1, c=0, 
  optimize.interval=c(-1,1)) # Logistic transport calculation

# Normal N(0,1) law truncated on [-1.87,+infty]
PoincareConstant(dfct=dnorm,qfct=qnorm,pfct=pnorm,mean=0,sd=1,logconcave=TRUE, 
  transport="double_exp", truncated=TRUE, min=-1.87, max=999) # log-concave hyp 
# Double-exponential transport approach
PoincareConstant(dfct=dnorm.trunc, qfct=qnorm.trunc, pfct=pnorm.trunc, 
  mean=0, sd=1, truncated=TRUE, min=-1.87, max=999,   transport="double_exp", 
    optimize.interval=c(-1.87,20)) 
# Logistic transport approach
PoincareConstant(dfct=dnorm.trunc, qfct=qnorm.trunc, pfct=pnorm.trunc, 
  mean=0, sd=1, truncated=TRUE, min=-1.87, max=999, optimize.interval=c(-1.87,20)) 


# Gumbel law (log-concave)
library(evd)
PoincareConstant(dfct=dgumbel, qfct=qgumbel, pfct=NULL, loc=0, scale=1, 
  logconcave=TRUE, transport="double_exp") # log-concave assumption
PoincareConstant(dfct=dgumbel, qfct=NULL, pfct=pgumbel, loc=0, scale=1, 
  transport="double_exp", optimize.interval=c(-3,20)) # Double-exp transport 
PoincareConstant(dfct=dgumbel, qfct=qgumbel, pfct=pgumbel, loc=0, scale=1, 
  optimize.interval=c(-3,20)) # Logistic transport approach

# Truncated Gumbel law (log-concave)
# Double-exponential transport approach
PoincareConstant(dfct=dgumbel, qfct=qgumbel, pfct=pgumbel, loc=0, scale=1, 
  logconcave=TRUE, transport="double_exp", truncated=TRUE, 
  min=-0.92, max=3.56) # log-concave assumption
PoincareConstant(dfct=dgumbel.trunc, qfct=NULL, pfct=pgumbel.trunc, loc=0, scale=1, 
  truncated=TRUE, min=-0.92, max=3.56, transport="double_exp", 
  optimize.interval=c(-0.92,3.56))
# Logistic transport approach
PoincareConstant(dfct=dgumbel.trunc, qfct=qgumbel.trunc, pfct=pgumbel.trunc, 
  loc=0, scale=1, truncated=TRUE, min=-0.92, max=3.56, 
  optimize.interval=c(-0.92,3.56)) 
  
# }

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