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

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Description

A DGSM is the product between a Poincare Constant (Roustant et al., 2014) and the integral (over the space domain of the input variables) of the squared derivatives of the model output with respect to one input variable. The DGSM is a maximal bound of the total Sobol' index corresponding to the same input (Lamboni et al., 2013).

This DGSM depends on the type of probability distribution of the input variable. In the particular case of log-concave distribution, analytical formulas are available 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 non-truncated distributions (including the non log-concave case), the Poincare constant is computed via a relatively simple optimization process (Lamboni et al., 2013).

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

  • uniform on [min,max] interval: The optimal Poincare constant is (max-min)^2/(pi^2).

  • normal with a standard deviation sd: The optimal Poincare constant is sd^2.

Usage

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PoincareConstant(densityfct=dnorm, qfct=qnorm, cdfct,
                 truncated=FALSE, min=0, max=1, 
                 logconcave=TRUE, optimize.interval=c(-100, 100), ...)

Arguments

densityfct

the probability density function of the input variable

qfct

the quantile function of the input variable

cdfct

the distribution function of the input variable

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

logconcave

logical value: TRUE (default value) for a log-concave distribution

optimize.interval

In the non-log concave case, a vector containing the end-points of the interval to be searched for the maximum of the function to be optimized

...

additional arguments

Value

PoincareConstant returns the value of the Poincare constant.

Author(s)

Jana Fruth and Bertrand Iooss

References

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.

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.

Examples

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# Exponential law (log-concave)
PoincareConstant(dexp,qexp,rate=1)

# Weibull law (non log-concave)
PoincareConstant(dweibull,cdfct=pweibull, logconcave=FALSE, 
optimize.interval=c(0, 15), shape=1, scale=1)

## Not run: 
# Triangular law (log-concave)
library(triangle)
PoincareConstant(dtriangle, qtriangle, a=49, b=51, c=50)

# Truncated Gumbel law (log-concave)
library(evd)
PoincareConstant(dgumbel, qgumbel, pgumbel, truncated=TRUE, 
min=500, max=3000, loc=1013.0, scale=558.0) 

## End(Not run)

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