sensitivity (version 1.28.0)

weightTSA: Weight-function to transform an output variable in order to perform Target Sensitivity Analysis (TSA)

Description

Transformation function of one variable (vector sample)

Usage

weightTSA(Y, c, upper = TRUE, type="indicTh", param=1)

Value

The vector sample of the transformed variable

Arguments

Y

The output vector

c

The threshold

upper

TRUE for upper threshold and FALSE for lower threshold

type

The weight function type ("indicTh", "zeroTh", logistic", "exp1side"):

• indicTh : indicator-thresholding

• zeroTh : zero-thresholding (keeps the variable value above (upper=TRUE case) or below the threshold)

• logistic : logistic transformation at the threshold

• exp1side : exponential transformation above (upper=TRUE case) or below the threshold (see Raguet and Marrel)

param

The parameter value for "logistic" and "exp1side" types

Author

B. Iooss

Details

The weight functions depend on a threshold \(c\) and/or a smooth relaxation. These functions are defined as follows

• if type = "indicTh": \(w = 1_{Y>c}\) (upper threshold) and \(w = 1_{Y<c}\) (lower threshold),

• if type = "zeroTh": \(w = Y 1_{Y>c}\) (upper threshold) and \(w = Y 1_{Y<c}\) (lower threshold),

• if type = "logistic": $$w = \left[ 1 + \exp{\left( -param\frac{Y-c}{|c|}\right)}\right]^{-1}$$ (upper threshold) and $$w = \left[ 1 + \exp{\left( -param\frac{c-Y}{|c|}\right)}\right]^{-1}$$ (lower threshold),

• if type = "exp1side": $$w = \left[ 1 + \exp{\left( -\frac{\max(c - Y, 0)}{\frac{param}{5} \sigma(Y)}\right)}\right]$$ (upper threshold) and $$w = \left[ 1 + \exp{\left( -\frac{\max(Y - c, 0)}{\frac{param}{5} \sigma(Y)}\right) }\right]$$ (lower threshold), where \(\sigma(Y)\) is an estimation of the standard deviation of Y and \(param = 1\) is a parameter tuning the smoothness.

References

H. Raguet and A. Marrel, Target and conditional sensitivity analysis with emphasis on dependence measures, Preprint, https://hal.archives-ouvertes.fr/hal-01694129

A. Marrel and V. Chabridon, 2021, Statistical developments for target and conditional sensitivity analysis: Application on safety studies for nuclear reactor, Reliability Engineering & System Safety, 214:107711.

A. Spagnol, Kernel-based sensitivity indices for high-dimensional optimization problems, PhD Thesis, Universite de Lyon, 2020

Spagnol A., Le Riche R., Da Veiga S. (2019), Global sensitivity analysis for optimization with variable selection, SIAM/ASA J. Uncertainty Quantification, 7(2), 417--443.

Examples

n <- 100  # sample size
c <- 1.5
Y <- rnorm(n)
Yt <- weightTSA(Y, c)