sensitivity (version 1.22.1)

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)

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

Value

The vector sample of the transformed variable

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. 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

# NOT RUN {
n <- 100  # sample size
c <- 1.5
Y <- rnorm(n)
Yt <- weightTSA(Y, c)
# }