shapleyLinearGaussian {sensitivity}

Computation of the Shapley effects in the linear Gaussian framework

Description

shapleyLinearGaussian implements the computation of the Shapley effects in the linear Gaussian framework, using the linear model (without the value at zero) and the covariance matrix of the inputs. It uses the block-diagonal covariance trick of Broto et al. (2019) which allows to go through high-dimensional cases (nb of inputs > 25). It gives a warning in case of dim(block) > 25.

Usage

shapleyLinearGaussian(Beta, Sigma, tol=10^(-6))

Arguments

Beta

a vector containing the coefficients of the linear model (without the value at zero).

Sigma

covariance matrix of the inputs. Has to be positive semi-definite matrix with same size that Beta.

tol

a relative tolerance to detect zero singular values of Sigma.

Value

shapleyLinearGaussian returns a numeric vector containing all the Shapley effects.

Author(s)

Baptiste Broto

References

B. Broto, F. Bachoc, M. Depecker, and J-M. Martinez, 2019, Sensitivity indices for independent groups of variables, Mathematics and Computers in Simulation, 163, 19–31.

B. Broto, F. Bachoc, L. Clouvel and J-M Martinez, 2022,Block-diagonal covariance estimation and application to the Shapley effects in sensitivity analysis, SIAM/ASA Journal on Uncertainty Quantification, 10, 379–403.

B. Iooss and C. Prieur, 2019, Shapley effects for sensitivity analysis with correlated inputs: comparisons with Sobol' indices, numerical estimation and applications, International Journal for Uncertainty Quantification, 9, 493–514.

A.B. Owen and C. Prieur, 2016, On Shapley value for measuring importance of dependent inputs, SIAM/ASA Journal of Uncertainty Quantification, 5, 986–1002.

See Also

shapleyBlockEstimation, shapleyPermEx, shapleyPermRand, shapleySubsetMc

Examples

library(MASS)
library(igraph)

# First example:

p=5 #dimension
A=matrix(rnorm(p^2),nrow=p,ncol=p)
Sigma=t(A)%*%A
Beta=runif(p)
Shapley=shapleyLinearGaussian(Beta,Sigma)
plot(Shapley)


# Second Example, block-diagonal:

K=5 #number of groups
m=5 # number of variables in each group
p=K*m
Sigma=matrix(0,ncol=p,nrow=p)

for(k in 1:K)
{
  A=matrix(rnorm(m^2),nrow=m,ncol=m)
  Sigma[(m*(k-1)+1):(m*k),(m*(k-1)+1):(m*k)]=t(A)%*%A
}
# we mix the variables:
samp=sample(1:p,p)
Sigma=Sigma[samp,samp]

Beta=runif(p)
Shapley=shapleyLinearGaussian(Beta,Sigma)
plot(Shapley)