sensitivity (version 1.14.0)

shapleyPermEx: Estimation of Shapley effects by examining all permutations of inputs (Agorithm of Song et al, 2016), in cases of independent or dependent inputs

Description

shapleyPermEx implements the Monte Carlo estimation of the Shapley effects (Owen, 2014) by examining all permutations of inputs (Song et al., 2016). It also estimates first order and total Sobol' indices. The function also allows the estimations of all these sensitivity indices in case of dependent inputs. The total cost of this algorithm is $Nv + d! * (d - 1) * No * Ni$ model evaluations.

Usage

shapleyPermEx(model = NULL, Xall, Xset, d, Nv, No, Ni = 3, colnames = NULL, ...)
"tell"(x, y = NULL, return.var = NULL, ...)
"print"(x, ...)
"plot"(x, ylim = c(0, 1), ...)

Arguments

model

a function, or a model with a predict method, defining the model to analyze.

Xall

Xall(n) is a function to generate a n-sample of a d-dimensional input vector (following the required joint distribution).

Xset

Xset(n, Sj, Sjc, xjc) is a function to generate a n-sample of a d-dimensional input vector corresponding to the indices in Sj conditional on the input values xjc with the index set Sjc (following the required joint distribution).

d

number of inputs.

Nv

Monte Carlo sample size to estimate the output variance.

No

Outer Monte Carlo sample size to estimate the expectation of the conditional variance of the model output.

Ni

Inner Monte Carlo sample size to estimate the conditional variance of the model output.

colnames

Optional: A vector containing the names of the inputs.

x

a list of class "shapleyPermEx" storing the state of the sensitivity study (parameters, data, estimates).

y

a vector of model responses.

return.var

a vector of character strings giving further internal variables names to store in the output object x.

ylim

y-coordinate plotting limits.

...

any other arguments for model which are passed unchanged each time it is called.

Value

shapleyPermEx returns a list of class "shapleyPermEx", containing all the input arguments detailed before, plus the following components:Users can ask more ouput variables with the argument return.var (for example, the list of permutations perms).

Details

This function requires R package "gtools".

The computations of the standard errors (and then the confidence intervals) is not realized with this function. The default values Ni = 3 is the optimal one obtained by the theoretical analysis of Song et al., 2016.

References

B. Iooss and C. Prieur, 2017, Analyse de sensibilite avec entrees dependantes : estimation par echantillonnage et par metamodeles des indices de Shapley, Submitted to 49emes Journees de la SFdS, Avignon, France, May 2017.

S. Kucherenko, S. Tarantola, and P. Annoni, 2012, Estimation of global sensitivity indices for models with dependent variables, Computer Physics Communications, 183, 937--946.

A.B. Owen, 2014, Sobol' indices and Shapley value, SIAM/ASA Journal of Uncertainty Quantification, 2, 245--251.

A.B. Owen and C. Prieur, 2016, On Shapley value for measuring importance of dependent inputs, submitted.

E. Song, B.L. Nelson, and J. Staum, 2016, Shapley effects for global sensitivity analysis: Theory and computation, SIAM/ASA Journal of Uncertainty Quantification, 4, 1060--1083.

See Also

shapleyPermRand

Examples

## Not run: 
# 
# ##################################
# # Test case : the Ishigami function (3 uniform independent inputs)
# # See Iooss and Prieur (2017)
# 
# library(gtools)
# 
# d <- 3
# Xall <- function(n) matrix(runif(d*n,-pi,pi),nc=d)
# Xset <- function(n, Sj, Sjc, xjc) matrix(runif(n*length(Sj),-pi,pi),nc=length(Sj))
# 
# x <- shapleyPermEx(model = ishigami.fun, Xall=Xall, Xset=Xset, d=d, Nv=1e4, No = 1e3, Ni = 3)
# print(x)
# plot(x)
# 
# ##################################
# # Test case : Linear model (3 Gaussian inputs including 2 dependent)
# # See Iooss and Prieur (2017)
# 
# library(gtools)
# library(mvtnorm) # Multivariate Gaussian variables
# library(condMVNorm) # Conditional multivariate Gaussian variables
# 
# modlin <- function(X) apply(X,1,sum)
# 
# d <- 3
# mu <- rep(0,d)
# sig <- c(1,1,2)
# ro <- 0.9
# Cormat <- matrix(c(1,0,0,0,1,ro,0,ro,1),d,d)
# Covmat <- ( sig %*% t(sig) ) * Cormat
# 
# Xall <- function(n) mvtnorm::rmvnorm(n,mu,Covmat)
# 
# Xset <- function(n, Sj, Sjc, xjc){
#   if (is.null(Sjc)){
#     if (length(Sj) == 1){ rnorm(n,mu[Sj],sqrt(Covmat[Sj,Sj]))
#     } else{ mvtnorm::rmvnorm(n,mu[Sj],Covmat[Sj,Sj])}
#   } else{ condMVNorm::rcmvnorm(n, mu, Covmat, dependent.ind=Sj, given.ind=Sjc, X.given=xjc)}}
# 
# x <- shapleyPermEx(model = modlin, Xall=Xall, Xset=Xset, d=d, Nv=1e4, No = 1e3, Ni = 3)
# print(x)
# plot(x)
# ## End(Not run)