# Sensitivity Analysis

### Description

Methods and functions for global sensitivity analysis.

### Details

The sensitivity package implements some global sensitivity analysis methods:

Linear regression coefficients: SRC and SRRC (

`src`

), PCC and PRCC (`pcc`

);Bettonvil's sequential bifurcations (Bettonvil and Kleijnen, 1996) (

`sb`

);Morris's "OAT" elementary effects screening method (

`morris`

);Derivative-based Global Sensitivity Measures:

Poincare constants for Derivative-based Global Sensitivity Measures (DGSM) (Roustant et al., 2014) (

`PoincareConstant`

),Distributed Evaluation of Local Sensitivity Analysis (DELSA) (Rakovec et al., 2014) (

`delsa`

);

Variance-based sensitivity indices (Sobol' indices):

Monte Carlo estimation of Sobol' indices (also called pick-freeze method):

Sobol' scheme (Sobol, 1993) to compute the indices given by the variance decomposition up to a specified order (

`sobol`

),Saltelli's scheme (Saltelli, 2002) to compute first order and total indices with a reduced cost (

`sobol2002`

),Mauntz-Kucherenko's scheme (Sobol et al., 2007) to compute first order and total indices using improved formulas for small indices (

`sobol2007`

),Jansen-Sobol's scheme (Jansen, 1999) to compute first order and total indices using improved formulas (

`soboljansen`

),Martinez's scheme using correlation coefficient-based formulas (Martinez, 2011; touati, 2016) to compute first order and total indices, associated with theoretical confidence intervals (

`sobolmartinez`

and`soboltouati`

),Janon-Monod's scheme (Monod et al., 2006; Janon et al., 2013) to compute first order indices with optimal asymptotic variance (

`sobolEff`

),Mara's scheme (Mara and Joseph, 2008) to compute first order indices with a cost independent of the dimension, via a unique-matrix permutations (

`sobolmara`

),Owen's scheme (Owen, 2013) to compute first order and total indices using improved formulas (via 3 input independent matrices) for small indices (

`sobolowen`

),Total Interaction Indices using Liu-Owen's scheme (Liu and Owen, 2006) (

`sobolTIIlo`

) and pick-freeze scheme (Fruth et al., 2014) (`sobolTIIpf`

),

Estimation of the Sobol' first order and closed second order indices using replicated orthogonal array-based Latin hypecube sample (Tissot and Prieur, 2012) (

`sobolroalhs`

),Estimation of the Sobol' first order and total indices with Saltelli's so-called "extended-FAST" method (Saltelli et al., 1999) (

`fast99`

),Estimation of the Sobol' first order and total indices with kriging-based global sensitivity analysis (Le Gratiet et al., 2014) (

`sobolGP`

);

Support index functions (

`support`

) of Fruth et al. (2015);Sensitivity Indices based on Csiszar f-divergence (

`sensiFdiv`

) (particular cases: Borgonovo's indices and mutual-information based indices) and Hilbert-Schmidt Independence Criterion (`sensiHSIC`

) of Da Veiga et al. (2014);Reliability sensitivity analysis by the Perturbed-Law based Indices (

`PLI`

) of Lemaitre et al. (2015);Sobol' indices for multidimensional outputs (

`sobolMultOut`

): Aggregated Sobol' indices (Lamboni et al., 2011; Gamboa et al., 2014) and functional (1D) Sobol' indices.Moreover, some utilities are provided: standard test-cases (

`testmodels`

) and template file generation (`template.replace`

).

### Model managing

The sensitivity package has been designed to work either models written in **R**
than external models such as heavy computational codes. This is achieved with
the input argument `model`

present in all functions of this package.

The argument `model`

is expected to be either a
funtion or a predictor (i.e. an object with a `predict`

function such as
`lm`

).

If

`model = m`

where`m`

is a function, it will be invoked once by`y <- m(X)`

.If

`model = m`

where`m`

is a predictor, it will be invoked once by`y <- predict(m, X)`

.

`X`

is the design of experiments, i.e. a `data.frame`

with
`p`

columns (the input factors) and `n`

lines (each, an
experiment), and `y`

is the vector of length `n`

of the
model responses.

The model in invoked once for the whole design of experiment.

The argument `model`

can be left to `NULL`

. This is refered to as
the decoupled approach and used with external computational codes that rarely
run on the statistician's computer. See `decoupling`

.

### Author(s)

Gilles Pujol, Bertrand Iooss, Alexandre Janon with contributions from Paul Lemaitre for the `PLI`

function, Laurent Gilquin for the `sobolroalhs`

function, Loic le Gratiet for the `sobolGP`

function, Khalid Boumhaout, Taieb Touati and Bernardo Ramos for the `sobolowen`

and `soboltouati`

functions, Jana Fruth for the `PoincareConstant`

and `support`

functions, `sobolTIIlo`

and `sobolTIIpf`

functions, Sebastien Da veiga for the `sensiFdiv`

and `sensiHSIC`

functions, Joseph Guillaume for the `delsa`

and `parameterSets`

functions, Frank Weber.

(maintainer: Bertrand Iooss biooss@yahoo.fr)

### References

R. Faivre, B. Iooss, S. Mahevas, D. Makowski, H. Monod, editors, 2013,
*Analyse de sensibilite et exploration de modeles. Applications aux
modeles environnementaux*, Editions Quae.

B. Iooss and A. Saltelli (in press). *Introduction: Sensitivity analysis.* In: *Springer Handbook on Uncertainty Quantification*, R. Ghanem, D. Higdon and H. Owhadi (Eds), Springer.

B. Iooss and P. Lemaitre, 2015, *A review on global sensitivity analysis methods*. In *Uncertainty management in Simulation-Optimization of Complex Systems: Algorithms and Applications*, C. Meloni and G. Dellino (eds), Springer.

A. Saltelli, K. Chan and E. M. Scott eds, 2000, *Sensitivity Analysis*,
Wiley.

A. Saltelli, P. Annoni, I. Azzini, F. Campolongo, M. Ratto and S. Tarantola, 2010, *Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index*, Computer Physics Communications 181, 259–270.