Sobol Indices for Models with Fixed and Stochastic Parameters.

Sobol4R, Sobol Indices for Models with Fixed and Stochastic Parameters

Frédéric Bertrand, Elizaveta Logosha and Myriam Maumy-Bertrand

Tools to design experiments, compute Sobol sensitivity indices, and summarise stochastic responses inspired by the strategy described by Zhu et Sudret (2021) https://doi.org/10.1016/j.ress.2021.107815. Includes helpers to optimise toy models implemented in C++, visualise indices with uncertainty quantification, and derive reliability-oriented sensitivity measures based on failure probabilities. It is further detailed in Logosha, Maumy and Bertrand (2022) and (2023) <https://doi.org/10.1063/5.0246024 or in Bertrand, Logosha and Maumy (2024) <https://hal.science/hal-05371795 and https://hal.science/hal-05371798.

This site was created by F. Bertrand and the examples reproduced on it were created by F. Bertrand, E. Logosha and M. Maumy.

Installation

You can install the latest version of the Sobol4R package from github with:

devtools::install_github("fbertran/Sobol4R")

Two complementary analysis paths

Sobol4R exposes two ways to compute sensitivity indices depending on your workflow:

• Reuse the sensitivity package estimators. Provide pre-built designs and call sensitivity::sobol() or sensitivity::sobol2007(); the autoplot() methods in Sobol4R will visualise those results without changing your existing code.
• Use the in-package Saltelli re-implementation. The sobol_design() and sobol_indices() helpers build the matrices, run the model, and return a sobol_result object that can be summarised or plotted directly, with optional bootstrap quantiles for noisy simulations.

The README examples below demonstrate the second path, while the earlier "Context and non random case" section illustrates interoperability with sensitivity.

Motivation

The methodology implemented in Sobol4R builds upon the stochastic Sobol analysis described by Lebrun et al. (2021) in Reliability Engineering & System Safety. The paper proposes to combine replicated simulator runs with Sobol estimators to account for intrinsic noise. The package mirrors this workflow:

1. Generate two Monte Carlo designs (A and B matrices).
2. Evaluate the stochastic simulator several times to estimate the mean response and the noise variance.
3. Derive first-order and total-order Sobol indices and visualise the results.

The package is also friendly with the sensitivity package and shows how to use the Sobol' indices estimators provided in this package to increase the capabilities of this Sobol4R package.

Basic usage

library(Sobol4R)
set.seed(123)
design <- sobol_design(n = 256, d = 3, lower = rep(-pi, 3), upper = rep(pi, 3),
                       quasi = TRUE)
result <- sobol_indices(ishigami_model, design, replicates = 4,
                        keep_samples = TRUE)
result$data
#>   parameter  first_order  total_order
#> 1        X1 0.0001508541 4.263399e-05
#> 2        X2 0.3407381134 3.506136e-01
#> 3        X3 0.2005719638 1.963439e-01

The resulting object stores the Monte Carlo variance estimate, the average noise variance across replications, and the Sobol indices. Diagnostic plots are available through the provided autoplot() method:

autoplot(result)

When keep_samples = TRUE, bootstrap resamples quantify the estimator uncertainty. The helper summarise_sobol() produces tidy quantiles that can be visualised directly:

autoplot(result, show_uncertainty = TRUE, probs = c(0.1, 0.9), bootstrap = 100)

Reliability metrics

The paper highlights the need to quantify failure probabilities associated with critical performance levels. The package exposes a helper for that task:

set.seed(321)
simulated <- ishigami_model(matrix(runif(3000, -pi, pi), ncol = 3))
estimate_failure_probability(simulated, threshold = -1)
#> $probability
#> [1] 0.087
#> 
#> $variance
#> [1] 7.9431e-05

When the simulator is stochastic and sobol_indices() stores the replicated samples (keep_samples = TRUE), the same Monte Carlo budget can be recycled to derive failure-indicator Sobol indices:

failure <- sobol_reliability(result, threshold = -1)
failure$failure_probability
#> [1] 0.08984375
autoplot(failure, show_uncertainty = TRUE, probs = c(0.1, 0.9), bootstrap = 200)

Combined usage with the sensitivity package

Test case : the non-monotonic Sobol g-function

The method of Sobol requires two samples. In this reference case there are eight variables, all following the uniform distribution on $[0,1]$.

if(require(sensitivity)){
n <- 50000
p <- 8
X1_1 <- data.frame(matrix(runif(p * n), nrow = n))
X2_1 <- data.frame(matrix(runif(p * n), nrow = n))
}

if(require(sensitivity)){
set.seed(4669)
gensol1 <- sobol4r_design(
  X1    = X1_1,
  X2    = X2_1,
  order = 2,
  nboot = 100
)

Y1 <- sobol_g_function(gensol1$X)
x1 <- sensitivity::tell(gensol1, Y1)
print(x1)
}
#> 
#> Call:
#> sensitivity::sobol(model = NULL, X1 = X1, X2 = X2, order = order,     nboot = nboot)
#> 
#> Model runs: 1850000 
#> 
#> Sobol indices
#>           original          bias  std. error
#> X1     0.714928599  0.0009558823 0.007223975
#> X2     0.183724278  0.0000624804 0.009787327
#> X3     0.027855443  0.0006997475 0.010403387
#> X4     0.010766859  0.0007602978 0.010419788
#> X5     0.004620120  0.0007537761 0.010448579
#> X6     0.004649758  0.0007462895 0.010452522
#> X7     0.004463461  0.0007438964 0.010427359
#> X8     0.004634047  0.0007602140 0.010421292
#> X1*X2  0.056723859 -0.0009475371 0.012211589
#> X1*X3  0.003935454 -0.0006296449 0.010796787
#> X1*X4 -0.002604387 -0.0007142755 0.010358372
#> X1*X5 -0.004544991 -0.0007418694 0.010437612
#> X1*X6 -0.004218935 -0.0007431264 0.010454798
#> X1*X7 -0.004514664 -0.0007320925 0.010439434
#> X1*X8 -0.004357070 -0.0007382679 0.010453573
#> X2*X3 -0.002000125 -0.0008414918 0.010476064
#> X2*X4 -0.004065270 -0.0007988678 0.010429953
#> X2*X5 -0.004516614 -0.0007620223 0.010449446
#> X2*X6 -0.004445576 -0.0007455077 0.010441943
#> X2*X7 -0.004463936 -0.0007408606 0.010444429
#> X2*X8 -0.004494686 -0.0007426589 0.010451796
#> X3*X4 -0.004482017 -0.0007738016 0.010445882
#> X3*X5 -0.004457124 -0.0007444663 0.010449294
#> X3*X6 -0.004480155 -0.0007438360 0.010445407
#> X3*X7 -0.004475348 -0.0007449987 0.010449367
#> X3*X8 -0.004435086 -0.0007448195 0.010447190
#> X4*X5 -0.004475161 -0.0007420839 0.010447688
#> X4*X6 -0.004469522 -0.0007452488 0.010447754
#> X4*X7 -0.004449099 -0.0007431436 0.010447623
#> X4*X8 -0.004464412 -0.0007423014 0.010445970
#> X5*X6 -0.004471409 -0.0007438270 0.010447393
#> X5*X7 -0.004471249 -0.0007439088 0.010447228
#> X5*X8 -0.004470571 -0.0007440524 0.010447349
#> X6*X7 -0.004469680 -0.0007437680 0.010447372
#> X6*X8 -0.004470664 -0.0007438057 0.010447396
#> X7*X8 -0.004472258 -0.0007439325 0.010447265
#>          min. c.i.  max. c.i.
#> X1     0.700146935 0.72663135
#> X2     0.161833558 0.20165596
#> X3     0.003977241 0.04506500
#> X4    -0.012120628 0.02849420
#> X5    -0.019076963 0.02308009
#> X6    -0.019033780 0.02316502
#> X7    -0.019093748 0.02292561
#> X8    -0.018936661 0.02305889
#> X1*X2  0.037376137 0.08179360
#> X1*X3 -0.016331343 0.02743327
#> X1*X4 -0.020601860 0.02117824
#> X1*X5 -0.023005656 0.01907945
#> X1*X6 -0.022671561 0.01932241
#> X1*X7 -0.022909245 0.01906720
#> X1*X8 -0.022923152 0.01925448
#> X2*X3 -0.019975577 0.02222722
#> X2*X4 -0.022468802 0.01906646
#> X2*X5 -0.022986048 0.01917049
#> X2*X6 -0.022902493 0.01917938
#> X2*X7 -0.022938548 0.01912497
#> X2*X8 -0.022999094 0.01916398
#> X3*X4 -0.022854213 0.01909924
#> X3*X5 -0.022950799 0.01916891
#> X3*X6 -0.022949312 0.01914533
#> X3*X7 -0.022956011 0.01914744
#> X3*X8 -0.022917907 0.01919287
#> X4*X5 -0.022953395 0.01914469
#> X4*X6 -0.022966075 0.01915940
#> X4*X7 -0.022941982 0.01916825
#> X4*X8 -0.022950242 0.01916666
#> X5*X6 -0.022958081 0.01915551
#> X5*X7 -0.022957662 0.01915454
#> X5*X8 -0.022956939 0.01915661
#> X6*X7 -0.022956400 0.01915675
#> X6*X8 -0.022958184 0.01915519
#> X7*X8 -0.022958435 0.01915437

if(require(sensitivity)){
autoplot(x1, ncol = 1)
}

You can also use the sobol_example_g_deterministic() wrapper for this example.

if(require(sensitivity)){ex1_results <- sobol_example_g_deterministic()
print(ex1_results)
}
#> 
#> Call:
#> sensitivity::sobol(model = NULL, X1 = X1, X2 = X2, order = order,     nboot = nboot)
#> 
#> Model runs: 1850000 
#> 
#> Sobol indices
#>            original          bias  std. error
#> X1     0.7245997507  1.318649e-04 0.006865099
#> X2     0.1852412158 -6.379462e-04 0.009725422
#> X3     0.0321041221 -3.943572e-04 0.009939738
#> X4     0.0150373622 -3.716233e-04 0.009571601
#> X5     0.0073639355 -5.240577e-04 0.009690646
#> X6     0.0073304377 -5.140176e-04 0.009697496
#> X7     0.0072934310 -5.369366e-04 0.009679297
#> X8     0.0070625492 -5.292390e-04 0.009661789
#> X1*X2  0.0459216617  8.939108e-05 0.010932749
#> X1*X3 -0.0006600465  6.814819e-04 0.010010933
#> X1*X4 -0.0056037444  4.901684e-04 0.009860488
#> X1*X5 -0.0070363484  5.187023e-04 0.009676301
#> X1*X6 -0.0071411552  5.319393e-04 0.009690812
#> X1*X7 -0.0072518362  5.303046e-04 0.009672163
#> X1*X8 -0.0070777721  5.186929e-04 0.009676468
#> X2*X3 -0.0051274794  5.279125e-04 0.009702252
#> X2*X4 -0.0060860874  5.210190e-04 0.009681757
#> X2*X5 -0.0071063957  5.147476e-04 0.009680715
#> X2*X6 -0.0071163219  5.256144e-04 0.009679855
#> X2*X7 -0.0070620281  5.299198e-04 0.009678650
#> X2*X8 -0.0071567767  5.166541e-04 0.009686683
#> X3*X4 -0.0073116996  5.314332e-04 0.009671714
#> X3*X5 -0.0071206761  5.265249e-04 0.009682280
#> X3*X6 -0.0071350887  5.241734e-04 0.009680955
#> X3*X7 -0.0071632203  5.264482e-04 0.009681046
#> X3*X8 -0.0071109279  5.241054e-04 0.009682418
#> X4*X5 -0.0071437469  5.248274e-04 0.009679986
#> X4*X6 -0.0071379129  5.276560e-04 0.009680886
#> X4*X7 -0.0071596546  5.255998e-04 0.009681341
#> X4*X8 -0.0071300368  5.260610e-04 0.009682262
#> X5*X6 -0.0071348129  5.263360e-04 0.009681340
#> X5*X7 -0.0071382804  5.262539e-04 0.009681217
#> X5*X8 -0.0071340327  5.262561e-04 0.009681110
#> X6*X7 -0.0071357204  5.261516e-04 0.009681295
#> X6*X8 -0.0071339651  5.264348e-04 0.009681123
#> X7*X8 -0.0071370385  5.263348e-04 0.009681299
#>          min. c.i.  max. c.i.
#> X1     0.711583661 0.73855259
#> X2     0.163919891 0.20792418
#> X3     0.012874359 0.05265936
#> X4    -0.002765501 0.03471590
#> X5    -0.010879190 0.02837068
#> X6    -0.010964117 0.02838793
#> X7    -0.010989042 0.02830642
#> X8    -0.010934969 0.02789333
#> X1*X2  0.026094148 0.06869013
#> X1*X3 -0.022980303 0.01844932
#> X1*X4 -0.026644889 0.01263961
#> X1*X5 -0.027999214 0.01120229
#> X1*X6 -0.028192638 0.01110205
#> X1*X7 -0.028265971 0.01093724
#> X1*X8 -0.028051234 0.01119551
#> X2*X3 -0.026495177 0.01365939
#> X2*X4 -0.027207071 0.01195456
#> X2*X5 -0.028066083 0.01115791
#> X2*X6 -0.028098511 0.01109647
#> X2*X7 -0.028017158 0.01116083
#> X2*X8 -0.028157933 0.01108562
#> X3*X4 -0.028275534 0.01102687
#> X3*X5 -0.028100457 0.01114042
#> X3*X6 -0.028110685 0.01112504
#> X3*X7 -0.028151981 0.01111060
#> X3*X8 -0.028109093 0.01117038
#> X4*X5 -0.028127750 0.01111732
#> X4*X6 -0.028127090 0.01112126
#> X4*X7 -0.028150168 0.01109302
#> X4*X8 -0.028120750 0.01114550
#> X5*X6 -0.028121862 0.01112445
#> X5*X7 -0.028124558 0.01111832
#> X5*X8 -0.028120227 0.01112288
#> X6*X7 -0.028121656 0.01112018
#> X6*X8 -0.028121004 0.01112360
#> X7*X8 -0.028124733 0.01112035

if(require(sensitivity)){
autoplot(ex1_results, ncol = 1)
}

Vignettes

There are more insights and examples in the vignettes.

vignette("Sobol_RV_five_examples", package = "Sobol4R")
vignette("Sobol4R_vignette_stochastic", package = "Sobol4R")
vignette("Sobol4R_vignette_process", package = "Sobol4R")
vignette("simmer_MM1_Sobol_example", package = "Sobol4R")