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) https://doi.org/10.1063/5.0246026 and (2023) https://doi.org/10.1063/5.0246024 or in Bertrand, Logosha and Maumy (2024) https://hal.science/hal-05371803, 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.
You can install the latest version of the Sobol4R package from github with:
devtools::install_github("fbertran/Sobol4R")Sobol4R exposes two ways to compute sensitivity indices depending on your workflow:
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.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.
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:
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.
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-01The 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)
plot of chunk unnamed-chunk-14
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)
plot of chunk unnamed-chunk-15
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
#>
#> $variance
#> [1] 7.9431e-05When the simulator is stochastic and sobol_indices()
stores the replicated samples (keep_samples = TRUE), the
same Monte Carlo budget can be recycled to derive failure-indicator
Sobol indices:
failure <- sobol_reliability(result, threshold = -1)
failure$failure_probability
#> [1] 0.08984375
autoplot(failure, show_uncertainty = TRUE, probs = c(0.1, 0.9), bootstrap = 200)
plot of chunk unnamed-chunk-17
sensitivity packageThe 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 0.01915437if(require(sensitivity)){
autoplot(x1, ncol = 1)
}
plot of chunk det-g-plot
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 0.01112035if(require(sensitivity)){
autoplot(ex1_results, ncol = 1)
}
plot of chunk unnamed-chunk-20
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")