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Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests

Matieyendou Lamboni, Bertrand Iooss, Anne-Laure Popelin, Fabrice Gamboa

arXiv:1202.0943v2math.STstat.TH

TL;DR

Variance-based sensitivity estimation can require many model evaluations, limiting its use for high-dimensional models. This paper links derivative-based measures to total Sobol’ indices for a broad class of probability distributions. Numerical tests report that the resulting bound is suited to screening expensive models and can provide correct input-sensitivity information.

  • Problem

    Variance-based sensitivity estimation can be prohibitively expensive for nonlinear models with interactions, limiting its application to fewer than tens of inputs despite models sometimes having hundreds or thousands of uncertain inputs.

    Source: §11 · pp. 1–2

  • Method

    The paper derives an inequality linking total Sobol’ indices to a derivative-based sensitivity measure for a large class of probability distributions.

    Source: 115. Conclusion · pp. 16–17

  • Results

    The conclusion reports that the bound Υ_j improves the factor-fixing setting using derivative-based sensitivities, and numerical tests confirm its suitability for screening expensive models.

    Source: 115. Conclusion · p. 17

  • Takeaways & Limitations

    For expensive models, Υ_j can provide correct information on input sensitivities and useful information about the presence or absence of interactions.

    Source: 115. Conclusion · p. 17

    The numerical tests focus on comparing differences in rankings.

    Source: §44.1 · p. 10

Abstract

from arXiv · show

The estimation of variance-based importance measures (called Sobol' indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the Derivative-based Global Sensitivity Measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol' indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.

1 1. Introduction

High-dimensional models may have hundreds or thousands of uncertain inputs, while variance-based sensitivity estimation can be too costly for models with nonlinear and interaction effects. This paper extends the DGSM–total Sobol’ inequality to Boltzmann measures, including the log-concave case.

1 1. Introduction

  • For nonlinear models with interactions, variance-based estimation can require enough evaluations to restrict its use to fewer than tens of input variables.This is a challenge for models that may have hundreds or thousands of uncertain inputs.Source: §11 · pp. 1–2
  • DGSM were presented as more computationally tractable for high-dimensional models, with an earlier inequality linking them to total Sobol’ indices for uniform or Gaussian inputs.The introduction describes DGSM as a generalization of the Morris screening method.Source: §11 · p. 2
  • This paper extends the inequality between DGSM and total Sobol’ indices to a large class of Boltzmann measures and also treats log-concave measures.The paper presents numerical simulations on two test models to illustrate practical use.Source: §11 · p. 3

9 2. Global sensitivity indices definition

The section defines total Sobol’ sensitivity in terms of the component of the model output associated with an input variable. Under independent inputs and finite second moment, the Hoeffding decomposition provides a unique decomposition of the output.

2. Global sensitivity indices definition

  • The Hoeffding decomposition represents the model output as a sum of components indexed by subsets of input variables, under independence and finite second moment.The decomposition is unique under assumptions A1 and A2.Source: 102.1. Variance-based sensitivity… · pp. 3–4
  • Regrouping terms involving X_j yields a component g that contains all information relating the output to X_j, with the remaining component depending on the other inputs.The section writes the output as f(X) = f0 + g(X_j, X∼j) + h(X∼j).Source: 102.1. Variance-based sensitivity… · pp. 3–4
  • The total Sobol’ index measures the total sensitivity associated with an input variable, and the section gives a further way to compute it under assumptions A1 and A2.The supplied passages identify the first-order and total indices but do not include their formulas.Source: 102.1. Variance-based sensitivity… · p. 4

5 2.2. Derivative-based sensitivity indices

Derivative-based global sensitivity measures use squared model derivatives as an importance measure. The section defines unweighted and weighted forms and notes that the relevant function can be lower-dimensional, reducing evaluations needed for their numerical computation.

2.2. Derivative-based sensitivity indices

  • DGSM use the second moment of model derivatives as an importance measure, motivated by the expectation that a large derivative signals a large output variation.The method is described as extending the Morris method and capturing small output variations due to inputs.Source: §52.2 · p. 5
  • The section defines two DGSM forms and a weighted version, assuming the relevant function is square-integrable and the weight is bounded and measurable.The supplied passages do not include the full index formulas.Source: §52.2 · p. 5
  • For a specific weighting function, τ_j equals the total Sobol’ index for a class of linear models with uniform inputs on [0, 1].This is a special case reported by Sobol and Kucherenko.Source: §52.2 · p. 5
  • Using g instead of f can reduce the dimension to d1 ≤ d, which can drastically reduce evaluations for computing ν or τ.The section states that DGSM estimations need much fewer model evaluations than total Sobol’ indices.Source: §52.2 · pp. 5–6

3. Variance-based sensitivity indices vs. derivative-based sensitivity in-

This section links derivative-based sensitivity measures to total Sobol’ indices for Boltzmann input measures, extending earlier results for uniform and Gaussian variables. It also gives a result for log-concave measures, with Cheeger constants relevant to the resulting bound.

3. Variance-based sensitivity indices vs. derivative-based sensitivity in-

  • The paper extends the Sobol’–DGSM inequality from uniform and Gaussian variables to Sobolev-space models with Boltzmann input marginals.The formal link is stated under assumptions A1–A4.Source: §3 · pp. 6–7
  • Finite-interval uniform measures are excluded from the continuous-on-R Boltzmann and log-concave classes discussed here.The text notes that such a uniform density is not continuous on R.Source: §3 · p. 7
  • The inequalities are based on a one-dimensional L2-Poincaré inequality applied to the model function.The Boltzmann result uses a Poincaré constant; the log-concave result involves the Cheeger constant and the input median.Source: §3 · p. 7
  • A second result establishes the Sobol’–DGSM link for log-concave inputs under assumptions A1–A3 and A5.The paper identifies log-concave measures as a subclass of Boltzmann measures.Source: §3 · pp. 6–8
  • For general log-concave measures, analytic Cheeger constants are unavailable, so the paper proposes estimating them numerically.Table 1 lists standard log-concave distributions with their potential function, median, and Cheeger constant.Source: §3 · pp. 8–9, Table 1 · p. 8

4 4.1. Derivative sensitivity indices estimates

This section describes empirical estimation of DGSM and compares the estimates and upper bounds with Sobol’ indices. The reported tests focus on rankings rather than estimator convergence speed.

4.1. Derivative sensitivity indices estimates

  • The classical empirical DGSM estimator averages squared model derivatives over sampled inputs.The section identifies the estimator as equation (4.17).Source: §44.1 · p. 9
  • DGSM estimates and Sobol’ index estimates appear to require approximately the same number of model evaluations to converge to their respective values.The estimators are based on the same function g(·).Source: §44.1 · p. 9
  • For gradient estimation, Monte Carlo evaluations n ≤ d can suffice; in the worst case, the proposed procedure requires as many model evaluations as finite differences.The procedure is described as very efficient for models with low effective dimension.Source: §44.1 · p. 9
  • The numerical tests compare sensitivity rankings and do not study the convergence speed of the estimates.Source: §44.1 · p. 10

6 4.2. Test on the Morris function

The Morris-function test finds that the DGSM-based bound can screen negligible inputs and help interpret total influence, but it can be uninformative for some inputs. Its performance varies with input interactions and, the authors suspect, with nonlinearities and input distributions.

4.2. Test on the Morris function

  • The test estimates indices for 20 independent factors using Monte Carlo Sobol’ estimates and finite-difference derivatives; the Sobol’ estimation requires 4.4 × 10^6 model evaluations.The authors state that comparing DGSM and Sobol’ computational cost is not the objective of this test.Source: §64.2 · p. 11
  • For X11–X20, first-order and total Sobol’ indices and bound Υj are negligible; for X7–X10, the indices significantly differ from zero and have approximately the same value.A negligible bound warrants that the input has no influence.Source: §64.2 · p. 12
  • For X1–X6, first-order Sobol’ indices are negligible while total indices differ significantly from zero; Υj largely overestimates total STj for X3–X5, and results for X3 and X5 are strongly unsatisfactory.For X4, Υ4 < 1 and the coarse information remains useful.Source: §64.2 · pp. 12–13
  • The authors suspect the poor bounds for X3 and X5 reflect model nonlinearity and exponential or Weibull inputs, whose extreme sampled values produce unusually large derivatives.They report particularly large derivative values for X3 and X5, but not for X1, X2, or X4.Source: §64.2 · p. 13
  • The authors conclude that Υj is suited to screening, and that pairing it with first-order Sobol’ indices can help assess whether interactions are present.They also report that the bound is especially efficient for uniform, normal, and exponential inputs, where it is the best bound and cannot be improved.Source: §64.2 · p. 13

23 4.3. A case study: a flood model

In the flood-model case study, the DGSM-based upper bound selects the same leading variables for overflow and cost as the Sobol’ analysis, while unnormalized derivative indices produce a different, misleading ranking. The Sobol’ results indicate small interactions and identify four influential inputs.

A case study: a flood model

  • The case study has 8 input variables, which are independent, and considers associated cost as another model output; computing first-order Sobol’ estimates requires 2 × 10^7 model evaluations.Source: 234.3. A case study: a flood model · pp. 14–15
  • The Sobol’ analysis shows small interactions and identifies Q, Hd, Ks, and Zv as drivers of both overflow and cost.This variable classification is used as the reference for comparison with derivative-based indices.Source: 234.3. A case study: a flood model · p. 15, Table 5 · p. 15, Table 6 · p. 16
  • The derivative indices νj and τj instead select Zv, Cb, Hd, and Zm, wrongly suggesting that Q has no impact on either output.The authors explain the discrepancy by noting that the indices are not renormalized by a distribution-dependent constant and that inputs have different units.Source: 234.3. A case study: a flood model · pp. 15–16
  • For both overflow and cost, Υj prioritizes Q, Zv, Hd, and Ks, providing a correct account of variance-based sensitivities and a similar influential-variable subset to the global sensitivity indices.The authors note slight differences in prioritization.Source: Table 5 · p. 15, 234.3. A case study: a flood model · p. 16

11 5. Conclusion

The paper links total Sobol’ indices to derivative-based sensitivity measures for a large class of probability distributions. Numerical tests support using the resulting bound for screening, including when expensive models make total-index estimation impractical.

11 5. Conclusion

  • A derivative-based global sensitivity method uses 10 to 100 fewer model evaluations, with larger reductions when few inputs control output and interactions are limited.Source: 115. Conclusion · p. 16
  • The paper establishes an inequality linking total Sobol’ indices and derivative-based sensitivity measures for a large class of probability distributions.Source: 115. Conclusion · pp. 16–17
  • The index Υj is a constant-scaled crude derivative-based sensitivity that bounds the total Sobol’ index and improves factors-fixing by replacing variance-based sensitivities with derivative-based ones.Source: 115. Conclusion · p. 17
  • The bound Υj was confirmed in two numerical tests as suitable for screening and informative when CPU cost prevents estimating total Sobol’ indices.The conclusion presents it as useful for identifying input sensitivities in expensive-model settings.Source: 115. Conclusion · p. 17

15 6. Acknowlegments

The acknowledgments record funding support and thank contributors for discussions and reviews.

15 6. Acknowlegments

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