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Global Sensitivity Analysis with Dependence Measures
Sébastien Da Veiga
TL;DR
Variance-based global sensitivity analysis is limited by its focus on output variance and restricted handling of multivariate variables. The paper constructs sensitivity indices by comparing output and conditional-output distributions, recovering established indices and dependence measures while extending the framework to distance correlation, HSIC, and feature selection. The resulting approach offers multivariate dependence-based alternatives to conventional screening, with limitations in high-dimensional estimation and some kernel-based independence guarantees.
Problem
Variance-based sensitivity measures focus only on output variance and handle multivariate variables in a limited way.
Method
The paper defines sensitivity by comparing the output distribution with its conditional counterpart and instantiates this framework with dissimilarity and dependence measures.
Results
The framework recovers Borgonovo’s index, mutual information, and other established measures, and motivates distance correlation, HSIC, and dependence-based feature selection.
Takeaways & Limitations
Dependence-based sensitivity indices provide multivariate alternatives to variance-based measures and feature-selection techniques can serve as alternatives to screening.
Takeaways & Limitations
Universal-kernel guarantees for functional-data semi-metrics cannot currently be checked, high-dimensional estimation can be infeasible, and the examples focus on independent inputs.
Abstract
from arXiv · showhide
Global sensitivity analysis with variance-based measures suffers from several theoretical and practical limitations, since they focus only on the variance of the output and handle multivariate variables in a limited way. In this paper, we introduce a new class of sensitivity indices based on dependence measures which overcomes these insufficiencies. Our approach originates from the idea to compare the output distribution with its conditional counterpart when one of the input variables is fixed. We establish that this comparison yields previously proposed indices when it is performed with Csiszar f-divergences, as well as sensitivity indices which are well-known dependence measures between random variables. This leads us to investigate completely new sensitivity indices based on recent state-of-the-art dependence measures, such as distance correlation and the Hilbert-Schmidt independence criterion. We also emphasize the potential of feature selection techniques relying on such dependence measures as alternatives to screening in high dimension.
1 Introduction
The paper frames variance-based global sensitivity analysis as informative but limited because it focuses on output variance and handles multivariate variables poorly. It introduces dependence-based sensitivity indices, connects them to established measures, and motivates feature selection as an alternative to screening.
- Variance-based indices study only input effects on output variance, a restricted summary of the output distribution.
- The proposed indices generalize global sensitivity analysis through dissimilarity measures and include Borgonovo’s density-based index as a special case.
- Csiszár f-divergence choices recover dependence measures including mutual information, while distance correlation and HSIC provide newer sensitivity indices.
- The approach supports multivariate random variables and motivates dependence-based feature selection as an alternative to standard screening.
- The paper develops the framework, estimation procedure, dependence-measure indices, feature-selection examples, and numerical experiments.
2 From dissimilarity measures to sensitivity indices
The paper defines sensitivity through dissimilarity between an output distribution and its conditional counterpart, then shows how different dissimilarities recover known indices and enable dependence-based extensions. Density-ratio estimation, multivariate generalization, and invariance are useful, but high-dimensional estimation and some alternative measures remain constrained.
- Sensitivity measures compare the distribution of Y with Y conditional on an input X_k, using a chosen dissimilarity measure.
- Mean-based dissimilarity recovers the unnormalized Sobol first-order sensitivity index S1.
- Csiszár f-divergences: Csiszár f-divergences include Pearson and Neyman χ2 choices and recover established sensitivity indices through the selected convex function f.
- Csiszár f-divergences: The indices are nonnegative, vanish under independence, and remain invariant under smooth uniquely invertible transformations of inputs and outputs.
- Csiszár f-divergences: Kullback-Leibler divergence yields mutual information, while Neyman χ2 divergence yields squared-loss mutual information.
- Estimating only the joint-to-marginal density ratio can avoid estimating full densities, although density estimation still suffers from the curse of dimensionality.
- Multivariate extensions: The framework extends to groups of inputs and multivariate outputs, but high-dimensional estimation becomes infeasible for screening.
- Other dissimilarity measures: Integral probability metrics offer alternative distributional dissimilarities, but their general substitution does not produce a closed-form sensitivity index.
3 Dependence measures and feature selection
The paper connects dependence measures to global sensitivity analysis and develops indices based on mutual information, distance correlation, and HSIC. It also shows how these measures support multivariate analysis and feature-selection approaches for screening.
- Dependence measures: Mutual information and squared-loss mutual information arise as sensitivity indices from specific Csiszár f-divergences.Both measures are nonnegative and equal zero exactly under independence.
- Distance correlation: Distance correlation provides a sensitivity index that detects nonlinear dependence and extends naturally to multivariate inputs and outputs.The construction is based on distance covariance and generalizes the pick-and-freeze perspective beyond correlation.
- HSIC: HSIC equals zero if and only if its variables are independent when the associated RKHSs are universal.Kernel choices also allow HSIC to handle categorical and functional inputs or outputs.
- HSIC: HSIC-based feature selection can evaluate categorical outputs through label kernels and threshold transformations of the output.This provides a route to assessing input effects on output level sets.
- Feature selection: Max-Dependency selects features jointly most dependent on the target, while Min-Redundancy addresses redundant selections when features are dependent.The paper relates these feature-selection methods to screening and reports strong performance on complex benchmarks with few output samples.
- Feature selection: Marginal feature-selection procedures do not eliminate redundant variables, and the paper studies the simpler iterative technique because its examples use independent inputs.A full mRMR treatment for correlated inputs is left for future work.
4 Experiments
The experiments evaluate dependence-based sensitivity indices against variance-based indices on benchmark functions, using repeated estimates and summarized index definitions.
- The study assesses new sensitivity indices on benchmark analytical functions and two industrial applications.
- Table 1 summarizes the proposed sensitivity indices for comparison.
- The first experiments consider fewer than 10 input variables and compare new indices with Sobol first-order and total indices.
- The Linkletter et al. (2006) decreasing function is included among the first benchmark cases.
- For the initial benchmark, indices are computed with n = 500 and 100 repetitions using a simple kernel density estimator for first-order indices.
X and SHSICF,GP F
On nonlinear benchmark functions, dependence-based indices identify influential and non-influential inputs, while distance correlation can detect interaction effects without input-count-dependent estimation.
- Loeppky et al. (2013) function: For the Loeppky et al. (2013) function, only the first three inputs have a large impact, with total and first-order indices nearly equal.
- Loeppky et al. (2013) function: Distance correlation and HSIC clearly identify inputs independent of the output on the Loeppky function.
- Ishigami function: On the Ishigami function, first-order indices miss X3’s interaction effect, whereas the other indices detect its impact.
- Ishigami function: Distance correlation is designed to detect nonlinear dependence, so it can detect interaction effects without estimation depending on the number of inputs.
- The pick-and-freeze indices retain the computational-cost limitation associated with the pick-and-freeze technique.
- Ishigami function: Using a categorical kernel on an Ishigami level-set output recovers X3 as more important than X1 and X2.
X and SHSICF,G
In high-dimensional screening experiments, dCor, HSIC, and HSIC Lasso generally identify influential factors more reliably than first-order indices, while iterative HSIC can select extra inputs at smaller sample sizes.
- Morris et al. (2006) function: On the Morris function, dCor and HSIC perfectly discriminate the first five factors and identify the remaining factors as independent from the output.
- Morris et al. (2006) function: HSIC Lasso performs very well on the Morris function, selecting the first five input factors almost every time.
- Sobol and Levitan (1999) function: For the Sobol and Levitan function, dCor and HSIC assign higher dependence to the first input factors than to the remaining ones.
- Sobol and Levitan (1999) function: HSIC Lasso never selects the last inputs as influential, while iterative HSIC performs well but tends to select more inputs than necessary.
- Sobol and Levitan (1999) function: With n = 100, dCor, HSIC, and HSIC Lasso almost perfectly identify the influential inputs, and iterative HSIC almost always discards non-influential inputs.
- Industrial applications: The industrial examples address reservoir characterization and contamination-migration screening using uncertain physical parameters and simulation data.
5 Conclusion
The paper introduces dependence-based sensitivity indices as an alternative to variance-based GSA and highlights feature selection for high-dimensional screening.
- The proposed indices compare output distributions with conditional counterparts through dependence measures, including Csiszár f-divergences, distance correlation, and HSIC.
- Feature selection techniques based on dependence measures are presented as alternatives to screening in high dimension.
- The new indices are described as robust to dimensionality, low-cost, and extendable to functional and categorical variables.
- These properties support tools for GSA and factor screening in high-dimensional and expensive computer codes.