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Calculations of Sobol indices for the Gaussian process metamodel
Amandine Marrel, Bertrand Iooss, Beatrice Laurent, Olivier Roustant
TL;DR
Sensitivity analysis for expensive computer codes requires too many evaluations, so the paper develops analytical Sobol-index calculations for Gaussian-process metamodels. It compares predictor-only and global stochastic approaches, finding the global approach more robust and useful for confidence intervals, especially with inaccurate metamodels or few data.
Problem
Sobol-index estimation for time-expensive computer codes requires a large number of model evaluations, limiting direct sensitivity analysis.
Method
The paper derives and compares Sobol-index formulations based on the Gaussian-process predictor and on the full global stochastic process, including a procedure to simulate the index distribution.
Results
The global stochastic approach yields more robust sensitivity indices, particularly when the Gaussian-process metamodel is inaccurate or few data are available, and can provide confidence intervals.
Takeaways & Limitations
Using the covariance structure makes Sobol-index estimates more informative about metamodel uncertainty than predictor-only calculations.
Takeaways & Limitations
Accounting for covariance provides a minimal bound for the influence of all variables and therefore tends to overestimate the lowest Sobol indices.
Abstract
from arXiv · showhide
Global sensitivity analysis of complex numerical models can be performed by calculating variance-based importance measures of the input variables, such as the Sobol indices. However, these techniques, requiring a large number of model evaluations, are often unacceptable for time expensive computer codes. A well known and widely used decision consists in replacing the computer code by a metamodel, predicting the model responses with a negligible computation time and rending straightforward the estimation of Sobol indices. In this paper, we discuss about the Gaussian process model which gives analytical expressions of Sobol indices. Two approaches are studied to compute the Sobol indices: the first based on the predictor of the Gaussian process model and the second based on the global stochastic process model. Comparisons between the two estimates, made on analytical examples, show the superiority of the second approach in terms of convergence and robustness. Moreover, the second approach allows to integrate the modeling error of the Gaussian process model by directly giving some confidence intervals on the Sobol indices. These techniques are finally applied to a real case of hydrogeological modeling.
1 INTRODUCTION
Sobol-index estimation is costly for complex computer codes, motivating metamodels. The paper studies Gaussian-process formulations that enable analytical indices, compare predictor-only and global stochastic approaches, and quantify index uncertainty.
- Large numbers of model evaluations make variance-based sensitivity analysis unacceptable for time-expensive computer codes.
- Metamodels approximate costly computer codes from relatively few experiments, enabling negligible-cost response prediction and straightforward Sobol-index estimation.
- Gaussian-process models provide analytical formulas for variance-based Sobol indices while offering exact interpolation and interpretability.
- The paper compares predictor-only indices, based on the Gaussian-process mean, with indices based on the complete global stochastic model.
- For the global stochastic model, the Sobol index is random, allowing its distribution and standard deviation to support confidence intervals.
- Applications include analytical test functions and a hydrogeological transport model, with uncertainty computed for the real data set.
MODEL
The model treats computer-code responses as realizations of a conditional Gaussian process with regression and covariance components. Sobol indices are computed either from its predictor or from the full stochastic process, whose index distribution quantifies accuracy.
- Gaussian process model: Gaussian-process modeling represents deterministic computer-code responses as realizations of a random function with regression and centered stochastic components.
- Gaussian process model: The regression component uses a linear combination of basis functions, with a one-degree polynomial adopted to simplify analytical sensitivity-index computations.
- Gaussian process model: The stochastic component is a centered Gaussian process characterized by variance and correlation functions, with stationary and product-form correlations assumed.
- Conditional distributions: Conditioning on the learning sample yields a Gaussian-process predictor and conditional variance, where the latter is the predictor's mean squared error.
- Sobol indices: The predictor-only approach applies Sobol formulas to the conditional mean, which is deterministic in the input variables.
- Sobol indices: The global approach incorporates the conditional covariance, producing a random Sobol index whose mean is an index and whose variance indicates accuracy.
- Study design: The study compares both approaches on test functions and uses the global-index distribution to construct confidence intervals.
3 IMPLEMENTATION OF SOBOL INDICES
The paper develops analytical and simulation-based procedures for Sobol indices using conditional and global Gaussian process models. These procedures exploit the Gaussian process covariance structure, reducing integration effort and enabling uncertainty quantification.
- 3.1 Estimation of Gp parameters: High-dimensional or complex applications can exceed the direct capabilities of standard Gaussian process software, motivating a sequential input-selection version of the DACE algorithm.The sequential method adds previously sorted input variables step by step for parameter estimation.
- 3.2 Computation of Sobol indices for the two approaches: The conditional Gaussian process metamodel provides analytical Sobol-index formulae that reduce the required calculations to one-dimensional and two-dimensional numerical integrals.These integrations are easier to control for accuracy and less computationally expensive than Monte Carlo simulations.
- 3.2 Computation of Sobol indices for the two approaches: The first approach computes indices from the Gaussian process predictor, whereas the second treats the global-process-based index ˜Si as a random variable.The mean of ˜Si can be used as a sensitivity index, while its distribution captures modeling uncertainty.
- 3.2 Computation of Sobol indices for the two approaches: For independent inputs, integrating the Gaussian process over all inputs except Xi yields a Gaussian process whose centered main-effect variance defines the Sobol index.The required covariance and mean are obtained analytically through one-dimensional integrations.
- 3.3 Simulation of the distribution of ˜Si: The distribution of ˜Si is simulated by discretizing a random integral, transforming the resulting Gaussian vector with a numerical-integration matrix, and repeating the evaluation ksim times.A single Cholesky factorization can be reused for all simulations, while convergence is assessed through the mean and variance of ˜Si.
4 APPLICATIONS
Applications compare predictor-based and global-Gaussian-process Sobol indices on analytical test functions and a hydrogeological model. The global approach generally converges more robustly, especially when metamodel accuracy is limited, while also providing confidence intervals.
- 4.2 Test on the g-function of Sobol: The global Gaussian process model outperforms the predictor-only model by showing smaller confidence intervals for the five sensitivity indices.This comparison is made using the predictor-based indices Si and global-model indices ˜Si.
- 4.2 Test on the g-function of Sobol: The global-model indices ˜Si are better in mean than predictor-only indices Si, with the difference especially significant for high Sobol indices.For lower indices, the two approaches give similar mean results.
- 4.2 Test on the g-function of Sobol: The global approach has lower sampling deviation and variability than the predictor-only approach, particularly when metamodel accuracy is weak (Q2 < 0.8).For Q2 > 0.8, the approaches give the same values, while the predictor-only method remains easier to compute.
- 4.5 Application to hydrogeological modeling: The hydrogeological Gaussian process model achieves Q2 = 0.92 versus Q2 = 0.83 for boosting regression trees, yielding a more accurate Sobol-index estimation.The global approach additionally supplies confidence intervals, which the boosting-model indices do not fall within.
- 4.5 Application to hydrogeological modeling: In the hydrogeological application, kd1, i3, and per1 explain nearly 90% of the output variance without interactions, with kd1 identified as the most influential input.The global Gaussian process model provides small standard deviations and little confidence-interval overlap, supporting the inferred ordering.
5 CONCLUSION
The study evaluates Gaussian process metamodels for Sobol-index estimation, comparing predictor-based and global stochastic approaches. Incorporating covariance improves robustness and supports confidence intervals, especially when data are limited or the metamodel is inaccurate, while very low indices remain problematic.
- The study uses Gaussian process metamodels to estimate Sobol indices and build associated confidence intervals.
- Gaussian process metamodels provide analytical sensitivity-index expressions involving one- and two-dimensional numerical integrals, avoiding many predictor evaluations.
- The global stochastic approach is particularly useful when the Gaussian process metamodel is inaccurate or few training data are available.
- The global stochastic approach accounts for the Gaussian process covariance structure, producing more robust and less variable sensitivity indices.
- Confidence intervals for Sobol indices can be simulated at any level from the distribution of the stochastic sensitivity index.
- For predictivity coefficients larger than 60%, the confidence intervals are satisfactory for the highest Sobol indices.
- Covariance-based estimates tend to impose a minimal influence on all variables, overestimating the lowest Sobol indices and producing inaccurate intervals near zero.
- The approach was also applied to a hydrogeological computer code simulating radionuclide groundwater transport, confirming the interest of the global approach.