Estimation of the Derivative-Based Global Sensitivity Measures Using a Gaussian Process Metamodel

Abstract

Physical phenomena are often studied using numerical simulators. Such computer codes are a function of uncertain input parameters and a global sensitivity analysis (GSA) can be performed to identify their impacts on the simulator outputs. Sobol' indices, based on output variance decomposition, are commonly used to perform quantitative GSA. For many years now, other tools have been studied, closer to physical practices such as the derivative-based global sensitivity measures (DGSM). However, numerical simulators rarely provide the output gradient and DGSM estimation is not directly possible. To address this limitation, we propose to estimate the DGSMs using a Gaussian process metamodel (GPM) which approximates the simulator. Based on this GPM, we propose two DGSM estimators: a plug-in one defined by the DGSM of the GPM predictor and another one defined by the expectation of the DGSM associated to the full-GPM. The latter is equal to the first one completed by a variance term and can be accompanied by a credibility interval. For Gaussian kernel and uniform input laws, analytical formulas are given for both DGSM estimators. For all other situations, Monte Carlo methods for the expectation approximations are proposed: a propagative version of the Gibbs sampler and a chi-square approximation. Moreover, a significance test for the full-GPM based estimator is proposed for screening. The convergence of the two GPM-based DGSM estimators and the Monte Carlo approaches is compared in analytical test cases. Finally, we apply our work to an environmental application.

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