Variance-based sensitivity analysis: The quest for better estimators and designs between explorativity and economy

Variance-based sensitivity indices have established themselves as a reference amongst practitioners of sensitivity analysis of model outputs. A variance-based sensitivity analysis typically produces the first-order sensitivity indices \(S_j\)and the so-called total-effect sensitivity indices \(T_j\) for the uncertain factors of the mathematical model under analysis.

Computational cost is critical in sensitivity analysis. This cost depends upon the number of model evaluations needed to obtain stable and accurate values of the estimates. While efficient estimation procedures are available for \(S_j\)  (Tarantola et al., 2006), this availability is less the case for \(T_j\) (Iooss and Lemaître, 2015). When estimating these indices, one can either use a sample-based approach whose computational cost depends on the number of factors or use approaches based on meta-modelling/emulators (e.g., Gaussian processes).

The present work focuses on sample-based estimation procedures for \(T_j\) for independent inputs and tests different avenues to achieve an algorithmic improvement over the existing best practices. To improve the exploration of the space of the input factors (design) and the formula to compute the indices (estimator), we propose strategies based on the concepts of economy and explorativity. We then discuss how several existing estimators perform along these characteristics.

Numerical results are presented for a set of seven test functions corresponding to different settings (few important factors with low cross-factor interactions, all factors equally important with low cross-factor interactions, and all factors equally important with high cross-factor interactions). We conclude the following from these experiments: a) sample-based approaches based on the use of multiple matrices to enhance the economy are outperformed by designs using fewer matrices but with better explorativity; b) amongst the latter, asymmetric designs perform the best and outperform symmetric designs having corrective terms for spurious correlations; c) improving on the existing best practices is fraught with difficulties; and d) ameliorating the results comes at the cost of introducing extra design parameters.