A comparison between Sobol's indices and Shapley's effect for global sensitivity analysis of systems with independent input variables

The model-based system engineering approach consists of assembling subsystems together to model a complete system. In this context, some functional blocks can have a considerable influence on the overall behaviour of the system. A preliminary identification of the influence of the subsystems on the output responses can help reducing the complexity of the overall system, with a negligible impact on the overall accuracy. Therefore, pertinent indicators must be introduced to achieve this goal. To this purpose, in this work, some well-established methods and algorithms for global sensitivity analysis (GSA) of linear and non-linear systems with independent input variables, i.e., approaches based on Sobol's indices (different algorithms are considered), and Shapley's effect, are compared on both benchmark functions and real-world engineering problems. Specifically, in this paper, real-world engineering problems dealing with linear and non-linear systems are modelled through commercial finite element software and/or dedicated programming languages for solving complex non-linear dynamics models, like Modelica. Regarding Modelica models, an efficient strategy based on functional mock-up units is presented to speed up the simulation of highly non-linear dynamic systems. All numerical models are interfaced with the algorithms used for GSA through ad-hoc routines coded in Python environment. For each problem, a systematic comparison between the results provided by the different algorithms making use of Sobol's indices and Shapley's indices is performed, in terms of reliability, accuracy and computational costs.