The regression importance index of a coherent system evaluates a component’s importance based on the system’s conditional mean lifetime when the component’s failure time is known (see Arriaza et al. [1]). We aim to introduce the ``regression importance signature”, a tool designed to identify, for a given number of components, the subgroup that should be prioritized in reliability analysis and failure localization. To achieve this, the concept of importance index is generalized for subgroups of components, taking into account the occurrence of failures within the subgroup. General results for systems with dependent components are provided, with a particular focus on system modules, as well as sufficient conditions for comparing the importance of individual components and subgroups. This analysis highlights how a component’s relevance depends not only on its reliability but also on its structural role within the system. As an application, we consider the ship control system already discussed in [1], extending the original analysis to the computation of the full signature. This allows us to identify the most influential subgroups of components and to explore how the dependence modeled by the FGM copula and the variation of its parameters affect the importance of different subsets of components.

References:

[1] Arriaza, A., Navarro, J., Sordo, M.A., Suárez-Llorens, A. A variance-based importance index for systems with dependent components. Fuzzy Sets and Systems 467, 108482 (2023). https://doi.org/10.1016/j.fss.2023.02.003

[2] Di Crescenzo, A., Pisano, G., Suárez-Llorens, A. Analysis of systems with dependent components through a variance-based index and regression importance signature. Reliability Engineering & System Safety, 273, 112357 (2026). https://doi.org/10.1016/j.ress.2026.112357