Consistent Sensitivity Indices for p-Boxes

Abstract

The main objective of global sensitivity analyses is to determine the influence of uncertain input parameters of a (simulation) model on the output parameters of the same model. Normally, for this purpose the uncertain input parameters are modelled using probabilistic approaches, i.e., statistical distributions. This works fine for input parameters featuring aleatory uncertainty. However, if the uncertainty is epistemic the prerequisites for probabilistic modelling are not always given. For example, if the uncertainty stems from limited, sparse data the requirement of a high number of data points is definitely violated. Such uncertain input parameters are frequently modelled using possibilistic approaches, e.g., intervals or fuzzy sets. If an input parameter features both, aleatory and epistemic uncertainty, imprecise probabilistic uncertainty models, e.g., p-boxes can be used to model it. For p-boxes, standard global sensitivity analyses do not work anymore. This is why recently, research on sensitivity analyses for p-boxes has been done. For example, an approach by Sch�bi and Sudret (2019) extended classical Sobol� indices defined for probabilistic input parameters to input parameters modelled using p-boxes. However, current sensitivity indices in the context p-boxes are not defined mathematically consistent. For example, the index for the epistemic uncertainty defined by Sch�bi and Sudret is zero if the aleatory uncertainty is zero, even if the epistemic uncertainty is greater than zero. Hence, it takes coupling effects of aleatory and epistemic uncertainty into account, but does not consider the effect of pure epistemic uncertainty.

Hence, in this contribution, more mathematically consistent sensitivity indices (Sal,T and Sep,T), which can be used for p-boxes, are proposed and tested using several mathematical test functions. The new sensitivity indices are based on the definitions by Sch�bi and Sudret. However, they are defined as a sum of a pure aleatory or pure epistemic term and a coupling term, e.g., Sal,T = Sal + Sal,ep. The latter one, i.e., the coupling term Sal,ep, becomes zero if either the aleatory or the epistemic uncertainty is zero. The former one does only depend on one of the two types of uncertainty. Hence, in the previously mentioned case of an aleatory uncertainty of zero and an epistemic uncertainty of greater than zero, the coupling term and the pure aleatory term are zero (Sal = Sal,ep = 0), whereas the pure epistemic term is greater than zero (Sep > 0) being mathematically consistent.