Section: New Results

Replication procedure for grouped Sobol' indices estimation in dependent uncertainty spaces

Sensitivity analysis studies how the uncertainty on an output of a mathematical model can be attributed to sources of uncertainty among the inputs. Global sensitivity analysis of complex and expensive mathematical models is a common practice to identify influent inputs and detect the potential interactions between them. Among the large number of available approaches, the variance-based method introduced by Sobol' allows to calculate sensitivity indices called Sobol' indices. Each index gives an estimation of the influence of an individual input or a group of inputs. These indices give an estimation of how the output uncertainty can be apportioned to the uncertainty in the inputs. One can distinguish first-order indices that estimate the main effect from each input or group of inputs from higher-order indices that estimate the corresponding order of interactions between inputs. This estimation procedure requires a significant number of model runs, number that has a polynomial growth rate with respect to the input space dimension. This cost can be prohibitive for time consuming models and only a few number of runs is not enough to retrieve accurate informations about the model inputs.

The use of replicated designs to estimate first-order Sobol' indices has the major advantage of reducing drastically the estimation cost as the number of runs becomes independent of the input space dimension. The generalization to closed second-order Sobol' indices relies on the replication of randomized orthogonal arrays. The motivation of this work is to extend this methodology in presence of dependent inputs. Indeed, the case of correlated parameters has to be tackled with caution, as the calculation of single input indices does not provide anymore a proper information, that can be easily interpreted. One strategy is thus to define grouped indices for groups of correlated variables. We address this issue by proposing an approach based on replicated designs and randomized orthogonal arrays that enables to take into account dependency within inputs. We suppose that this dependency can be expressed through constraints. This approach can be used facing any set of constraints at the condition that one is able to provide points in the input space that verify the considered constraints. Guided by our application on a land-use and transport integrated model (LUTI) where some economical parameters are linked by order relations, we focus on the case of sets of linear ordered constraints. Thus we propose a sampling strategy based on the simplex geometric structure, that ensures a proper input space filling.

This work has been done in collaboration with Laurent Gilquin and Clementine Prieur (members of Moise Team), and belongs to the work program of CiTIES project. It is described in [18] . The proposed procedure will be soon applied to study the sensitivity of TRANUS model.