Section: New Results

Estimation of Sobol' indices combining nested designs and replication method

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. However the replication method still requires a large number of model evaluations. By rendering this method iterative, the required number of evaluations can be controlled. The estimation procedure is therefore stopped when the convergence of estimates is considered reached. The key feature of this approach is the construction of nested designs. For the estimation of first-order indices, we exploit a nested Latin Hypercube already introduced in the litterature. For the estimation of closed second-order indices, two methods are proposed to construct a nested orthogonal array. One of the two leads to a partition of the coordinate space over a Galois field.

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. The proposed procedure will be soon applied to study the sensitivity of TRANUS model.