A Trade-Off Between Explorations and Repetitions for Estimators of Two Global Sensitivity Indices in Stochastic Models Induced by Probability Measures

摘要

Sobol sensitivity indices assess how the output of a given mathematical model is sensitive to its inputs. If the model is stochastic, then it cannot be represented as a function of the inputs, thus raising questions about how to do a sensitivity analysis in those models. Practitioners have been using an approach that exploits the availability of methods for deterministic models. For each input, the stochastic model is repeated and the outputs are averaged. These averages are seen as if they came from a deterministic model and hence Sobol's method can be used. We show that the estimator so obtained is asymptotically biased if the number of repetitions goes to infinity too slowly. With limited computational resources, the number of repetitions of the stochastic model and the number of explorations of the input space cannot be large together and hence some balance must be found. We find the pair of numbers that minimizes a bound on some rank-based error criterion, penalizing bad rankings of the inputs' sensitivities. Also, under minimal distributional assumptions, we derive a functional relationship between the output, the input, and some random noise; the Sobol--Hoeffding decomposition can be applied to it to define a new sensitivity index, which asymptotically is estimated without bias even though the number of repetitions remains fixed. The theory is illustrated on numerical experiments.