A tradeoff between explorations and repetitions for estimators of two global sensitivity indices in stochastic models induced by probability measures
Résumé
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 as 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.
Origine : Fichiers produits par l'(les) auteur(s)