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

Abstract

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 a method that exploits the availability of softwares 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. However, in the context of limited computational resources, one must ensure that the number of repetitions of the stochastic model multiplied by the number of explorations of the input space is less than a fixed threshold. The problem of finding an optimal tradeoff between the repetitions and the explorations is addressed. A bound on an error criterion that penalizes bad rankings of the inputs' sensitivities is minimized. The estimator induced by the empirical method described above is shown to be asymptotically biased if the number of repetitions goes to infinity too slowly. A functional relationship between the output, the input and some random noise is derived from minimal distributional assumptions, which leads to a new sensitivity index with better statistical properties. The theory is illustrated on numerical experiments.

Maximizing Regional Sensitivity Analysis Indices to Find Sensitive Model Behaviors

“…In order to avoid ending up with a stochastic model (which would raise additional issues for the definition and estimation of the Sobol' indices [22], outside the scope of the present study), the perturbation is defined in a deterministic way by taking the row of index ⌊100•X 4 ⌋ of a pre-defined matrix Υ. Matrix Υ is of size (100, 2) and made of small 2D displacements sampled from random distributions. (Y…”

Test comparison for Sobol Indices over nested sets of variables

“…where X := (Z j ) j∈v represents the inputs of interest to explain Y and W := (Z j ) j / ∈v contains all the variables that are irrelevant to the analysis. Alternatively, W may contain hidden random inputs encountered in the context of stochastic codes [15,6]. For our purposes however, the precise nature of W is not important as both situations are dealt with in the same way.…”