Sobol Sensitivity Indices

Variance-based measures that apportion the output variance among the inputs and their interactions, the standard global sensitivity tool.

Apportioning the Variance

Sobol indices are the standard variance-based measure of global sensitivity. The idea is to decompose the total variance of the output into contributions from each input and from each combination of inputs. The fraction of output variance explained by varying one input alone is its first-order Sobol index; the fraction explained by that input including all its interactions is its total-effect index.

First-Order and Total-Effect

• First-order index: the effect of an input acting on its own, averaged over the others.
• Total-effect index: the input's full contribution, including every interaction it participates in.
• A large gap between the two for an input signals that it acts mainly through interactions, not alone.

Reading the Indices

The first-order indices of all inputs sum to at most one; whatever is left over is due to interactions. If the first-order indices sum close to one, the model is essentially additive and each input can be understood in isolation. If they sum well below one, interactions dominate and the inputs cannot be reasoned about separately, which itself is important design knowledge.

Computing Them

Sobol indices are estimated by structured sampling, running the code on carefully paired input sets so that the variance contributions can be separated. This is expensive, often requiring many runs, which is why surrogate models and polynomial chaos, from which Sobol indices can be read almost directly, are frequently used to make the computation affordable. Screening methods such as Morris are run first to discard negligible inputs before the expensive Sobol computation.

For a design study, the Sobol ranking answers the practical question of where to spend effort: the inputs with the largest total-effect indices are the ones whose measurement and modeling most tighten the prediction, and the ones a robust design must be least sensitive to.