Variance-Based Sensitivity

Sobol indices decompose output variance into contributions from each input and their interactions, giving a rigorous global sensitivity ranking.

Decomposing variance

Variance-based sensitivity, or Sobol analysis, splits the total variance of a model output into the parts attributable to each input alone and to combinations of inputs. The framework assumes independent inputs and yields interpretable, dimensionless indices between 0 and 1 that sum to 1 over all terms.

First-order and total indices

Kronos motion — uncertainty

First-order index S_i = Var(E[Y|X_i]) / Var(Y): the effect of X_i alone, averaging over the rest
Total-effect index S_Ti: the effect of X_i including all its interactions
S_i = S_Ti means no interactions for that input; a large gap signals strong interaction

Estimation

Sobol indices are estimated by Monte Carlo using specially structured sample matrices (the Saltelli scheme), or from a fitted polynomial-chaos surrogate whose coefficients give the indices analytically. The surrogate route is far cheaper when many evaluations would otherwise be needed, provided the surrogate is validated.

Reading the results

Inputs with negligible total-effect indices can be fixed at nominal values, reducing problem dimension with confidence. Inputs with high first-order indices are the priorities for uncertainty reduction. A large total-effect index with a small first-order index tells you the input matters mainly through interactions, which changes how you act on it.

Cautions

The standard indices assume independent inputs; correlated inputs require Shapley-value or grouped extensions to avoid misattribution. Estimation variance can be large for small samples, so report confidence intervals on the indices. Variance-based measures also focus on the second moment and can miss effects that alter tails without changing variance.