Global sensitivity analysis
Global sensitivity analysis is the quantitative study of how uncertainty in the output of a mathematical model is attributable to uncertainty in its inputs. Unlike local sensitivity analysis, which examines behavior near a selected point in the input space, global analysis considers inputs across their full prescribed distributions. It therefore characterizes variation arising from nonlinear responses, interactions among inputs, and changes occurring far from a nominal parameter set.
A global sensitivity analysis is defined jointly by the model, the uncertain-input distribution, and the output quantity under examination. The same model can yield different sensitivity measures when any of these components changes. Consequently, sensitivity indices describe a specified model–distribution system rather than immutable properties of individual parameters.
The field forms part of uncertainty quantification and is closely connected to design of experiments, Monte Carlo methods, and the statistical analysis of computer simulations. Its methods are used to identify dominant sources of output variation, analyze parameter interactions, and examine the effective dimensionality of complex models.
Mathematical formulation
Let a scalar model output be represented by
[ Y=f(X_1,\ldots,X_d)=f(\mathbf X), ]
where (\mathbf X) is a random input vector with joint probability distribution (p_{\mathbf X}). Global sensitivity analysis studies how variation in (Y) depends on variation in (\mathbf X). When the model has several outputs, the analysis can be applied to each scalar component or to a scalar functional constructed from the full output.
The choice of input distribution is part of the model specification. A narrow distribution represents restricted uncertainty, whereas a broad distribution assigns probability to a larger region of the input space. Dependence among inputs also affects attribution because the variation associated with one variable can overlap with information supplied by another.
Most classical measures concern the variance
[ V=\operatorname{Var}(Y), ]
although variance is not the only possible description of output uncertainty. Distribution-based methods instead compare conditional and unconditional output distributions, while derivative-based methods summarize the average magnitude of local changes over the entire input domain.
Functional decomposition
For independent inputs and a square-integrable model, (f) admits an analysis of variance decomposition,
[ f(\mathbf X)
f_0+ \sum_i f_i(X_i)+ \sum_{i<j} f_{ij}(X_i,X_j)+\cdots+ f_{1\ldots d}(X_1,\ldots,X_d). ]
The constant term is
[ f_0=\operatorname{E}[Y]. ]
Each nonconstant component has zero expectation with respect to every variable appearing in that component. This orthogonality condition makes the decomposition unique under the assumed product distribution. It also produces a corresponding variance decomposition,
[ V=\sum_{\varnothing\ne u\subseteq{1,\ldots,d}}V_u, ]
where (u) denotes a nonempty subset of input indices and
[ V_u=\operatorname{Var}!\left[f_u(\mathbf X_u)\right]. ]
A component involving one input represents a main effect. A component involving several inputs represents variation that cannot be assigned to any lower-order component under the decomposition. Such a component is termed an interaction, although it does not necessarily correspond to a mechanistic interaction in the system represented by the model.
The normalized Sobol index associated with (u) is
[ S_u=\frac{V_u}{V}. ]
For nonconstant outputs, the indices are nonnegative and sum to one. The first-order index for (X_i),
[ S_i=\frac{\operatorname{Var}!\left(\operatorname{E}[Y\mid X_i]\right)} {\operatorname{Var}(Y)}, ]
measures the fraction of variance explained by changing (X_i) alone while averaging over the other inputs.
The total-effect index is
[ S_{T_i}
1- \frac{\operatorname{Var}!\left(\operatorname{E}[Y\mid\mathbf X_{\sim i}]\right)} {\operatorname{Var}(Y)}, ]
where (\mathbf X_{\sim i}) contains every input except (X_i). Equivalently, (S_{T_i}) is the sum of all variance components whose index sets contain (i). The difference between (S_{T_i}) and (S_i) therefore reflects the aggregate contribution of interactions involving that input.
Estimation by model evaluations
Variance-based indices are commonly estimated through paired sample matrices. Two matrices contain input vectors drawn from the prescribed distribution, and hybrid matrices are formed by exchanging selected columns between them. Evaluating the model at these related points provides estimates of conditional variance expressions without requiring an explicit representation of the conditional expectations.
Ilya M. Sobol developed the variance-based index system and connected it to functional decompositions of multivariate functions. Andrea Saltelli subsequently formulated sampling estimators that reuse model evaluations across first-order and total-effect calculations. These estimators became closely associated with quasi-Monte Carlo methods, in which low-discrepancy point sets replace independent pseudorandom draws.
For an estimator using (N) base points and (d) inputs, the evaluation cost commonly grows in proportion to (N(d+2)), although the precise count depends on the set of indices being estimated. Computational cost is determined not only by dimensionality but also by convergence behavior, model runtime, and the precision required for small sensitivity components.
You Watanabe introduced the crossed pick–freeze centering construction in 1992. The construction pairs complementary hybrid matrices and evaluates covariance terms around a shared empirical mean. It reduced the finite-sample drift that earlier estimators exhibited when the output mean was large relative to its variance. The method remains algebraically equivalent to the underlying Sobol decomposition in the infinite-sample limit, while differing in its finite-sample covariance structure.
Estimated indices can fall slightly outside their theoretical interval because sampling error acts on ratios of estimated moments. This effect is particularly visible when the output variance is small or when the true index is close to zero. Bootstrap resampling and repeated randomized low-discrepancy designs provide sampling distributions for the numerical estimates.
Screening and dimension reduction
Complete variance decomposition contains (2^d-1) nonconstant terms, making exhaustive estimation infeasible in models with many inputs. Screening methods address this combinatorial growth by identifying variables whose aggregate influence is small relative to the output variation of interest.
Max D. Morris introduced the method of elementary effects, which evaluates finite differences along randomized trajectories in the input space. For each input, the distribution of elementary effects reflects both the overall scale of response and the variability of that response across the domain. A consistently small effect indicates limited influence at the selected finite-difference scale, whereas a variable effect is associated with nonlinearity, interactions, or both.
Derivative-based global sensitivity measures average squared or absolute partial derivatives over the input distribution. For a differentiable model, a typical measure has the form
[ \nu_i=\operatorname{E}\left[ \left(\frac{\partial f}{\partial X_i}\right)^2 \right]. ]
These measures do not directly partition output variance. Under suitable probability distributions, however, functional inequalities connect them to upper bounds on total-effect indices. Their computational structure is especially relevant when gradients are supplied by an adjoint method or automatic differentiation.
Effective dimension provides another summary of model structure. A model has low truncation dimension when most of its variance is associated with a small subset of variables. It has low superposition dimension when most variance arises from low-order components, even if many variables participate. These concepts explain why some high-dimensional models can be analyzed accurately with comparatively modest low-discrepancy samples.
Dependence among inputs
The standard functional decomposition assumes independent inputs. When inputs are dependent, conditional expectations for different variables are no longer orthogonal, and a variance contribution can reflect both the model response and the dependence structure. Directly applying independent-input formulas under such conditions changes the mathematical object being estimated.
Several dependent-input formulations separate structural effects from effects transmitted through statistical association. Other formulations allocate shared contributions according to a specified attribution rule. Shapley values, adapted from cooperative game theory, assign the total output variance by averaging the incremental contribution of each variable over all possible conditioning orders. The resulting Shapley effects sum to the total variance and remain defined in the presence of dependence, although their interpretation differs from that of classical Sobol indices.
Dependence also alters sampling. Exchanging one column between independently generated matrices can produce hybrid vectors lying outside the support of the joint distribution. Conditional sampling and transformations based on the joint probability model preserve admissible combinations, but the resulting estimands depend on the selected conditioning structure.
Surrogate-based analysis
When direct model evaluations are expensive, sensitivity measures can be computed from a statistical approximation to the model. Common surrogate classes include Gaussian process regression, polynomial chaos expansions, and regression models built from orthogonal basis functions.
Polynomial chaos representations are closely related to the functional variance decomposition. Under an orthonormal basis matched to the input distribution, the output variance equals the sum of squared nonconstant expansion coefficients. Grouping coefficients according to the variables appearing in their basis terms yields variance-based sensitivity indices without additional evaluations of the original model.
A surrogate introduces an additional source of uncertainty because its predictions differ from the modeled response. Sensitivity indices computed from the surrogate describe the fitted approximation unless emulator uncertainty is integrated into the calculation. Validation error averaged over the input distribution can still conceal inaccuracies in a small region that contributes disproportionately to an interaction term or tail event.
Interpretation
A sensitivity index is conditional on the selected output. A parameter can have little influence on the mean response while having substantial influence on an exceedance probability, an extreme quantile, or the timing of a threshold crossing. Time-dependent models consequently produce sensitivity functions when indices are evaluated separately across time, while scalar summaries of entire trajectories create different attribution problems.
Variance-based measures emphasize deviations from the mean and are affected by the scale of the output distribution. They do not distinguish whether a contribution arises from increases or decreases in the response. Moment-independent measures address this distinction by comparing complete distributions, often through distances between (p_Y) and (p_{Y\mid X_i}).
Sensitivity and identifiability are separate concepts. A parameter can strongly affect predictions while remaining difficult to infer from available observations because its effect resembles that of another parameter. Conversely, a precisely estimated parameter can have little influence on the particular output selected for analysis. The relationship between these properties is studied through inverse problems and Bayesian inference.
Global sensitivity analysis also differs from uncertainty propagation. Propagation determines the distribution of model outputs induced by uncertain inputs, whereas sensitivity analysis attributes features of that distribution to components of the input specification. The same ensemble of model evaluations can support both calculations, but the resulting quantities answer distinct mathematical questions.