1 Introduction to Global Sensitivity Analysis

1.1 What “global” means versus local sensitivity

Global sensitivity analysis evaluates how uncertainty in a model’s inputs affects uncertainty in its outputs over an entire region of plausible input values. In contrast, local sensitivity methods study changes around a single operating point (or small neighborhood), typically answering how the output responds to infinitesimal perturbations near that point. Because realistic models often operate across a range of conditions, global approaches aim to characterize influence in a way that is representative of the full uncertainty context.

1.2 Goals and typical applications

A central goal of GSA is to identify which inputs most strongly drive output variability when the inputs are uncertain. This information supports model validation (whether a model behaves plausibly under uncertainty), experimental design (where to invest measurement effort), calibration prioritization (which parameters to estimate carefully), and risk assessment (which uncertainties dominate performance uncertainty). In many workflows, GSA is used as a decision tool: either to focus computation and data collection or to justify model simplifications.

1.3 Output types: scalar, vector, and time-dependent responses

Sensitivity analysis can target outputs of different forms. Scalar responses are common when a single quantity summarizes model behavior. Vector outputs can be analyzed component-wise (with separate indices per component) or via multivariate summaries. For time-dependent responses, a typical approach is to compute sensitivity measures at multiple time points or to use aggregate measures across time (e.g., variance over a time window), depending on the scientific question.

1.4 Uncertainty representation for inputs

Inputs in GSA are treated as uncertain quantities with specified probability distributions. These distributions may be derived from measurement data, expert judgment, or prior studies. The choice of distributional assumptions is important: it defines the “plausible space” over which sensitivity is assessed. Joint distributions can be used when inputs are not independent, and sampling-based methods rely on the ability to generate draws consistent with the assumed uncertainty model.

2 Mathematical Foundations

2.1 Model and uncertainty notation

Consider a model \(Y = f(X)\), where \(X = (X_1,\dots,X_d)\) is a vector of input variables and \(Y\) is a quantity of interest. In GSA, \(X\) is random, representing uncertainty. Often, \(Y\) is scalar, but the notation extends to vector-valued outputs by applying the analysis separately or using aggregated metrics. Inputs are characterized by distributions, which may be independent or jointly specified.

2.1.1 Random inputs and joint distributions

If inputs are independent, the joint distribution factorizes into marginals; otherwise, dependence must be modeled explicitly. Joint distributions allow correlation structures and other constraints (for example, parameters that must satisfy physical relationships). The mathematical objects used in variance decomposition and conditional expectations depend on this distributional setup, which in turn affects how sensitivity indices are interpreted.

2.2 Variance decomposition principles

Variance-based GSA decomposes the variance of the output into contributions associated with subsets of inputs. Under appropriate conditions, the total output variance can be partitioned into terms representing main effects (individual inputs), interaction effects (combined influence of multiple inputs), and higher-order interactions (involvement of larger input subsets). This decomposition forms the foundation of Sobol’ indices and functional ANOVA interpretations.

2.2.1 Sobol’ decomposition and functional ANOVA

Functional ANOVA expresses the model response as a sum of components, where each component depends on a particular subset of inputs and is orthogonal to other components in a variance sense. Sobol’ indices then correspond to ratios of variances of these components relative to the total variance. This yields interpretable sensitivity measures: a main-effect index quantifies how much variability remains when averaging over all other inputs, while an interaction index quantifies additional variability that cannot be attributed to lower-order effects.

2.3 Sensitivity indices: interpretations

Sensitivity indices summarize the strength of input effects on output variance. They can be interpreted as normalized contributions to output variability under the chosen input distribution. While different families of indices exist, variance-based indices are especially popular because they link directly to variance decomposition and allow clear accounting for interactions.

2.3.1 Main effects and total effects

A main-effect index for an input \(X_i\) represents the fraction of output variance explained by that input alone, with other inputs averaged out. Total-effect indices capture not only the marginal contribution of \(X_i\) but also any interactions involving \(X_i\). As a result, total-effect indices are often larger or equal to main-effect indices, reflecting the presence of interaction-driven influence.

2.3.2 Interaction effects among inputs

Interaction effects measure how pairs (or larger sets) of inputs jointly affect the output beyond what would be expected from their separate contributions. In practice, interaction indices are useful for detecting non-additive behavior, where a parameter might appear unimportant on its own yet become influential when combined with another uncertain factor.

3 Variance-Based Methods (Sobol’ Indices)

3.1 Basic Sobol’ estimators

Sobol’ indices are estimated through Monte Carlo sampling combined with variance decomposition identities. The canonical estimators use repeated evaluations of the model at systematically constructed combinations of input samples. For practical computation, estimators separate terms associated with conditioning and complement events, allowing approximate recovery of main and total variance contributions from sampled model outputs.

3.2 Estimating main effects

Main-effect estimation targets the portion of output variance explained by one input while treating other inputs as nuisance variables. The core idea is to use conditional expectation: for a given \(X_i\), compute the expected output over the remaining inputs, then quantify how much this conditional mean varies with \(X_i\).

3.2.1 Conditional expectation formulation

Let \(m_i(x_i)=\mathbb{E}[Y\mid X_i=x_i]\). The main-effect variance is \(\mathrm{Var}(m_i(X_i))\). Normalizing by the total output variance yields a Sobol’ main index. Estimators of \(\mathbb{E}[Y\mid X_i]\) are obtained by averaging model evaluations over random draws of the other inputs, which introduces sampling variability into the estimated index.

3.3 Estimating total effects

Total-effect estimation captures all variance contributions involving a given input, including interactions with any subset of other inputs. This is typically formulated using an equivalent “complement” perspective: total-effect variance can be expressed using conditional variance given all other inputs.

3.3.1 Complement variance idea

If \(X_{\sim i}\) denotes all inputs except \(X_i\), then the total-effect contribution can be represented via \(\mathbb{E}[\mathrm{Var}(Y\mid X_{\sim i})]\). Intuitively, if knowing all other variables largely determines the output, then the remaining uncertainty attributable to \(X_i\) is small. Monte Carlo estimators approximate this conditional variance through repeated sampling and averaging.

3.4 Higher-order interaction indices

Beyond main and total effects, Sobol’ methods can estimate variance contributions from interactions among two or more inputs. Higher-order indices quantify the incremental importance of a joint subset beyond all lower-order terms. These indices are informative for understanding synergistic effects, but they can be more expensive to estimate because they require additional structured sampling to disentangle overlapping contributions.

3.5 Practical considerations and estimator variance

Estimated Sobol’ indices have uncertainty due to finite sampling. Estimator variance depends on the model’s nonlinearity, the dimensionality of input space, and the sampling scheme. Practitioners often compute confidence bounds or monitor estimator stability by repeating computations, using increasing sample sizes, or relying on diagnostics that measure convergence of index estimates.

3.6 Sample-efficiency and convergence diagnostics

Sobol’ estimation typically needs many model evaluations, especially for higher-order effects or large input dimension. Convergence diagnostics help determine whether additional samples are likely to materially change the estimated indices. Sample-efficiency considerations motivate the use of screening methods, surrogate models, or alternative estimators that target key inputs first and refine only where necessary.

4 Screening and Efficient Approximation Methods

4.1 Motivation for screening in high dimensions

When the number of uncertain inputs is large, full variance-based Sobol’ analysis becomes computationally demanding. Screening methods aim to quickly identify inputs that are likely to have negligible impact on output variability, allowing the analyst to reduce dimensionality before applying more expensive techniques.

4.2 One-at-a-time (OAT) versus global screening

A naive strategy is one-at-a-time perturbation: vary one parameter while holding others fixed. While OAT can reveal local tendencies, it may miss interactions and depends heavily on the chosen baseline. Global screening approaches instead sample across the plausible space, capturing effects that emerge only through combinations of uncertain inputs, at the cost of additional computation compared with purely deterministic perturbations.

4.3 Morris (elementary effects) method

The Morris method provides an efficient way to assess the relative importance of each input. It constructs trajectories through the input space and measures “elementary effects,” which are finite-difference-like changes in the output when moving along each input direction. Aggregating these changes over multiple trajectories yields summary statistics used for ranking factors.

4.3.1 Interpreting trajectories and elementary effects

Two common summary measures are the mean of absolute elementary effects (often interpreted as overall impact) and the standard deviation (often associated with nonlinearity and interaction complexity). Inputs with large mean but small variability tend to influence the output in a relatively monotonic way, whereas large variability suggests that the effect depends strongly on where in the input space the model is evaluated.

4.4 FAST and frequency-based strategies (conceptual overview)

Frequency-based approaches, such as Fourier amplitude methods, connect sensitivity to spectral content of the model output viewed as a function of inputs. Conceptually, inputs are associated with frequency components induced by structured sampling. These strategies can be efficient for certain classes of models, but practical performance depends on the sampling design and the form of model nonlinearity.

4.5 Emulators as acceleration tools

In many engineering settings, each model evaluation may be expensive. Building an emulator—a fast surrogate approximating the original model—can make global sampling feasible. Screening and ranking can be performed on the surrogate, either to narrow down influential parameters or to guide where to allocate expensive evaluations for higher-fidelity analysis.

5 Surrogate-Model and Emulator-Based GSA

5.1 Why emulation is used

Emulators replace costly simulations with cheaper approximations, enabling large numbers of evaluations that would otherwise be impractical. The emulator is trained on a set of input-output pairs produced by the original model. Once trained, global sensitivity indices can be estimated by sampling the emulator over the input uncertainty distributions.

5.2 Common surrogate types (high level)

Surrogate models can be constructed using various statistical or numerical techniques. The choice typically balances accuracy, interpretability, and computational cost of training and evaluation.

5.2.1 Gaussian process emulators

Gaussian process (GP) emulators provide a probabilistic surrogate, where predictions include both a mean response and uncertainty estimates. This uncertainty can be propagated into sensitivity analysis to reflect emulator error. GP models are often effective for smooth functions and can adapt well when data are moderately sized.

5.2.2 Polynomial chaos and regression surrogates

Polynomial chaos expansions approximate the model output as a series of orthogonal polynomials in the inputs, chosen to match the input distributions. Regression-based surrogates fit flexible functional forms directly from data, which can range from linear models to nonlinear basis expansions. These approaches can be efficient and enable analytical or semi-analytical computation of sensitivity measures.

5.3 Sensitivity analysis on surrogates

Once the surrogate is available, one can compute sensitivity indices using either the surrogate’s analytic structure (when available) or Monte Carlo sampling through the surrogate. Indices derived from the emulator are only as reliable as the surrogate quality, so analysts commonly validate emulator predictions in regions relevant to the sensitivity task.

5.4 Managing approximation error

Approximation error can bias sensitivity results if the surrogate underfits or extrapolates into regions unsupported by training data. Remedies include increasing the design size, using adaptive sampling (adding points where the surrogate uncertainty is high), cross-validation, and techniques that propagate emulator uncertainty into sensitivity estimates, thereby separating model-driven effects from surrogate artifacts.

6 Derivative-Based and Statistical Learning Approaches

6.1 Gradient-based sensitivity measures (conceptual)

When differentiable models are available, gradient information can be used to quantify sensitivity. Derivative-based measures relate output variability to local changes in the output with respect to inputs. While these are often “local” in their raw form, various strategies attempt to aggregate gradient information across the input space to produce a more global view.

A common challenge is that gradients are computed at specific input locations. Globalization strategies evaluate gradients at many sampled points and aggregate their magnitude or directional influence according to the input distribution. This yields sensitivity rankings that can complement variance-based indices, particularly for smooth systems.

6.3 Sensitivity via mutual information (overview)

Mutual information methods treat sensitivity as a measure of statistical dependence between inputs and outputs. Unlike variance-based approaches that focus on second moments, mutual information captures more general nonlinear relationships. In practice, estimating mutual information can be more complex, especially in high dimensions, but it offers an alternative when variance decomposition is less informative or when dependencies are strongly nonlinear.

6.4 Active learning for sensitivity exploration

Active learning iteratively selects new model evaluations to reduce uncertainty in sensitivity estimates. The selection rule is typically guided by an acquisition function, which may depend on surrogate uncertainty or on the expected value of improved sensitivity ranking. This reduces the number of expensive simulations needed to reach stable conclusions, particularly when the influential set of inputs is not known in advance.

7 Handling Correlated Inputs and Constraints

7.1 Correlation-aware sampling

Standard Sobol’ indices are most straightforward under independent inputs. When inputs are correlated, sampling must reflect the joint dependence structure. Correlation-aware sampling ensures that the investigated combinations of inputs are consistent with how the system is expected to vary, avoiding misleading sensitivity results that would arise from unrealistic independence assumptions.

7.2 Conditional sensitivity formulations

With dependence, conditioning-based formulations become central. Sensitivity measures may be defined in terms of conditional expectations or conditional variances, capturing how much output variability remains after accounting for the statistical structure linking inputs. These definitions can differ from independence-based indices, so careful interpretation is required.

7.3 Impacts of dependent distributions

Dependence can blur the separation between inputs. For instance, an input may appear influential because it is correlated with a truly driving variable, or it may seem less important because much of its variability is shared with others. This makes it crucial to use indices whose definitions match the dependence assumptions, and to report the assumed joint distribution when presenting results.

7.4 Constrained parameter spaces

Constraints arise when parameters must satisfy bounds, conservation laws, or other feasibility conditions. In GSA, constraints define the feasible region for sampling. Approaches that respect constraints generate valid input draws and compute sensitivity indices in a way that reflects the restricted space the model actually occupies.

7.5 Interpreting indices under dependence

Under dependence, “importance” is not always attributable to an input in isolation. Indices may represent contributions conditional on the dependence structure, rather than purely marginal effects. Transparent reporting—joint distribution assumptions, conditioning conventions, and constraints—is therefore necessary for correct interpretation and fair comparison across studies.

8 Experimental Design and Sampling Strategies

8.1 Selecting distributions for input uncertainty

Defining input uncertainty distributions is a prerequisite for GSA. Analysts often combine measurement error models, calibration priors, and physical constraints to build distributions that represent plausible variability. Sensitivity results should be viewed as conditional on these distributional choices, making distribution elicitation and documentation an integral part of the workflow.

8.2 Sampling schemes

Most GSA estimators require repeated sampling of inputs and model evaluations. Efficient sampling strategies aim to reduce estimator variance while maintaining fidelity to the assumed distributions.

8.2.1 Monte Carlo and quasi–Monte Carlo (QMC)

Monte Carlo sampling uses random draws and yields consistent estimates as sample size grows. Quasi–Monte Carlo methods use low-discrepancy sequences to cover the space more uniformly, often improving convergence for sufficiently smooth integrands. The practical benefit depends on the model structure and how the estimator’s underlying assumptions interact with the sampling method.

8.3 Stratified and Latin hypercube designs (overview)

Stratified sampling partitions the input space into regions and samples within each, reducing variability from uneven coverage. Latin hypercube sampling ensures that marginal distributions are well represented by design, while still providing multi-dimensional space-filling behavior. These designs are popular because they can improve coverage with fewer samples than purely random draws.

8.4 Number of samples and computational budgets

The required sample size depends on the estimator type, the number of inputs, the desired precision, and the cost of model evaluation. Analysts commonly start with a pilot study to gauge variability, then scale up sample sizes to meet convergence targets. Budget constraints often motivate hybrid workflows: screening first, then deeper variance-based estimation on a reduced set of variables.

8.5 Variance reduction techniques

Variance reduction can improve efficiency by decreasing estimator noise without changing the underlying model. Techniques include antithetic variates, control variates, and improved estimators designed for variance-based indices. When combined with intelligent sampling designs, these methods can make sensitivity analysis more reliable at a fixed computational budget.

9 Uncertainty Quantification Coupling

9.1 Relationship between sensitivity and UQ

Uncertainty quantification (UQ) quantifies how uncertainties in inputs produce uncertainties in outputs. Sensitivity analysis can be viewed as a structural component of UQ: it explains which uncertainties drive which parts of the output uncertainty. In workflows where output uncertainty is computed, sensitivity indices help interpret the sources and guide actions to reduce uncertainty meaningfully.

9.2 Uncertainty decomposition and model discrepancy (conceptual)

In many modeling contexts, output uncertainty arises from both input uncertainty and model discrepancy (differences between the model and reality). Conceptually, GSA focuses on the former while leaving discrepancy characterization to other components of UQ. Some integrated frameworks allow sensitivity analysis to inform calibration of parameters and separate effects from discrepancy terms.

9.3 Propagating input uncertainty through GSA pipelines

A typical pipeline uses sampling to propagate input uncertainty through the model or surrogate, then computes sensitivity indices from the resulting output ensemble. When surrogates are used, the pipeline may propagate emulator uncertainty as well. The final reported indices therefore reflect both the uncertainty model and the estimation procedure.

9.4 Case studies: sensitivity-guided calibration (light overview)

Sensitivity-guided calibration uses GSA results to determine which parameters to tune and how to allocate measurement resources. For example, parameters with high total-effect indices on key outputs may be targeted for more careful calibration. Conversely, parameters with negligible indices might be fixed at nominal values to simplify the calibration problem, reducing dimensionality and computational burden.

10 Visualization and Reporting

10.1 Tornado charts and bar plots for main/total effects

Bar plots and tornado charts are common for presenting main-effect and total-effect indices. Tornado charts order inputs by their influence magnitude, often highlighting the difference between main and total effects as an indicator of interaction importance. Clear labeling and consistent scaling are important to avoid confusion between different index types.

10.2 Heatmaps for pairwise interactions

Pairwise interaction strength is often visualized with heatmaps that display indices or derived measures for input pairs. This format makes it easy to spot combinations that produce strong joint effects. Accompanying text should clarify whether the heatmap represents an interaction index, a difference between total and main effects, or another interaction proxy.

10.3 Partial dependence–style summaries (conceptual)

Some surrogate-based approaches support summaries akin to partial dependence, showing how expected outputs change as one input varies while others are averaged. While these are not identical to variance-based indices, they provide intuitive insight into functional form and can complement index-based rankings, especially when nonlinearity is present.

10.4 Reporting uncertainty in sensitivity estimates

Sensitivity results should include uncertainty quantification for the estimated indices, such as confidence intervals or error bars derived from repeated sampling or estimator variance formulas. Reporting also benefits from stating sample size, sampling design type, convergence diagnostics used, and assumptions about input distributions. These details help readers judge the stability and transferability of conclusions.

11 Practical Workflow

11.1 Step-by-step GSA pipeline

A typical workflow begins with problem definition: identify the model outputs of interest and the uncertain inputs. The analyst then selects sensitivity indices aligned with the objective—screening versus precise decomposition—and specifies the input uncertainty model.

11.1.1 Choosing indices and sampling plan

Index choice depends on whether main effects, total effects, or interaction patterns are needed. The sampling plan is then designed to support those indices, balancing estimator requirements with available computational resources. In high-dimensional cases, screening indices or surrogate-assisted approaches are often used first.

11.1.2 Running estimates and checking convergence

Estimation proceeds by generating input samples, evaluating the model (or emulator), and computing indices. Convergence checking typically involves increasing sample size, monitoring index stability, and verifying that estimated uncertainties shrink as expected. If indices change materially with additional samples, more computation or an improved strategy (e.g., variance reduction or emulation refinement) may be required.

11.2 Model screening to reduce dimensionality

Screening identifies a subset of influential inputs, reducing dimensionality for subsequent analysis. This step can prevent wasted effort on parameters that have minimal contribution to output variance. Screening results also help interpret complex models by pointing to the most relevant directions of uncertainty.

11.3 Iterative refinement of sensitivity studies

Many studies are iterative: initial screening suggests which variables matter; a refined analysis then estimates indices more precisely and explores interactions. If surrogate models are used, new training points may be added after inspecting residual errors or regions where sensitivity conclusions appear unstable. The iterative process continues until the analyst has sufficient confidence to support decisions.

12 Software Ecosystem and Reproducibility

12.1 Common tool types (framework categories)

Software for GSA typically falls into categories such as: variance-based index estimators, screening method implementations, surrogate modeling packages, and workflow toolkits that combine sampling, model evaluation, and post-processing. Some libraries provide ready-made estimators for Sobol’ indices and elementary effects, while others emphasize generic probabilistic modeling and require custom sensitivity computations.

12.2 Verification and validation of implementations

Verification checks that a method is implemented correctly (e.g., consistency with known identities on test functions). Validation evaluates performance and interpretability on representative models, possibly comparing results across methods (e.g., screening versus variance-based indices) to ensure conclusions are robust. Careful handling of random seeds and sampling designs supports both verification and validation.

12.3 Reproducible workflows (inputs, seeds, data provenance)

Reproducibility involves recording the model version, input distributions, sampling parameters, random seeds, emulator training settings, and post-processing code. Data provenance should track where model outputs came from, how surrogates were trained, and which indices were computed from which datasets. Such documentation enables others to replicate results and evaluate differences in assumptions.

12.4 Performance considerations

Performance issues include the computational cost of model evaluations, memory use for large sample sets, and time spent training surrogates. Parallelization is commonly used, especially for Monte Carlo sampling where model evaluations are independent. Efficient data handling and batching can reduce overhead and improve throughput in large-scale sensitivity studies.

13 Limitations and Best Practices

13.1 Curse of dimensionality and computational cost

Many GSA methods scale poorly with the number of inputs, particularly when estimating higher-order interactions or using expensive simulations. The curse of dimensionality manifests as increased sampling demands to achieve stable estimates. Screening, surrogate modeling, and careful selection of indices are common strategies to mitigate these limitations.

13.2 Sensitivity versus identifiability (conceptual)

Sensitivity indicates how uncertainty in inputs affects output variability, but it does not automatically imply that input parameters are identifiable from data. Identifiability depends on the structure of the model and the available observations, including whether distinct parameter combinations produce distinguishable outputs. A parameter can have low sensitivity to a chosen output yet still be relevant for identifiability when other outputs or measurements are considered.

13.3 Robustness to distribution choices

Because sensitivity indices depend on the assumed input distributions, results can change when distributions are modified. Best practice is to perform robustness checks—such as sensitivity to distributional parameters or alternative plausible ranges—to determine whether conclusions about “important” inputs are stable or highly assumption-driven.

13.4 Avoiding misinterpretation of interaction indices

Interaction indices can be misread if one assumes additive contributions. Large interaction values indicate non-additive effects under the assumed sampling distribution, not necessarily causal mechanisms. It is also important to remember that interactions depend on how inputs are defined and distributed; reparameterizations or changed uncertainty models can alter the apparent interaction structure.

13.5 Ethical and responsible communication of uncertainty

Reporting sensitivity results responsibly involves clarifying assumptions, communicating uncertainty in estimates, and avoiding overconfident statements about precision. Analysts should describe the limitations of the sampling design, the surrogate quality (if used), and convergence diagnostics. Clear documentation supports informed decision-making and prevents the misuse of sensitivity findings as certainty.