Global sensitivity analysis concepts

Global sensitivity analysis (GSA) asks how uncertainty in a model's inputs relates to uncertainty in its outputs. Unlike a local derivative, it considers the full input distributions rather than behaviour at one nominal point.

The practical questions are usually:

• Which parameters account for most of the output variation?
• Which parameters can be fixed with little effect?
• Which parameters matter mainly through interactions?
• Does a parameter change only the mean, or the whole output distribution?
• How does dependence between parameters affect the answer?

There is no single sensitivity index that answers all of these questions. Choose the quantity you want to interpret before choosing an estimator. See Choosing a method for the package-wide comparison.

Variance-based indices

Variance-based methods express importance as a fraction of output variance. For independent inputs, a square-integrable model can be decomposed into main effects and interactions:

$$f(\mathbf{X})=f_0+\sum _i{f}_i(X_i)+\sum _{i<j}{f}_{ij}(X_i,X_j)+\cdots$$

The corresponding variance components add to the total output variance:

$$\mathrm{Var}(Y)=\sum _i{V}_i+\sum _{i<j}{V}_{ij}+\cdots$$

This gives three commonly reported Sobol' indices.

Index	Meaning
$S_1(i)$	Variance attributed to parameter $i$ alone.
$S_2(i,j)$	Additional variance attributed to the interaction between $i$ and $j$.
$S_T(i)$	Variance involving parameter $i$, including all of its interactions.

More formally,

$$S_1(i)=\frac{{\mathrm{Var}}_{X_i}(\mathbb{E}[Y\mid X_i])}{\mathrm{Var}(Y)},$$$$S_2(i,j)=\frac{V_{ij}}{\mathrm{Var}(Y)},$$

and

$$S_T(i)=\frac{{\mathbb{E}}_{{\mathbf{X}}_{\sim i}}[\mathrm{Var}(Y\mid {\mathbf{X}}_{\sim i})]}{\mathrm{Var}(Y)}.$$

Here, ${\mathbf{X}}_{\sim i}$ means every input except $X_i$. For the population indices, $S_T(i)\ge S_1(i)$. The gap $S_T(i)-S_1(i)$ shows how much of the parameter's influence is associated with interactions. A value of $S_T(i)\approx 0$ supports fixing that parameter within the distributions and model configuration that were analysed.

Finite-sample estimates can be noisy. Small negative estimates, or an estimated $S_1$ slightly above $S_T$, do not change the mathematical definitions. Use confidence intervals and larger designs before interpreting small differences. See the Sobol' API for estimators and sampling requirements.

Sobol' and eFAST estimate variance indices from dedicated model evaluations. PCE and HDMR derive them from fitted surrogates. Shapley effects instead divide the variance into one allocation per parameter.

Dependent inputs

The usual Sobol' decomposition assumes independent inputs. Under dependence, "importance" can include the model's structural response, information carried through correlation, or both. These are different estimands and need not give the same ranking.

jaxgsa provides several routes for dependent inputs:

• Kucherenko estimates conditional-variance indices from a dedicated design.
• VKOGA estimates correlated and uncorrelated variance contributions through a kernel surrogate.
• HDMR separates each fitted component into structural and correlative contributions.
• Distribution-based methods can measure dependence-inclusive influence without constructing a Sobol' decomposition.

Do not compare these values as though they were interchangeable versions of the same index. The methods guide explains which dependent- input question each route answers.

Screening measures

Screening methods look for inputs that are negligible or worth investigating further. They are useful when the parameter count is large or the model-run budget is too small for a precise variance decomposition.

Morris summarizes elementary effects with ${\mu }^{\ast }$ and $\sigma$. A small ${\mu }^{\ast }$ suggests little overall influence; a large $\sigma$ suggests a nonlinear response or interactions. DGSM uses derivatives across the input domain and can provide bounds on total effects.

These quantities are not variance fractions. Use them to remove clearly inert parameters or to plan a second analysis, not to read off percentages of explained variance. A common workflow is to screen first, then apply a variance-based method to the remaining parameters.

Distribution-based measures

Variance can be an incomplete description of skewed, multimodal, or heavy-tailed outputs. Distribution-based methods compare the unconditional output distribution with the output conditioned on an input. They detect changes in location, spread, and shape, but their scores are not Sobol' indices.

• HSIC measures statistical dependence between an input and the output using kernels.
• PAWN compares conditional and unconditional cumulative distributions with the Kolmogorov--Smirnov distance.
• Borgonovo delta compares output densities.
• Optimal transport uses Wasserstein distance and separates mean-shift and shape-change contributions.

Use these methods when the question is "does this input change the output distribution?" rather than "what fraction of variance does it explain?"

Allocation and surrogate-based methods

Shapley effects allocate importance across parameters so the reported shares sum to one. Interactions are divided among their participants, which gives one number per parameter but no separate pairwise interaction index.

PCE, HDMR, and VKOGA fit a surrogate and derive sensitivity measures from that fitted model. This is valuable when you have an existing set of input-output pairs or also need a fast emulator. However, the reported sensitivity is sensitivity of the fitted surrogate. Poor fit, sparse coverage, or a restrictive basis can produce confident-looking indices that do not describe the original model.

Check predictive error on held-out data where possible, inspect diagnostics, and include surrogate refitting in uncertainty estimates. A bootstrap that only resamples a fixed surrogate understates fit uncertainty.

Interpreting an analysis

Keep four points attached to every result:

1. The estimand. A screening score, variance fraction, dependence measure, and distributional distance answer different questions.
2. The input distribution. Sensitivity is defined over the ranges, marginals, and correlations supplied to the analysis. Change them and the result can change.
3. The output. Indices may differ across scalar outputs or across time. A single aggregation can hide that structure.
4. The uncertainty. Sampling error and, where applicable, surrogate error matter most when indices are small or rankings are close.

Once the question and estimand are clear, use Choosing a method to select an implementation and follow its API page for sampling, assumptions, and result fields.