Global sensitivity analysis concepts

Global sensitivity analysis (GSA) asks which uncertain inputs matter to a model's output. It looks across the input ranges and distributions you specify, rather than at one chosen set of input values.

A result is always relative to that choice of inputs. If you change their ranges, probabilities, or correlations, you may change the ranking of important inputs.

What does “important” mean?

Imagine a model with two uncertain inputs. If the output is x1 + x2, each input contributes on its own: changing x1 has the same effect whatever x2 is. If the output is x1 * x2, the effect of x1 depends on x2. That is an interaction.

For independent inputs, Sobol' indices separate these effects.

This decomposition follows Sobol' (1993); the Sobol' references also distinguish the estimators used to calculate the indices.

Index	Read it as
$S_1(i)$, first order	The share of output variance due to input $i$ on its own.
$S_T(i)$, total order	The share involving input $i$, on its own or through interactions.
$S_2(i,j)$, second order	The share due to the interaction between inputs $i$ and $j$, beyond their separate effects.

For an additive model, $S_1$ and $S_T$ agree. A gap between them points to interactions. For example, an input with $S_1$ near zero and $S_T$ around 0.25 matters through interactions even though it has little effect alone. A small $S_T$ supports fixing that input within the model and input distributions you analysed.

These are shares of variance, not shares of the mean output or probabilities that an input matters. Sobol' and eFAST estimate them from planned model runs; PCE and HDMR can derive them from a fitted surrogate.

Estimates from a finite number of runs can be noisy. Small negative estimates, or an estimated $S_1$ slightly above $S_T$, do not change the definitions. Check confidence intervals or increase the design before interpreting small differences. See the Sobol' API for sampling requirements. For a study of convergence and validation, see Sarrazin, Pianosi & Wagener (2016).

Other questions need other measures

A variance share may not answer your question. Use Choosing a method to match the result to the decision you need to make.

Which inputs can I screen out?

Morris measures how much the output changes in short moves across the input space. Its ${\mu }^{\ast }$ summarizes the size of those changes; a large $\sigma$ suggests a nonlinear effect or interactions. DGSM measures derivatives across the input domain and can give bounds on total effects.

See Morris (1991) and Campolongo et al. (2007) for elementary-effects screening, and Sobol' & Kucherenko (2009) for derivative-based measures.

These are screening measures, not variance percentages. They are useful for finding clearly inactive inputs before a more detailed analysis. A large $\sigma$ alone does not tell you whether nonlinearity or interactions caused it.

Does an input change the whole output distribution?

Two inputs can have similar variance effects but change different parts of an output distribution. This matters for skewed, multimodal, or heavy-tailed outputs. Distribution-based methods compare outputs across input values:

• PAWN compares cumulative distributions.
• Borgonovo delta compares densities.
• Optimal transport measures how far distributions move and separates changes in mean from changes in shape.
• HSIC measures statistical dependence with kernels and can report permutation p-values.

Their scores are not Sobol' indices. They can rank inputs differently because they measure different changes.

For the definitions and implemented estimators, see the references for PAWN, Borgonovo delta, optimal transport, and HSIC.

What if inputs are dependent?

If inputs are correlated, one input can carry information about another even when the model uses only the latter. The usual Sobol' variance decomposition assumes independent inputs, so there is no single replacement index under dependence.

Kucherenko, VKOGA, and HDMR report different conditional or structural and correlative variance quantities. Distribution-based scores can include influence carried through correlation. Do not compare these values as though they were interchangeable. See Sensitivity with dependent inputs for a worked choice.

Can I have one share per input?

Shapley effects allocate variance across inputs so that the shares sum to one. They divide interactions among participating inputs; they do not show a separate pairwise interaction index.

For independent inputs, the allocation follows Owen (2014). The dependent-input guide explains the different ANCOVA allocation used with the HDMR backend.

Before trusting a result

1. Check the input distribution. Indices refer to the specified ranges, marginal distributions, and correlations.
2. Check the output. Different outputs or time points can have different rankings; averaging them can hide that difference.
3. Check uncertainty. Sampling error matters when estimates are small or close together.
4. Check surrogate fit. PCE, HDMR, and VKOGA report sensitivity of a fitted model. Poor fit or sparse coverage can make its indices misleading. Check predictions on held-out data where possible. A bootstrap that keeps the fitted surrogate fixed does not include uncertainty from refitting it.

Formal definitions of variance indices

For independent inputs and a square-integrable model, the output 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 variance components then add to the total output variance:

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

The first-order, second-order, and total-order indices are

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

Here, ${\mathbf{X}}_{\sim i}$ contains every input except $X_i$. For population indices, $S_T(i)\ge S_1(i)$, and their difference is the contribution involving interactions with input $i$. For a model with nonzero output variance, an input's indices lie between zero and one. Finite-sample estimates may fall outside these bounds.

References and further reading

• Introduction to GSA and study design: Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., & Tarantola, S. (2008). Global Sensitivity Analysis. The Primer. John Wiley & Sons. ISBN 978-0-470-05997-5.
• Convergence and validation: Sarrazin, F., Pianosi, F., & Wagener, T. (2016). Global Sensitivity Analysis of environmental models: Convergence and validation. Environmental Modelling & Software, 79, 135–152.

For the definitions of variance indices and total-order importance, see Sobol' (2001) and Homma & Saltelli (1996).