Sensitivity Analysis

Learning objectives

• Calculate a local sensitivity coefficient and a normalized sensitivity, stating units and sign.
• Distinguish a sensitivity from an uncertainty contribution.
• Interpret first-order and total-effect Sobol indices and interaction effects under the standard mutually independent-input decomposition.
• Determine whether an input may be considered for factor fixing using its total-effect index and a stated decision tolerance, not its first-order index alone.

This module ranks inputs by influence. Tick only what you can do closed-notes.

• Take a partial derivative of a function.
• Recall a normalized or relative quantity.
• Recall variance as a measure of spread.
• Add fractions that should sum to one.
• Recall that a first-order uncertainty contribution is |c_i|u(x_i), not c_i alone.

Seven skills build local sensitivity, its relationship to uncertainty contribution, global variance-based analysis, and what a ranking does and does not support.

8.1 Local sensitivity coefficients

The local sensitivity of the output to an input is the partial derivative c_i = ∂y/∂x_i evaluated at a stated point, with the other inputs held fixed. It measures the output response to a small perturbation of one input.

The coefficient is signed: a positive c_i means the output rises locally as the input rises, a negative one that it falls, and zero means no first-order effect at that point, which is not the same as no influence elsewhere. It carries output-per-input units, so raw coefficients from different inputs are not comparable, and its magnitude changes if the model is reparameterized.

Where no analytic derivative is available, finite differences are the usual estimate, and the step is a real choice. For a sufficiently smooth deterministic model, a central difference has second-order truncation error, whereas a forward difference has first-order truncation error. Central differences usually provide better truncation accuracy for the same step, but require two evaluations and may still be degraded by roundoff, solver noise, parameter bounds, or discontinuities. There is no universal perturbation size: a blanket rule such as changing every input by 1% is undefined at zero and may be physically meaningless for a bounded, signed, or categorical parameter.

8.2 Normalized sensitivity

Dividing by the output and multiplying by the input gives a dimensionless local coefficient that keeps its sign:

read as the local percent output change per percent input change. It lets inputs of different units be compared on one scale. State the point at which it was evaluated, because normalising removes the units, not the locality.

The normalized coefficient requires a nonzero output y. Its percent-to-percent interpretation also requires a nonzero input x_i, because a relative input change is undefined at zero. Near an output zero the coefficient may become singular; near an input zero the algebraic value may remain finite, but the percentage interpretation is not meaningful.

8.3 Sensitivity versus uncertainty contribution

A sensitivity says how strongly the output responds. It does not say how much uncertainty an input contributes, because that also depends on how uncertain the input is. Module 7 supplies the pieces: the signed first-order effect d_i = c_iu(x_i), the non-negative standard-uncertainty component u_i(y) = |d_i|, and the diagonal variance contribution d_i².

For nonzero x_i and y these carry over to relative form. The signed relative first-order effect is

and the non-negative relative standard-uncertainty component is

Rank uncertainty contributions with the magnitude expression. Keep the signed form wherever covariance is in play, since Module 7's cross term 2ρ_ijd_id_j needs the signs. A signed relative effect is not itself a standard uncertainty.

The two rankings can disagree. An input with twice the normalized sensitivity but a quarter of the relative uncertainty contributes half as much. Ranking by c_i answers what the model responds to; ranking by u_i(y) answers where the output uncertainty is coming from.

Normalising the diagonal terms gives a share of the first-order output variance:

Provided Σ_jd_j² > 0, P_i is a normalized local first-order variance share and lies between zero and one. If all first-order effects vanish at the evaluation point, the shares are undefined and the local ranking provides no first-order discrimination. This does not establish that the inputs are globally non-influential.

This is valid only under the zero-covariance first-order approximation of Module 7. It is not a Sobol index. Where covariance is present the total also contains cross terms, which cannot generally be assigned to individual inputs without adopting an allocation convention.

8.4 Global sensitivity analysis

Global methods vary all inputs across a stated joint distribution and measure the effect on the output over the whole range, not just at a point. They capture nonlinearity and interactions that local derivatives miss, at the cost of many model evaluations.

A global result is conditional throughout. It depends on the input ranges and distributions adopted, the dependence assumptions, the output quantity analysed, the region of the model explored, and the parameterization, and it requires the output variance to exist. Report those choices with the ranking, because a ranking without them is not reproducible and is not a property of the model.

Variance-based Sobol indices require a probabilistic representation of the inputs and output. Aleatory and epistemic components may be analysed jointly, conditionally, or separately depending on the adopted uncertainty framework. The indices reported here are conditional on the chosen distributions and independence assumptions; they are not automatically appropriate for interval-only or otherwise non-probabilistic epistemic representations.

8.5 Variance-based Sobol indices

This section and the next describe the standard Sobol, or functional-ANOVA, decomposition, which assumes mutually independent inputs. Every property stated below belongs to that framework. The first-order index is

so V_i is the main-effect variance: the spread of the output's conditional mean as X_i varies over its distribution. Within this framework 0 ≤ S_i ≤ 1. For an additive model the first-order indices sum to one; otherwise Σ_iS_i < 1, and the shortfall is interaction variance. Where the inputs are not mutually independent this decomposition is not unique in this form, and a specialised dependent-input formulation is required.

In practice these indices are estimated by sampling, so they carry sampling error. Report the estimator and the sample count, and check that the ranking has settled rather than only the point estimates. Two indices that are close may be unresolved at the sample size used. Estimators can also return slightly negative index estimates from noise alone, which is an estimation artefact rather than a negative share.

8.6 Total-effect indices and interactions

The first-order index deliberately excludes interactions. The total-effect index includes them:

where X_~i is every input except i. It captures everything input i does, alone and through interaction, so within this framework 0 ≤ S_Ti ≤ 1 and S_Ti ≥ S_i always. The gap S_Ti − S_i is evidence of interactions involving input i. Summing those gaps double counts, because a shared interaction appears in the total effect of every input it involves; for the same reason the total-effect indices generally sum above one, with equality only for a purely additive model.

Interaction is not a property of the algebra. For y = x₁x₂ with mutually independent inputs, the interaction share is about 0.001 when each input has a 5% coefficient of variation and about 0.111 when each has 50%. The same product model, two very different decompositions, because the input distributions differ.

Local measures are cheap and narrow. Only the last two answer a global question, and only the total-effect index accounts for what an input does through interaction. Both Sobol rows assume mutually independent inputs.

8.7 What a ranking supports, and choosing a method

Name the question before reading the ranking, because two different decisions call for two different indices.

Choosing what to learn. In the idealized factor-prioritization setting, suppose one input can be learned perfectly and fixed at its true value, with comparable investigation costs across inputs. The largest first-order index identifies the input expected to produce the greatest reduction in output variance, averaged over its possible true values. Real measurement or research programmes usually reduce uncertainty only partially and at different costs. The achievable uncertainty reduction, investigation cost, and decision value must therefore be considered alongside S_i.

Choosing what to leave out. Under the standard independent-input Sobol decomposition, S_Ti = 0 identifies a non-influential input. A sufficiently small estimated total-effect index may support fixing an input only relative to a stated tolerance for the resulting output-variance or decision impact. The first-order index cannot serve here: for y = x₁ + x₂x₃ with independent inputs symmetric about zero, x₂ and x₃ each have a first-order index of exactly zero, yet together they carry a quarter of the output variance.

Reconsider before fixing anything when the decision concerns tails or rare events rather than variance, when the input acts on a constraint or a threshold, when the estimated ranking is itself uncertain, when the inputs are dependent, when the model region may shift, or when the decision is safety-critical.

Finally, separate the ranking from the decision. Uncertainty importance, design leverage, controllability, physical influence, and the cost of reducing an uncertainty are five different orderings. A high-sensitivity input you cannot control and a controllable input you already know well are different engineering situations, and no single index settles either. Sensitivity analysis reports that, within the adopted model and variation ranges, changes in an input are associated with the stated output response; causal conclusions need physical reasoning and experiment as well. In very high-dimensional problems, cheaper screening methods may be preferable for a first pass before variance-based indices are estimated.

Engineering connection: a sensitivity study turns a broad UQ result into a focused test plan, but only once you have said which decision it serves: which parameter to measure better, or which to leave out of the analysis.

Source. This module uses the variance-based sensitivity definitions and factor-prioritization/factor-fixing settings described by Saltelli et al. in Global Sensitivity Analysis: The Primer. It does not claim normative compliance.

For a power law y ∝ x³, what is the normalized sensitivity to x?

An output variance of 200 has main-effect variances V₁ = 150 and V₂ = 40, for two inputs treated as mutually independent. Find the first-order indices and the interaction share, then say what you can conclude about the total effects.

(a) For y = x₁·x₂ at x₁ = 4, x₂ = 5, find the normalized sensitivity to x₁. (b) The same model is analysed globally with X₁ and X₂ mutually independent. Does the interaction share follow from the algebra?

A model has ten uncertain inputs but a limited test budget. Explain how sensitivity analysis focuses the effort, which index answers which question, and what the result would not establish.

Closed notes. Answer out loud, then reveal.