VVUQ · Module 8 of 10
Sensitivity Analysis
Uncertainty tells you how much the output could vary; sensitivity tells you which inputs are to blame. Ranking the inputs by influence shows where to spend effort reducing uncertainty, and where it would be wasted.
Readiness check
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 uncertainty propagation weights inputs by sensitivity.
The core idea
Local sensitivity measures how the output responds to a small change in one input, often normalised to a percent-for-percent elasticity. Global sensitivity apportions the output variance among the inputs, and Sobol indices give each input's share.
local: ∂y/∂x, normalised (∂y/∂x)(x/y)global: variance share of each inputSobol index Si = Vi/VSensitivity analysis identifies which inputs drive the output, complementing the uncertainty of the previous module. Local sensitivity is the partial derivative ∂y/∂xi at a nominal point: how much the output moves for a small change in one input, holding the others fixed. Normalising it, (∂y/∂xi)(xi/y), gives a dimensionless elasticity, the percent change in output per percent change in input, which is comparable across inputs of different units. Local measures are cheap but valid only near the nominal point and only for one input at a time. Global sensitivity analysis instead varies all inputs across their full ranges and apportions the total output variance among them. Variance-based Sobol indices make this precise: the first-order index Si = Vi/V is the fraction of output variance explained by input i alone, and what remains after summing the first-order indices is due to interactions between inputs. Ranking inputs by their indices shows where reducing uncertainty will actually shrink the output spread, the practical payoff of the analysis.
The skills, taught in order
Five skills build local and global sensitivity and the Sobol decomposition.
8.1 Local sensitivity coefficients
The local sensitivity of the output to an input is the partial derivative ∂y/∂xi evaluated at the nominal point. It measures the output response to a small perturbation of one input and is cheap to compute by finite differences or directly.
8.2 Normalized sensitivity
Dividing by the output and multiplying by the input, (∂y/∂xi)(xi/y), gives a dimensionless normalized sensitivity, the percent output change per percent input change. It lets inputs of different units be compared on one scale.
| Measure | Scope | Handles interactions? |
|---|---|---|
| Local derivative | near the nominal point | no |
| Normalized sensitivity | local, comparable | no |
| Sobol first-order | global, full range | separates them out |
Local measures are cheap but narrow; global Sobol indices cover the whole input range and reveal interactions.
8.3 Global sensitivity analysis
Global methods vary all inputs across their distributions 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.
8.4 Variance-based Sobol indices
Sobol analysis decomposes the total output variance into contributions from each input and their interactions. The first-order index Si = Vi/V is the share due to input i acting alone; the first-order indices sum to less than one when interactions are present.
8.5 Ranking and factor prioritisation
Ranking inputs by their sensitivity indices identifies the few that dominate the output uncertainty. Reducing the uncertainty of a high-index input shrinks the output spread; effort on a low-index input is largely wasted, the practical purpose of the analysis.
Engineering connection: sensitivity ranking tells you which one or two uncertain parameters to measure better, turning a broad UQ result into a focused test plan.
Worked example 1: normalized sensitivity of drag
Aerodynamic drag scales with the square of speed, D ∝ v2. Find the normalized sensitivity of drag to speed and interpret it.
- ProblemFind and interpret the normalized sensitivity of drag to speed in Figure 1.
- Given / findD = c·v2. Find the normalized sensitivity S = (∂D/∂v)(v/D).
- AssumptionsDrag follows the quadratic law over the range of interest.
- ModelNormalized sensitivity S = (∂D/∂v)(v/D); for a power law D ∝ vn, S = n.
- Equations∂D/∂v = 2c·vS = (2c·v)(v/(c·v2)) = 2
- SolveS = (2c·v)(v)/(c·v2) = 2. A 1% change in speed produces about a 2% change in drag.
- CheckFor any power law y ∝ xn the normalized sensitivity equals the exponent n, so the quadratic drag law gives exactly 2, independent of the constant or the speed.
- ConclusionDrag is twice as sensitive to speed in relative terms, so speed uncertainty matters more than a linearly acting input. The normalized sensitivity makes that comparison immediate.
Worked example 2: variance-based Sobol indices
A global sensitivity study finds a total output variance of 100, with input x1 contributing V1 = 64 and input x2 contributing V2 = 25 (first-order effects). Find the first-order Sobol indices and the interaction share.
- ProblemFind the Sobol indices and interaction share for the study in Figure 2.
- Given / findTotal variance V = 100, V1 = 64, V2 = 25. Find S1, S2, and the interaction share.
- AssumptionsV1 and V2 are first-order effects; the remainder is interaction.
- ModelSi = Vi/V; interaction share = 1 − ΣSi.
- EquationsS1 = V1/V, S2 = V2/Vinteraction = 1 − (S1 + S2)
- SolveS1 = 64/100 = 0.64, S2 = 25/100 = 0.25. Interaction = 1 − (0.64 + 0.25) = 0.11.
- CheckThe shares 0.64 + 0.25 + 0.11 sum to 1, accounting for all the variance; the nonzero interaction (0.11) shows the inputs are not purely additive.
- ConclusionInput x1 explains 64% of the output variance and x2 a quarter, with 11% from their interaction. Reducing the uncertainty of x1 would shrink the output spread the most.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Local derivative over a wide range | Ranking wrong for large variations | "Is the input range small?" | Use global sensitivity for wide ranges. |
| Comparing raw derivatives | Different units confuse the ranking | "Did I normalise?" | Use normalized sensitivity to compare inputs. |
| Ignoring interactions | First-order indices do not sum to 1 | "Where did the rest of the variance go?" | The remainder is interaction, not error. |
| Reducing a low-index input | Effort with little payoff | "Does this input dominate the variance?" | Target the high-index inputs. |
Practice ladder
For a power law y ∝ x3, what is the normalized sensitivity to x?
Show answer
The normalized sensitivity equals the exponent, so it is 3: 1% in x gives about 3% in y.
An output variance of 200 has V1 = 150 and V2 = 40. Find the first-order Sobol indices.
Show answer
S1 = 150/200 = 0.75, S2 = 40/200 = 0.20; interaction = 1 − 0.95 = 0.05.
For y = x1·x2 at x1 = 4, x2 = 5, find the normalized sensitivity to x1.
Show answer
∂y/∂x1 = x2 = 5; normalized = 5 × (4/20) = 1. A product is linear in each factor, so the normalized sensitivity is 1.
A model has ten uncertain inputs but limited test budget. Explain how sensitivity analysis focuses the effort.
What good work looks like
Run a global sensitivity analysis, rank the inputs by Sobol index, and spend the test budget measuring the two or three that dominate the output variance, since reducing their uncertainty shrinks the output spread while the low-index inputs can be left at nominal.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
For a real model, compute normalized local sensitivities or Sobol indices, rank the inputs, and identify the one whose better measurement would most reduce the output uncertainty.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What is a local sensitivity coefficient?
The partial derivative ∂y/∂xi at the nominal point.
2. What is normalized sensitivity?
(∂y/∂xi)(xi/y), the percent output change per percent input change.
3. What does a Sobol first-order index give?
The fraction of output variance explained by that input alone, Si = Vi/V.
4. What does the remainder after first-order indices represent?
Interactions between inputs.
5. Why rank inputs by sensitivity?
To reduce the uncertainty of the inputs that dominate the output spread.
Standards mapping
This module follows the ASME Verification, Validation, and Uncertainty Quantification standards. Use these references to read further.
| Topic in this module | Where to read more |
|---|---|
| Sensitivity in uncertainty quantification | ASME VVUQ 10.2, UQ in Solid Mechanics |
| Local and global sensitivity | ASME V&V 20, CFD and Heat Transfer |
| Variance-based Sobol indices | Saltelli et al., Global Sensitivity Analysis |
Standard designations refer to the ASME V&V series; variance-based sensitivity methods are standard across the UQ literature.