Sensitivity and Uncertainty Quantification

The Central Idea

Suppose a model has inputs

and output

Sensitivity analysis asks how the output changes when one or more components of \(\mathbf{x}\) change. Uncertainty quantification asks what values of \(y\) become plausible when the uncertain inputs vary according to specified ranges or probability models.

In compact form:

Why We Need Both

A wide output distribution tells us that the model result is uncertain, but it does not tell us which input is responsible. A large sensitivity measure tells us that an input can strongly affect the output, but it does not tell us how uncertain that input actually is.

An input may be highly influential but known very precisely. Another may be only moderately influential but poorly known. The second input can contribute more to overall output uncertainty.

For this reason, sensitivity and uncertainty should be interpreted together.

Local Sensitivity

A local sensitivity measures how the output changes near a particular operating point. For a differentiable model, the most direct local sensitivity of \(y\) to input \(x_i\) is

This derivative measures the instantaneous rate of change of the model output with respect to that input while the other inputs are held fixed.

Example: Local Sensitivity of Revenue

Suppose revenue is modeled by

where \(p\) is price and \(q\) is quantity sold.

The local sensitivities are

and

At \(p=40\) and \(q=500\),

Near that operating point, a one-unit increase in price changes revenue by about \(500\) units, while a one-unit increase in quantity changes revenue by about \(40\) units.

Raw Sensitivities Depend on Units

Raw derivatives can be difficult to compare because different inputs may use different units and scales. A derivative measured in dollars per degree is not directly comparable with one measured in dollars per kilogram.

This motivates normalized or dimensionless sensitivity measures.

Normalized Sensitivity

One common normalized sensitivity coefficient is

This quantity measures the approximate percentage change in output associated with a one-percent change in the input.

It is often called an elasticity when used in economics, business, and related models.

Example: Elasticity in a Power Model

Consider

Differentiate with respect to \(x\):

The normalized sensitivity is

The exponent is therefore the elasticity of the output with respect to the input.

Finite-Difference Sensitivity

If a derivative is unavailable or inconvenient, sensitivity can be estimated numerically. A forward finite difference is

A centered finite difference often provides a better local approximation:

The Step Size Matters

In finite-difference calculations, the perturbation \(h\) should be small enough to represent local behavior but not so small that numerical roundoff or solver noise dominates the calculation.

A useful practice is to compute the sensitivity using several nearby step sizes and check whether the estimate remains reasonably stable.

One-at-a-Time Sensitivity

In one-at-a-time analysis, one input is varied while all others are held fixed at baseline values.

If the baseline is \(\mathbf{x}^{(0)}\), then the sensitivity of input \(x_i\) might be explored by evaluating

over a selected range of \(x_i\).

This method is simple and interpretable, but it can miss interactions between inputs.

Tornado-Style Comparisons

A practical one-at-a-time method is to move each input from a low value to a high value while holding all other inputs at baseline levels. The resulting output range is recorded for every input.

Inputs can then be ranked by the width of the resulting output change. This produces the logic behind a tornado chart even when no chart is drawn.

The method is useful for screening, but the ranking depends on the chosen low and high input values.

Limits of Local Sensitivity

Local methods describe behavior near one point. They can be misleading when the model is strongly nonlinear, contains thresholds, changes regime, or has important interactions among inputs.

In such cases, sensitivity should be examined over the full uncertainty region rather than only near a single baseline.

Global Sensitivity Analysis

Global sensitivity analysis studies model behavior over the full range or probability distribution assigned to uncertain inputs.

Instead of asking only

What happens near the baseline?

it asks

Across all plausible input combinations, which inputs and interactions account for the variability in the output?

Forward Uncertainty Quantification

In forward uncertainty quantification, uncertainty in the inputs is specified first and propagated through the model.

If

and

then the goal is to characterize the induced distribution or range of \(Y\).

Typical output summaries include the mean, variance, quantiles, confidence or credible ranges, exceedance probabilities, and probabilities of failure.

Input Uncertainty Must Be Characterized First

Before propagating uncertainty, each uncertain input must be described in a way that matches the available information. An input may be represented by a fixed interval, an empirical distribution, a parametric probability distribution, or a set of competing assumptions.

The uncertainty model is part of the overall mathematical model. Poorly justified input distributions can create a false impression of rigor.

Aleatory and Epistemic Uncertainty

Two broad forms of uncertainty are often distinguished.

• Aleatory uncertainty represents inherent variability or randomness.
• Epistemic uncertainty represents limited knowledge about parameters, mechanisms, assumptions, or model structure.

The distinction matters because the management response may differ. More data may reduce epistemic uncertainty, while inherent variability may remain even with perfect measurement.

First-Order Uncertainty Propagation

For a smooth model, uncertainty near a baseline can be approximated using a first-order Taylor expansion:

If the input covariance matrix is \(\Sigma\), then the output variance is approximately

This connects derivative-based sensitivity directly to uncertainty propagation.

Independent Inputs

If the uncertain inputs are independent, the covariance matrix is diagonal. The first-order variance approximation becomes

Each term combines two pieces of information: how strongly the model responds to an input and how uncertain that input is.

Example: Sensitivity Is Not the Same as Uncertainty Contribution

Suppose

with independent inputs satisfying

The raw sensitivities are \(4\) and \(1\), so \(X_1\) is locally more influential per unit change. But the variance contributions are

and

Therefore, \(X_2\) contributes more to output variance because it is much more uncertain.

Dependence Between Inputs

If uncertain inputs are correlated, changing or sampling them independently can misrepresent the model. For two inputs, the first-order variance formula includes a covariance term:

where

Positive dependence can increase uncertainty in some models, while negative dependence can reduce it.

Monte Carlo Uncertainty Quantification

For nonlinear models or complicated input distributions, Monte Carlo simulation provides a flexible propagation method.

1. Specify uncertainty models for the inputs.
2. Generate a joint sample \(\mathbf{x}^{(1)},\ldots,\mathbf{x}^{(N)}\).
3. Evaluate \(y^{(j)}=f(\mathbf{x}^{(j)})\) for every sample.
4. Summarize the empirical distribution of the outputs.

The resulting sample can estimate means, variances, quantiles, exceedance probabilities, and reliability measures.

Quantiles Often Communicate Uncertainty Better Than a Single Standard Deviation

If an output distribution is skewed or heavy-tailed, a mean and standard deviation may not summarize it well. Quantiles can be more informative.

For example, a central 90% simulation interval can be estimated using the empirical 5th and 95th percentiles:

The interpretation should remain tied to the assumptions and probability model used to generate the simulation.

Exceedance Probabilities

A model may be used to estimate the probability that an uncertain output exceeds a threshold \(c\):

From Monte Carlo output \(y^{(1)},\ldots,y^{(N)}\), the natural estimate is

This directly connects uncertainty quantification to risk and reliability analysis.

Variance Decomposition

One major goal of global sensitivity analysis is to decompose output variance into contributions from individual inputs and their interactions.

Conceptually,

where \(V_i\) represents the contribution associated with input \(X_i\), and terms such as \(V_{12}\) represent interactions between inputs.

First-Order Sobol Sensitivity

For independent uncertain inputs, a first-order Sobol index for \(X_i\) is

It measures the fraction of output variance explained by varying \(X_i\) alone, averaged over the uncertainty in the other inputs.

Values near zero indicate little first-order influence. Larger values indicate a stronger direct contribution to output variance.

Total-Order Sobol Sensitivity

An input can matter through interactions even if its direct contribution is modest. The total-order Sobol index includes the main effect of \(X_i\) and every interaction involving \(X_i\).

One equivalent expression is

where \(X_{-i}\) denotes all inputs except \(X_i\).

Interactions Matter

Suppose

The effect of changing \(X_1\) depends on the value of \(X_2\). Likewise, the effect of changing \(X_2\) depends on \(X_1\).

This is an interaction. One-at-a-time methods can miss or understate such behavior because they hold the other inputs fixed.

Global methods are designed to capture these joint effects more systematically.

Screening Large Models

Some models contain dozens, hundreds, or thousands of uncertain inputs. A complete variance decomposition may be computationally expensive.

Screening methods identify inputs that are likely to be important and inputs that appear negligible. The modeler can then concentrate more expensive analysis on the smaller set of influential inputs.

Morris-Style Elementary Effects

One useful screening idea is to perturb an input at many locations in the uncertainty region and record the resulting elementary effect

Large average absolute effects suggest an influential input. Large variation among the elementary effects can indicate nonlinearity or interaction with other variables.

The purpose is usually screening rather than precise variance attribution.

Correlation-Based Sensitivity Measures

Correlation between an input and model output can provide a quick indication of monotonic influence. For linear relationships, Pearson correlation may be useful. For monotonic but nonlinear relationships, rank correlation can be more robust.

These measures are easy to compute from simulation samples, but they do not generally provide the same interpretation as variance-based global sensitivity indices.

Correlation Can Miss Nonmonotonic Influence

Consider

with \(X\) symmetrically distributed around zero. Positive and negative values of \(X\) can produce the same value of \(Y\), so the ordinary correlation between \(X\) and \(Y\) may be near zero even though \(X\) completely determines \(Y\).

A small correlation therefore does not automatically imply a small global influence.

Derivative-Based Global Measures

Local derivatives can be extended into global summaries by averaging their magnitude over the uncertain input space. One example is

This measure is large when the model is frequently steep in the direction of \(x_i\). It can be useful when derivatives are inexpensive to compute.

Surrogate Models

If a full model is expensive to evaluate, uncertainty analysis may require a simpler approximation called a surrogate, metamodel, or response surface.

A surrogate is trained on selected evaluations of the original model and then used for large simulation or sensitivity studies.

Common surrogate forms include regression models, polynomial response surfaces, Gaussian-process models, neural approximations, and polynomial chaos expansions.

A Surrogate Adds Another Source of Error

The uncertainty analysis is only as reliable as the surrogate in the region where it is used. A fast approximation that is inaccurate in the tails can badly distort failure probabilities, extreme quantiles, or sensitivity rankings.

Surrogate validation should therefore be treated as part of the uncertainty analysis rather than as a separate optional step.

Uncertainty Budgets

An uncertainty budget organizes the major sources of output uncertainty and estimates how much each contributes.

For a nearly linear model with independent inputs, a contribution can be approximated by

A normalized contribution is then

The values \(w_i\) approximately partition the modeled output variance.

Example: A Simple Uncertainty Budget

Suppose a first-order calculation gives variance contributions

The total is

Therefore,

and

Under the approximation, the first source is the largest contributor to output variance.

Which Uncertainty Should We Reduce?

A sensitivity study becomes operationally useful when it helps determine where better data, measurement, calibration, or model development would have the greatest value.

An input with both high uncertainty and high sensitivity is a natural target for improvement. Reducing uncertainty in an input that barely affects the model may produce little practical benefit.

Sensitivity and the Value of Information

Suppose a decision depends on an uncertain parameter. If better information about that parameter would rarely change the decision, collecting more data may have little operational value. If reducing uncertainty could frequently change the preferred action, the information may be much more valuable.

Sensitivity analysis therefore supports not only model diagnosis but also prioritization of data collection and research.

Sensitivity Is Not Identifiability

A parameter can strongly influence model output and still be difficult to estimate from available data. Conversely, a parameter may be precisely estimated but have little influence on the particular output or decision of interest.

Sensitivity concerns how output responds to a parameter. Identifiability concerns whether the parameter can be reliably inferred from observations.

Structural Uncertainty

Not all uncertainty can be represented by uncertain numerical parameters. Competing equations, boundary conditions, mechanisms, functional forms, or policy assumptions may define different model structures.

Structural uncertainty can be studied by comparing alternative model forms, scenario families, or model ensembles rather than by varying only the parameters of one fixed equation.

Model Ensembles

An ensemble evaluates multiple plausible models or multiple parameterized versions of a model. The spread of ensemble outputs can reveal sensitivity to structural choices that would be invisible inside a single specification.

Ensemble spread should not automatically be interpreted as a formal probability distribution unless the weighting and probabilistic assumptions justify that interpretation.

Sensitivity Rankings Can Change

There is no universal sensitivity ranking independent of modeling choices. Rankings can change when the operating point, uncertainty ranges, input distributions, dependence assumptions, output metric, or time horizon changes.

A good analysis therefore reports the conditions under which each ranking was obtained rather than presenting the ranking as an intrinsic property of the model.

Models Often Have Multiple Outputs

An input that strongly influences cost may have little effect on reliability. Another may dominate emissions but barely affect profit. Sensitivity should therefore be evaluated relative to the output or decision metric that matters.

For a vector output

different components of \(\mathbf{y}\) can have different sensitivity structures.

Time-Dependent Sensitivity

In dynamic models, an input may matter strongly during one period and weakly during another. Sensitivity can therefore be treated as a function of time:

This is especially useful in differential-equation, reliability, ecological, financial, and control models.

Decision Sensitivity Can Differ from Output Sensitivity

Suppose a decision depends only on whether \(Y\) exceeds a threshold \(c\). Then the most important uncertainty may be the uncertainty that changes

not necessarily the uncertainty that contributes most to \(\operatorname{Var}(Y)\).

Sensitivity analysis should therefore be aligned with the actual decision criterion whenever possible.

Uncertainty Estimates Have Sampling Error

Monte Carlo estimates of means, quantiles, failure probabilities, and global sensitivity indices are themselves uncertain because they are estimated from finite samples.

Increasing the simulation size generally improves precision, but convergence rates differ among statistics. Tail probabilities and high-order sensitivity measures may require much larger samples than the mean.

Assessing Stability

Repeated simulation runs, bootstrap resampling, confidence intervals for sensitivity indices, and convergence plots can help determine whether the reported uncertainty and ranking estimates are stable enough for the intended use.

Reporting a sensitivity coefficient to many decimal places is not meaningful when its simulation error is large.

A Practical Sensitivity and UQ Workflow

1. Define the model output or decision metric of interest.
2. Identify uncertain inputs, parameters, and structural assumptions.
3. Specify ranges, distributions, and dependence relationships.
4. Run a preliminary screening analysis if the input set is large.
5. Propagate uncertainty through the model.
6. Summarize the output distribution or uncertainty range.
7. Estimate local or global sensitivity measures.
8. Check interactions, nonlinearities, and threshold effects.
9. Assess numerical and sampling convergence.
10. Use the results to prioritize data, model refinement, and decisions.

Communicating Sensitivity Results

A useful report should state what was varied, over what range or distribution, what was held fixed, which dependence assumptions were used, what output was measured, and what sensitivity definition was applied.

Statements such as “input \(X_1\) is the most sensitive parameter” are incomplete unless the method and uncertainty domain are clear.

Communicating Uncertainty Quantification

Uncertainty should be reported with an interpretation that matches the method. A simulated 90% interval, a confidence interval, a Bayesian credible interval, and a scenario range are not interchangeable even if they happen to have similar endpoints.

The assumptions behind the uncertainty statement are part of the result.

Common Errors

• Varying inputs over arbitrary ranges without explaining why those ranges are plausible.
• Comparing raw derivatives across inputs with very different units and scales.
• Calling a local derivative a complete global sensitivity analysis.
• Ignoring correlations or dependence among uncertain inputs.
• Using one-at-a-time methods when interactions dominate the model.
• Interpreting an uncertain-input simulation as objective reality rather than as a conditional result based on assumptions.
• Reporting a sensitivity ranking without specifying the output metric.
• Ignoring simulation error in estimated sensitivity indices or tail probabilities.
• Assuming a surrogate is accurate without validating it in the region of interest.
• Confusing parameter uncertainty with structural uncertainty.

Practice 1: Local Derivatives

Let

Find \(\partial y/\partial x_1\) and \(\partial y/\partial x_2\), then evaluate them at \(x_1=4\).

Practice 2: Normalized Sensitivity

Suppose

Compute the normalized sensitivity

Practice 3: Centered Finite Difference

A model gives

Estimate \(f'(10)\) using a centered finite difference.

Practice 4: First-Order Variance

Let

where \(X_1\) and \(X_2\) are independent with

Find \(\operatorname{Var}(Y)\).

Practice 5: Uncertainty Contributions

Using Practice 4, determine the variance contribution from each input and the percentage of total variance attributable to each one.

Practice 6: Correlated Inputs

Suppose

with

Find \(\operatorname{Var}(Y)\).

Practice 7: Monte Carlo Exceedance Probability

In a simulation of \(5000\) model runs, \(375\) outputs exceed a critical threshold. Estimate the exceedance probability.

Practice 8: Sobol Interpretation

A model has first-order Sobol index

and total-order index

What does the difference suggest?

Practice 9: Sensitivity Versus Uncertainty

Input \(A\) has sensitivity magnitude \(10\) and standard deviation \(0.1\). Input \(B\) has sensitivity magnitude \(2\) and standard deviation \(2\). Assuming independence and first-order propagation, which input contributes more to output variance?

Practice 10: Method Choice

A model contains 80 uncertain inputs, is strongly nonlinear, and requires several minutes per run. You want to identify a small group of influential inputs before performing an expensive global analysis.

What general strategy is appropriate?

Practice Answers

Practice 1

At \(x_1=4\),

Practice 2

Since

we have

Practice 3

Here \(h=0.1\), so

Practice 4

Because the inputs are independent,

Practice 5

The contribution from \(X_1\) is

and the contribution from \(X_2\) is

Therefore,

Practice 6

Practice 7

The estimated exceedance probability is \(7.5\%\).

Practice 8

The first-order effect of \(X_1\) explains about \(18\%\) of the modeled output variance, while its total contribution including interactions is about \(47\%\). The large gap suggests that \(X_1\) participates in important interactions with other inputs.

Practice 9

Under first-order propagation, compare

with

Therefore, \(\boxed{B}\) contributes much more to output variance even though its raw sensitivity is smaller.

Practice 10

Begin with a computationally efficient screening method, such as one-at-a-time range screening, Morris-style elementary effects, or another low-cost design. Use the screening results to reduce the dimensionality, then perform the more expensive global uncertainty and sensitivity analysis on the influential subset. If the original model remains too expensive, construct and validate a surrogate model before the large global calculation.

What to Remember

• Sensitivity analysis measures how model outputs respond to changes in inputs.
• Uncertainty quantification characterizes the uncertainty in model outputs created by uncertain inputs and assumptions.
• Local derivatives describe behavior near a chosen operating point.
• Normalized sensitivities allow more meaningful comparison across inputs with different scales.
• Finite differences estimate local sensitivities when analytic derivatives are unavailable.
• One-at-a-time methods are useful for simple screening but can miss interactions.
• Global sensitivity methods examine the full uncertainty region rather than only one baseline.
• Input uncertainty and model sensitivity must be considered together when assessing output uncertainty.
• Dependence among inputs can materially change uncertainty propagation and sensitivity conclusions.
• First-order propagation links the gradient of the model to the covariance of the inputs.
• Monte Carlo simulation provides a flexible method for forward uncertainty propagation.
• Variance-based methods can separate direct input effects from interaction effects.
• First-order Sobol indices measure direct variance contributions; total-order indices include interactions.
• Screening methods help reduce large input sets before expensive global analysis.
• Correlation-based methods can be useful but may miss strong nonmonotonic relationships.
• Surrogate models can reduce computational cost but introduce approximation error that must be validated.
• An uncertainty budget identifies the dominant sources of modeled output uncertainty.
• Sensitivity rankings depend on the chosen output, operating region, uncertainty assumptions, and method.
• Structural uncertainty requires comparing alternative model forms, not merely varying parameters inside one fixed structure.
• Simulation-based uncertainty and sensitivity estimates have their own numerical and sampling uncertainty.

The essential principle is: do not report only how uncertain the model output is; determine which assumptions create that uncertainty, how strongly they influence the result, and whether reducing them would improve the decision.

Where We Go Next

Sensitivity and uncertainty quantification show how uncertain assumptions propagate through a model and which parameters contribute most to the result. The next step is to treat uncertain parameters as quantities about which we can update our beliefs when new data arrive.

The next lesson, Bayesian Parameter Treatment, develops a probability-based framework for representing prior parameter uncertainty, combining it with observed evidence, and producing posterior parameter and predictive distributions.