Uncertainty Quantification and Sensitivity Analysis Formula Sheet

This formula sheet collects uncertainty quantification and sensitivity analysis formulas used to turn model outputs, measurements and simulations into engineering decisions. Use it when a nominal result is not enough: the question is whether uncertainty can change the action.

The formulas are most useful when the decision boundary is explicit. State the output metric, acceptance limit, input evidence, correlation assumptions, validation basis and action if the uncertainty is larger than the margin.

Symbols and Conventions

Common symbols:

Symbol	Meaning
x_i	uncertain input
y	model output or decision metric
f(x)	model mapping inputs to output
\mu_x	mean or expected value of input x
\sigma_x	standard deviation of input x
u_x	standard uncertainty of input x
u_y	standard uncertainty of output y
\Sigma_x	input covariance matrix
\nabla f	gradient of the model output with respect to inputs
N	Monte Carlo sample count
P_f	probability of failure
g(x)	limit-state function; positive is usually safe
S_i	first-order variance-based sensitivity index
S_{T_i}	total-effect sensitivity index

Keep units attached to physical quantities. Sensitivity coefficients may have units; normalized sensitivities and variance fractions are dimensionless.

Deterministic Margin

For an upper-bound requirement:

the deterministic margin is:

For a lower-bound requirement:

the deterministic margin is:

If m is small compared with output uncertainty, the nominal pass is not stable.

Decision Guard

A simple guarded upper-bound rule is:

where k is a coverage or engineering guard factor. The margin after uncertainty is:

If m_g<0, the design should be held, redesigned, tested further or justified with stronger evidence.

Mean, Variance and Standard Uncertainty

For sampled input data:

and:

The standard deviation is:

The standard uncertainty of a mean estimate is:

Do not confuse spread of individual parts or operating cases with uncertainty in the estimated mean. They answer different engineering questions.

Standardization and Z-Score

For a normally distributed or approximately standardized quantity:

For a limit x_{\lim}:

A larger positive z_{\lim} means the limit is farther above the mean in standard-deviation units. This is useful for quick screening, but it depends on the distribution assumption.

Local Linear Propagation

For:

the first-order approximation is:

For independent inputs:

The contribution from input i is:

The fractional contribution is:

This identifies which input dominates the output uncertainty near the nominal point.

Matrix Covariance Propagation

Let:

With input covariance matrix \Sigma_x:

For multiple outputs y=f(x) with Jacobian J:

Use the matrix form when inputs are correlated or when several outputs share the same uncertain inputs.

Two-Input Correlation

For two inputs:

where:

Using correlation coefficient \rho_{xz}:

Positive correlation increases uncertainty when the two sensitivities have the same sign. It can reduce uncertainty when the sensitivities have opposite signs.

Relative Uncertainty for Products and Powers

For a product:

with independent uncertainties, the relative standard uncertainty is approximately:

For a general product:

the relative form is:

This is useful for first-pass sizing equations, power laws, heat-transfer correlations and material indices. It fails near zero or when inputs are strongly nonlinear over the uncertainty range.

Interval Propagation by Bounds

If only bounds are defensible:

then evaluate:

and:

For monotonic models, the extrema occur at interval endpoints. For non-monotonic or constrained models, endpoint checking may miss the true worst case.

Coverage Interval

For approximately normal output uncertainty:

Common first-pass values are:

and:

The coverage factor should match the consequence of the decision, the distribution shape and whether u_y is well supported by evidence.

Monte Carlo Propagation

For sampled inputs:

evaluate:

The sample mean is:

The sample standard deviation is:

Monte Carlo is appropriate when the model is nonlinear, threshold-based, discontinuous, correlated or too complex for closed-form propagation.

Monte Carlo Exceedance Probability

For an upper limit:

define an indicator:

The estimated exceedance probability is:

Approximate sampling standard error is:

Use this uncertainty when comparing a simulated failure probability with a release threshold.

Percentiles from Simulation

The pth percentile of simulated output is:

For a guarded upper-bound release:

or:

may be more meaningful than checking only the mean. The chosen percentile must be tied to the consequence and requirement.

Monte Carlo Convergence

For the sample mean:

For a target half-width h using multiplier z:

This formula estimates sample count for a mean. Tail percentiles and rare failures usually require many more samples or specialized methods.

One-at-a-Time Sensitivity

For a baseline input x_i and perturbation \Delta x_i:

The normalized sensitivity is:

This elasticity tells the percent change in output for a percent change in input near the baseline. It is local and can miss interactions.

Finite-Difference Sensitivity with Noise

Forward difference:

Central difference:

Choose \Delta x_i large enough to overcome numerical noise and small enough to represent the local response. In simulation models, sensitivity can be corrupted by solver tolerance, mesh changes, event switches or discontinuities.

Regression-Based Sensitivity

For a fitted linear surrogate:

a standardized coefficient is:

Large |\beta_i^*| indicates a strong linear effect in the sampled range. Regression coefficients are not reliable when inputs are strongly collinear or when the response is nonlinear without an adequate surrogate.

Variance-Based Sensitivity

For independent inputs, a first-order sensitivity index is:

It measures the fraction of output variance explained by input X_i alone.

The total-effect index is:

It includes direct effects and interactions involving X_i. If S_{T_i} is much larger than S_i, interactions are important.

Limit-State Reliability

Define a limit-state function:

and:

The probability of failure is:

From Monte Carlo:

where N_f is the number of failed samples.

For a normally distributed safety margin g:

and:

where \Phi is the standard normal cumulative distribution function.

Chance Constraint

A probabilistic requirement may be written:

Equivalently:

For an upper-bound output:

Chance constraints should not be used unless the input distributions and tail behavior are credible enough for the consequence.

Robust Objective

A simple risk-aware objective can penalize spread:

For a lower-is-better objective, this discourages designs that look good only at nominal inputs. For a performance metric that must be high, the conservative form may be:

The value of k should be a policy or requirement choice, not a tuning knob chosen after seeing the result.

Model Residual Metrics

For validation data y_i and model predictions \hat{y}_i:

Mean bias:

Root mean square error:

Normalized residual:

where u_i is the expected standard uncertainty for the comparison. Large structured residuals indicate model-form error, sensor bias, missing physics or operating-range drift.

Coverage of Prediction Intervals

If a model reports prediction interval [L_i,U_i], empirical coverage is:

For a nominal 95\% interval, measured coverage far below 95\% means the uncertainty model is overconfident or the validation data are outside the intended envelope.

Evidence-Weighted Decision Record

A compact uncertainty decision record should include:

Item	Formula or check
Nominal margin	m=y_{\lim}-y or m=y-y_{\min}
Combined uncertainty	u_y^2\approx J\Sigma_xJ^T
Guarded margin	m_g=y_{\lim}-(y+k u_y)
Failure probability	\hat{P}_f=N_f/N
Dominant input	largest F_i or S_{T_i}
Validation residual	RMSE, \bar{r}, normalized residuals
Action	release, hold, test, redesign, monitor or retire model

The action should follow from the decision metric, not from the fact that an analysis was performed.

Practical Checklist

Before accepting an uncertainty or sensitivity analysis, check:

1. The engineering decision and acceptance threshold are stated.
2. Input uncertainty evidence is named: measurement, supplier data, field data, test, judgement or standard.
3. Units, distributions, intervals and correlations are explicit.
4. Local propagation is used only where linearization is credible.
5. Monte Carlo convergence is checked on the decision metric, not only the mean.
6. Sensitivity ranks the quantity that controls the decision.
7. Tail probabilities and percentiles have enough evidence for the consequence.
8. Validation residuals are independent of calibration data when possible.
9. Model-form uncertainty is not hidden inside arbitrary input spread.
10. The decision record states what new evidence would change the conclusion.

Common Mistakes

• Reporting a nominal result without comparing uncertainty to the decision margin.
• Assuming independence when inputs share calibration, supplier lot, environment or operating mode.
• Adding probability distributions that have no evidence basis.
• Trusting Monte Carlo sample count without checking convergence of tails or failures.
• Running sensitivity analysis on a convenient output instead of the release metric.
• Treating RSS propagation as valid through discontinuities, saturations or threshold logic.
• Using a robust objective after tuning k to prefer a desired design.
• Validating against the same data used for calibration and calling it independent evidence.
• Reporting percentiles without stating distribution assumptions and sample uncertainty.
• Letting an early concept model become production evidence without a model-authority boundary.

The central habit is to make uncertainty decision-focused. The calculation should reveal whether the engineering action is stable, what input can change it, and what evidence would reduce risk most efficiently.