Sensitivity & Uncertainty Analysis

Highly specialised requests are first checked for feasibility from this information. Please do not send confidential data through the public form; describe it first and agree secure transfer afterwards.

1. 01Enquiry and requirement assessment
2. 02Technical scoping and proposal
3. 03Execution
4. 04Quality review and delivery, with an agreed round of revisions

Read the full project workflow.

In depth: how the work is done

1. Characterising input uncertainty

   The analysis begins with describing the uncertainty of each input: ranges, probability distributions, correlations and the evidence behind them. Distributions might come from measurement statistics, manufacturer tolerances, literature values or expert elicitation, and the source is recorded. Correlated inputs must not be treated as independent. Inputs for which there is no basis for a distribution can be handled with ranges or bounding analysis, and the effect of these choices is itself examined.

2. Local and global sensitivity

   Local sensitivity, based on derivatives at a nominal point, is cheap and informative for small perturbations of smooth models. Global sensitivity explores the full input ranges and captures nonlinearity and interactions. Screening methods such as Morris elementary effects identify unimportant inputs economically, while variance-based methods apportion output variance to individual inputs and their interactions through first-order and total indices. The appropriate method depends on model cost, number of inputs and the question asked.

3. Sampling and propagation

   Monte Carlo sampling propagates input distributions through the model to estimate output statistics, with accuracy improving with the number of samples but independent of the dimension. Latin hypercube and quasi-random designs achieve better coverage with fewer runs. When the model is expensive, surrogate models trained on a limited set of runs allow thousands of cheap evaluations, provided the surrogate is validated. Convergence of statistics with sample size is checked and reported.

4. Using the results

   Results are used to rank inputs, to decide which parameters to measure more precisely or to fix, to construct credible intervals for predictions and to assess the probability of exceeding a limit. In calibration, sensitivity results show which parameters can be estimated from data. We present them in forms that engineers can act on, such as tornado charts and exceedance probabilities, and we clearly distinguish uncertainty in inputs from uncertainty in the model form, which this analysis does not capture.

5. Reporting uncertainty in publications

   Journals and reviewers increasingly expect uncertainty to be reported, not just a single curve. We help present results with intervals or ensembles, state the sources of uncertainty included and excluded, and describe the sampling and convergence checks in the methods. Figures should make clear what the shaded regions represent, whether input uncertainty, statistical uncertainty or both. Careful wording prevents readers from interpreting an interval as something it is not.

Common pitfalls this service helps avoid

• Assuming independent inputs. Correlation between inputs, such as strength and density, changes output uncertainty materially.
• Too few samples. Estimates of tails and small probabilities need many samples or specialised methods.
• Invented distributions. Distributions with no evidential basis create a false impression of rigour.
• Ignoring model-form uncertainty. Parameter uncertainty analysis does not account for errors in the model's structure.

What good reporting looks like

An uncertainty study states the model and outputs of interest, input uncertainties with their basis, the sensitivity and sampling methods, the number of samples and convergence checks, the ranking of inputs, output statistics or intervals, any surrogate and its validation, and the limitations, including model-form uncertainty. Results are presented in charts and tables suitable for publication, and the sampling plan and seeds are recorded for reproducibility.

An illustrative scenario

Hypothetical example written for explanation. It is not a client case, a result or a testimonial.

Consider a structural model with twelve uncertain inputs and a runtime of several minutes. A screening analysis identifies four influential inputs, and a surrogate trained on a space-filling sample of the model reproduces held-out runs closely. Variance-based indices show that one material parameter accounts for most of the output variance. The researcher therefore invests in measuring that parameter more precisely and presents the predicted displacement with a credible interval and the probability of exceeding a limit.

Questions to settle before you begin

• Which inputs are most uncertain, and what evidence do I have for their ranges?
• Are any inputs correlated?
• How expensive is one model run, and how many can I afford?
• What output statistic, interval or probability do I need?
• How will I check that the number of samples is sufficient?

Key terms

Monte Carlo method

Estimating statistics by repeated random sampling of inputs.

Latin hypercube sampling

A stratified sampling scheme that covers each input range evenly.

Sobol indices

Variance-based measures of each input's contribution to output variance.

Tornado chart

A bar chart ranking the effect of each input on an output.

Surrogate

A fast approximation of an expensive model.

Working together on Computational Engineering & Modelling projects

Computational engagements begin by agreeing the question, the quantities of interest and the accuracy that matters, so that modelling effort goes where it changes the answer. We ask for geometry or system descriptions, operating conditions, material or property data and any measurements for validation. Early on, a small pilot case often checks feasibility and cost. Models are verified and, where data allow, validated before parametric results are produced. Software, versions and settings are recorded so results can be reproduced. Where inputs are missing, assumptions are listed and agreed rather than quietly filled in.

Few projects need a single service in isolation. These are the services most often combined with this one.

• Computational Engineering

  Application of numerical and computational methods to engineering problems where closed-form solutions are not available, with emphasis on model formulation, verification and validation.

• Engineering Simulation

  Simulation-based analysis of engineering components and systems, from problem definition and model set-up through post-processing and interpretation, with a clear account of what the simulation can and cannot represent.

• Regression & Predictive Modelling

  Development of regression and predictive models for engineering responses, from interpretable linear models to more flexible approaches, with validation designed to detect overfitting and extrapolation risk.

• Mathematical Modelling

  Formulation of mathematical models for engineering and physical systems: choosing state variables and governing relations, simplifying sensibly, and checking behaviour against limiting cases and data.