Sensitivity Analysis · aiwiki.page

Sensitivity analysis is the study of how variations in a model’s inputs, parameters, or assumptions affect its outputs. In statistics and mathematical modelling, it helps identify influential factors, explain variation in results, and test whether conclusions depend strongly on particular choices. It is closely related to uncertainty analysis: uncertainty analysis characterizes the variability of outputs, whereas sensitivity analysis investigates the sources of that variability. (publications.jrc.ec.europa.eu)

Scope and purpose

A model can be represented by a function $Y=f(X_1,\ldots,X_d)$ , where the inputs may include measured quantities, uncertain parameters, or numerical representations of assumptions. Sensitivity analysis can examine a numerical prediction, a ranking of alternatives, or whether an output crosses a specified threshold. Its scope therefore depends on both the model and the question being investigated. (publications.jrc.ec.europa.eu)

Applications include engineering, physical sciences, economics, and model-based policy assessment. Typical purposes are to identify inputs requiring better information, screen out relatively unimportant factors, and understand model behaviour. An input’s importance is conditional on the chosen output, input ranges, and sensitivity measure; it is not an intrinsic property of the input alone. (publications.jrc.ec.europa.eu)

Local sensitivity analysis

Local analysis examines changes near a specified reference point $\mathbf{x}_0$ . For a differentiable scalar output, the sensitivity coefficient for input $i$ is the partial derivative

c_i=\left.\frac{\partial f}{\partial x_i}\right|_{\mathbf{x}_0}.

A first-order approximation gives

\Delta Y\approx\sum_i c_i\Delta x_i.

Thus, a coefficient describes the output change associated with a small input change, with other inputs held fixed. Its numerical magnitude depends on the units used. A large coefficient does not necessarily identify the largest uncertainty contribution, because the extent of input uncertainty also matters. (nist.gov)

In measurement uncertainty calculations, these coefficients combine with input variances and covariances:

u_Y^2\approx \sum_i c_i^2u_i^2+ 2\sum_{i<j}c_ic_j\,\operatorname{Cov}(X_i,X_j).

This propagation formula follows from a first-order Taylor expansion. It incorporates dependence between input estimates rather than assuming that all uncertainty contributions are independent. (nist.gov)

A common local approach varies one factor at a time around a baseline. Such an analysis can miss important regions of the input space and interactions, in which the effect of one input depends on another. Alternatives draw on experimental design, including factorial designs, and regression-based analysis. (publications.jrc.ec.europa.eu)

Global methods

Global analysis explores input variation across a specified domain, rather than only near one reference point. Inputs may be represented as random variables with assigned probability distributions. The analysis then examines their influence over the resulting range of model behaviour. (publications.jrc.ec.europa.eu)

Several approaches serve different purposes:

Regression and correlation methods. Standardized coefficients from linear regression and measures of correlation summarize input–output associations. Their usefulness depends on whether linear or monotonic relationships adequately describe the response.
Elementary-effects screening. The Morris method changes one input at each step along trajectories distributed across the input space. The average absolute elementary effect measures influence, while its standard deviation indicates nonlinearity or interactions, without distinguishing between them.
Variance-based methods. These allocate output variability to individual inputs and combinations of inputs.
Derivative-based global measures. These aggregate derivative information across the input domain and can support screening in high-dimensional models. (snl-dakota.github.io)

Global analysis can require many model evaluations. Statistical emulators offer computational savings, but their construction can itself become demanding as the number of inputs increases. Comparative studies show that screening performance depends on the model: no single sensitivity measure is uniformly best across different functions and dimensionalities. (publications.jrc.ec.europa.eu)

Sobol’ indices

Sobol’ indices are variance-based global sensitivity measures. For inputs satisfying statistical independence and a square-integrable output with positive variance, a functional decomposition separates main effects from interaction effects. The first-order index is

S_i= \frac{\operatorname{Var}\!\left(\mathbb{E}[Y\mid X_i]\right)} {\operatorname{Var}(Y)}.

Here the conditional expectation averages over the other inputs. The index measures the share of output variance attributable to input $i$ alone. (snl-dakota.github.io)

The total-effect index includes every interaction involving that input:

S_{T_i}= \frac{\mathbb{E}\!\left[\operatorname{Var}(Y\mid X_{-i})\right]} {\operatorname{Var}(Y)},

where $X_{-i}$ denotes all inputs except $X_i$ . Under the independent-input framework, $S_{T_i}\geq S_i$ ; the difference captures interaction contributions. These indices can be estimated using Monte Carlo methods. (snl-dakota.github.io)

As an illustrative calculation, let $Y=X_1+2X_2$ , with independent inputs each having variance one. Then $\operatorname{Var}(Y)=5$ , giving $S_1=1/5$ and $S_2=4/5$ . Because the function is additive, the corresponding total-effect indices equal the first-order indices.

Interpretation and limitations

Sensitivity results must be interpreted alongside the assumptions that generated them. With dependent inputs, changing one factor while holding others fixed may not reflect plausible input combinations. Dependence also complicates the separation of individual and shared variance contributions. Shapley effects provide one approach to allocating contributions arising jointly from correlation and interaction. (arxiv.org)

A sensitivity ranking describes behaviour within the specified model and input distribution; it does not establish causation in the real system. Reporting should identify the output studied, uncertain inputs, their ranges or distributions, dependence assumptions, computational design, and estimation accuracy. These details are necessary for interpreting the results and supporting reproducibility. (sciencedirect.com)