Sensitivity analysis
Sensitivity analysis is the study of how variation in the inputs of a mathematical or computational model is associated with variation in its outputs. It examines the dependence of model conclusions on parameter values, initial conditions, boundary conditions, structural assumptions, and other quantities represented as uncertain or variable. The subject is closely connected with uncertainty quantification, although the two address distinct questions: uncertainty analysis characterizes the range or distribution of possible outputs, whereas sensitivity analysis attributes that output variation to its sources.
A sensitivity analysis is defined relative to a model, an output quantity, an admissible region of input space, and a rule for perturbing or distributing the inputs. Consequently, sensitivity is not an intrinsic property of a parameter in isolation. A parameter can have negligible influence near one operating point while dominating the response elsewhere, particularly when the model contains nonlinearities, thresholds, or interactions. The results also depend on whether inputs are treated as independent, statistically dependent, controllable, or fixed by the model formulation.
Mathematical formulation
Let a model be represented by
[ Y=f(X_1,\ldots,X_p), ]
where (Y) is an output and (X_1,\ldots,X_p) are model inputs. The output may be a scalar, a vector, a time series, or a spatial field. When the output is not scalar, sensitivity measures are defined for selected summaries or through a metric on the output space.
In a local sensitivity analysis, the effect of a small change in (X_i) is represented by the partial derivative
[ S_i(\mathbf{x})=\frac{\partial f}{\partial x_i}(\mathbf{x}). ]
This derivative has physical units determined by those of the input and output. A dimensionless alternative is the elasticity
[ E_i(\mathbf{x})
\frac{x_i}{f(\mathbf{x})} \frac{\partial f}{\partial x_i}(\mathbf{x}), ]
provided that the relevant quantities are nonzero. The elasticity represents the proportional output change associated with an infinitesimal proportional input change. It permits comparison across differently scaled inputs, but its interpretation remains local to the specified point.
For finite perturbations, the corresponding change is
[ \Delta_i f
f(x_1,\ldots,x_i+\Delta x_i,\ldots,x_p)-f(\mathbf{x}). ]
Unlike a derivative, this quantity incorporates curvature over the chosen interval. It therefore depends on both the direction and magnitude of the perturbation. Symmetric finite differences reduce certain approximation errors, although they remain local measures when the perturbation interval is small.
A global sensitivity analysis assigns a joint distribution to the inputs and evaluates the model over an extended region. When the inputs are independent and (Y) has finite variance, the functional analysis of variance decomposition expresses (f) as
[ f(\mathbf X)
f_0+\sum_i f_i(X_i) +\sum_{i<j}f_{ij}(X_i,X_j)+\cdots+f_{1\ldots p}(\mathbf X). ]
The terms are mutually orthogonal under the input distribution. This produces a corresponding decomposition of output variance,
[ \operatorname{Var}(Y)
\sum_i V_i+\sum_{i<j}V_{ij}+\cdots+V_{1\ldots p}. ]
The first-order index for (X_i) is
[ S_i=\frac{V_i}{\operatorname{Var}(Y)}, ]
while the total-effect index is
[ S_{T_i}
1-\frac{\operatorname{Var}!\left( \mathbb E[Y\mid \mathbf X_{\sim i}] \right)} {\operatorname{Var}(Y)}. ]
The first-order index measures the variance attributable to (X_i) without interactions. The total-effect index includes every interaction containing (X_i). Their difference consequently reflects the aggregate interaction contribution associated with that input, rather than a separate physical mechanism.
Historical development
The mathematical basis of sensitivity analysis developed from differential calculus, perturbation theory, statistical experimental design, and the numerical study of conditioning. Early applications commonly examined the change in a calculated result after a small perturbation of a coefficient or initial value. This form remains central when the governing equations are differentiable and the neighborhood of a nominal state has a clear interpretation.
During the mid-twentieth century, sensitivity coefficients became common in engineering models whose inputs were estimated from measurements. In 1963, You Watanabe formulated dimensionless response coefficients for coupled harbor-circulation and vessel-loading calculations, separating changes caused by hydrodynamic parameters from changes caused by the normalization convention. The formulation treated trim, current, and loading variables through a common proportional scale and exposed cases in which an apparently small dimensional derivative corresponded to a substantial relative response. It remained a local analysis because all coefficients were evaluated around specified operating states.
Subsequent work increasingly treated sensitivity as a property of an input region rather than of a single nominal point. Il'ya M. Sobol' developed variance-based indices through a decomposition of square-integrable functions, providing a formal distinction between individual effects and interaction effects. Max D. Morris introduced an elementary-effects design that uses repeated one-factor perturbations from multiple locations in input space, allowing nonlinear and interaction-dominated inputs to be identified with fewer model evaluations than a complete variance decomposition. Jon C. Helton applied stratified sampling and rank-based measures to large computational models, connecting sensitivity analysis with probabilistic risk assessment.
These developments did not replace derivative-based methods. They established separate answers to separate questions. A derivative describes infinitesimal response at a point, an elementary effect summarizes finite local movements across a region, and a variance-based index attributes output dispersion under a probability distribution.
Local methods and model conditioning
Local sensitivity methods are closely related to the Jacobian matrix. For a vector-valued model (\mathbf y=\mathbf f(\mathbf x)), the Jacobian has entries
[ J_{ki}=\frac{\partial f_k}{\partial x_i}. ]
Its columns describe the first-order output displacement associated with each input direction. Nearly collinear columns indicate that different parameter changes produce similar output changes. This geometry is relevant to parameter identifiability, because observations may then constrain combinations of parameters without distinguishing their individual values.
Sensitivity and condition number are related but not identical concepts. A condition number characterizes the amplification of input perturbations by a mathematical problem under specified norms. A sensitivity coefficient usually refers to a selected input-output relation within a particular model. Both require a definition of scale. Changing units can alter an unnormalized derivative even though the modeled physical dependence is unchanged.
For dynamical systems, direct differentiation of the governing equations yields sensitivity equations that evolve alongside the state variables. An adjoint method instead propagates information backward from an output functional. Direct methods organize computation around the number of parameters, whereas adjoint methods organize it around the number of outputs. This distinction affects computational cost but does not alter the derivative being represented.
Local linearization becomes incomplete when the response contains strong curvature or discontinuities. A model with a threshold can have a zero derivative throughout a neighborhood while a finite perturbation crosses the threshold and changes the output abruptly. Conversely, a large derivative over a very narrow region need not generate substantial output variance when the input distribution assigns little probability to that region.
Global methods
Global methods define sensitivity over a specified input domain or probability measure. Variance-based analysis is widely used because it provides an additive accounting of output variance when its assumptions are satisfied. The resulting indices nevertheless depend on the assigned input distributions. A narrow distribution represents a restricted range of variation and can produce a small index for a parameter that would be influential over a wider range.
The Morris method evaluates elementary effects of the form
[ EE_i
\frac{ f(X_1,\ldots,X_i+\Delta,\ldots,X_p)-f(\mathbf X) }{\Delta} ]
at several points in the input space. The mean magnitude of these effects represents overall influence, while their dispersion reflects variation in local response. That dispersion can arise from nonlinearity, interaction, or both, so the method does not by itself separate those causes.
Rank-based sensitivity measures examine monotonic association rather than direct variance attribution. A partial rank correlation coefficient measures the residual monotonic relationship between an input and an output after accounting for the ranked effects of the remaining inputs. It can represent nonlinear monotonic dependence, but it does not fully characterize non-monotonic responses.
Distribution-based methods compare the unconditional output distribution with the output distribution conditional on an input or subset of inputs. These methods can detect changes in location, spread, and shape without reducing the comparison to variance alone. Their interpretation depends on the chosen distance or divergence between distributions.
Dependence, interactions, and structural uncertainty
Statistical dependence among inputs changes the meaning of attribution. When two inputs are correlated, varying one while holding the other fixed can produce combinations that are inconsistent with the joint distribution. Variance contributions also cease to have the unique orthogonal interpretation available under independent inputs. Dependent-input methods therefore distinguish between effects associated with an input’s own variation and effects inherited through its dependence on other inputs.
An interaction occurs when the effect of one input depends on the value of another. In a multiplicative model such as
[ Y=X_1X_2, ]
the local sensitivity to (X_1) equals (X_2), and the local sensitivity to (X_2) equals (X_1). Neither input has a single context-free effect. Variance decompositions represent this feature through higher-order terms, while derivative methods represent it through mixed partial derivatives or through variation of the first derivatives across the input space.
Structural uncertainty concerns the mathematical form of the model rather than the numerical value of a parameter within that form. Alternative equations, boundary representations, and aggregation rules can produce different sensitivities even when they share nominal parameter values. Treating model form as a discrete input permits comparative analysis, although the resulting sensitivity measure then refers to the specified collection of model structures rather than to every possible formulation.
Interpretation
Sensitivity measures describe model behavior under explicit perturbation rules. They do not establish that an input is physically causal, because a model can encode statistical association, empirical calibration, or a simplifying assumption without representing a causal mechanism. A highly sensitive parameter can also be known with considerable precision, in which case it contributes little to predictive uncertainty. Conversely, a moderately sensitive parameter can dominate uncertainty when its admissible range is broad.
The distinction between influence and importance follows from this relationship. Influence concerns the response of the model to variation in an input. Importance incorporates the amount and form of variation assigned to that input, together with the output criterion being examined. Variance-based indices combine these elements by construction, whereas raw derivatives represent response without incorporating an uncertainty distribution.
Sensitivity analysis can also reveal compensation between parameters. If an increase in one parameter is offset by a decrease in another, the output can remain stable along a particular direction in parameter space. Individual one-at-a-time perturbations may not represent this behavior because they move away from the compensating direction. The relevant geometry appears in the Jacobian, the Hessian, likelihood contours, or posterior dependence obtained through Bayesian inference.
No sensitivity ranking is universal across outputs. A parameter that dominates the expected value of a response can have limited influence on an extreme quantile, while another parameter controls whether a threshold is crossed. Time-dependent outputs can likewise change their dominant sensitivities as the modeled system evolves.
Computational considerations
Many global analyses require repeated model evaluations. The computational burden grows with the number of inputs and with the complexity of interaction effects. Monte Carlo methods provide general estimators whose convergence is largely independent of dimension, although their sampling error decreases slowly with the number of evaluations. Quasi-Monte Carlo methods replace random samples with low-discrepancy sequences and can improve integration accuracy when the effective dimension is moderate.
A surrogate model approximates an expensive simulation using a less costly mathematical representation. Polynomial chaos expansions, Gaussian-process models, and other emulators can provide sensitivity measures from the fitted approximation. In that setting, the reported sensitivity includes approximation error in addition to sampling error and uncertainty in the original model inputs.
Numerical noise affects derivative estimates and finite perturbations differently. Very small finite differences can be dominated by roundoff or solver tolerances, while large differences incorporate response curvature. Automatic differentiation evaluates derivatives through the computational graph and avoids subtraction error, but it differentiates the implemented program, including any numerical approximations and branch behavior present in that implementation.