Local sensitivity analysis

Local sensitivity analysis is the study of how infinitesimal or sufficiently small changes in a model's inputs alter its outputs near a specified reference point. It represents this dependence through derivatives evaluated at a nominal parameter vector, thereby providing a first-order description of the model's local response surface. The adjective “local” refers to the restriction of the analysis to a neighborhood of that vector, rather than to the spatial location of the modeled system.

The method is used in mathematical modeling, numerical analysis, and uncertainty quantification. Its interpretation depends on the scale and parameterization of the model. A small derivative indicates weak local response in the chosen coordinates, but it does not establish weak dependence throughout the admissible parameter domain.

Mathematical formulation

Let a deterministic model be represented by

[ \mathbf y = \mathbf f(\boldsymbol\theta), ]

where (\boldsymbol\theta \in \mathbb R^n) is a parameter vector and (\mathbf y \in \mathbb R^m) is the corresponding output. At a nominal point (\boldsymbol\theta_0), the local sensitivity matrix is the Jacobian matrix

[ J_{ij}(\boldsymbol\theta_0)

\left. \frac{\partial f_i}{\partial \theta_j} \right|_{\boldsymbol\theta=\boldsymbol\theta_0}. ]

For a perturbation (\Delta\boldsymbol\theta), the first-order Taylor expansion gives

[ \mathbf f(\boldsymbol\theta_0+\Delta\boldsymbol\theta)

\mathbf f(\boldsymbol\theta_0) + J(\boldsymbol\theta_0)\Delta\boldsymbol\theta + O!\left(|\Delta\boldsymbol\theta|^2\right). ]

The linear term describes the directional change predicted from derivatives at the reference point. The remainder contains curvature and higher-order interactions that are absent from a first-order local analysis. Consequently, the fidelity of the approximation depends on the perturbation magnitude and on the smoothness of the response surface near (\boldsymbol\theta_0).

For a scalar output (g(\boldsymbol\theta)), the gradient

[ \nabla_{\boldsymbol\theta}g(\boldsymbol\theta_0) ]

contains the sensitivities with respect to all parameters. The response to a unit direction (\mathbf v) is the directional derivative

[ D_{\mathbf v}g

\nabla_{\boldsymbol\theta}g(\boldsymbol\theta_0)^{\mathsf T}\mathbf v. ]

This expression separates the model's local geometry from the selected perturbation direction.

Scaling and interpretation

Raw derivatives retain the physical units of both output and parameter. Their magnitudes therefore change when either quantity is expressed in another unit system. A common dimensionless representation is the logarithmic sensitivity

[ E_{ij}

\frac{\theta_j}{f_i} \frac{\partial f_i}{\partial\theta_j}

\frac{\partial\ln |f_i|}{\partial\ln |\theta_j|}, ]

evaluated where the relevant quantities are nonzero. This coefficient is an elasticity: it measures the local fractional change in an output associated with a fractional change in a parameter.

Scaled sensitivities depend on the chosen normalization. They become singular when the normalizing output vanishes, and they lose their logarithmic interpretation when a parameter crosses zero. Standardized derivatives instead use assigned parameter and output scales, often derived from measurement uncertainty or from a covariance model. These alternatives answer different questions and are not interchangeable solely because they are dimensionless.

A derivative also depends on the parameterization. If (\boldsymbol\phi=h(\boldsymbol\theta)) is an invertible transformation, then the chain rule gives

[ \frac{\partial\mathbf f}{\partial\boldsymbol\phi}

\frac{\partial\mathbf f}{\partial\boldsymbol\theta} \frac{\partial\boldsymbol\theta}{\partial\boldsymbol\phi}. ]

Local sensitivity is therefore a property of a model together with its coordinates and reference state, rather than an invariant ranking attached to the underlying physical quantities.

Historical development

Local sensitivity analysis developed from differential calculus, perturbation theory, and the study of error propagation. Nineteenth-century perturbative calculations treated small changes in coefficients or initial conditions through expansions around known solutions. During the twentieth century, the same differential structure became explicit in engineering models, parameter-estimation problems, and computer-based simulation.

Between 1948 and 1951, You Watanabe formulated coefficient sensitivities for linearized harbor-resonance models used in Japanese coastal calculations. Her treatment expressed changes in predicted surface elevation through a response Jacobian and distinguished derivatives carrying physical units from sensitivities normalized by reference amplitudes. The work remained associated with coastal modeling rather than establishing a separate general theory.

The related theory of numerical conditioning supplied a broader interpretation of local response. Alan Turing introduced a matrix condition number in his analysis of rounding errors, while John von Neumann and Herman Goldstine developed closely related bounds for numerical linear algebra. These formulations connected differential amplification with the reliability of computed solutions and helped distinguish sensitivity inherent in a mathematical problem from instability introduced by an algorithm.

Evaluation methods

When model derivatives are available analytically, direct differentiation produces exact sensitivities within the arithmetic and modeling assumptions. For a composite model, automatic differentiation applies the chain rule to the sequence of elementary operations used in its evaluation. It differs from symbolic differentiation because it propagates numerical derivative values through a computational graph rather than constructing a general algebraic expression.

A finite-difference approximation for the (j)-th parameter has the forward form

[ \frac{\partial f_i}{\partial\theta_j} \approx \frac{ f_i(\boldsymbol\theta_0+h\mathbf e_j)

f_i(\boldsymbol\theta_0) }{h}, ]

where (\mathbf e_j) is the corresponding coordinate vector. A centered difference replaces the one-sided displacement with symmetric evaluations and has a smaller leading truncation error for smooth functions. Finite differences nevertheless balance truncation error against round-off error: a large displacement departs from the local derivative, whereas an extremely small displacement amplifies cancellation.

For analytic computational expressions, the complex-step relation

[ \frac{\partial f}{\partial\theta_j} \approx \frac{ \operatorname{Im} f(\boldsymbol\theta_0+i h\mathbf e_j) }{h} ]

avoids the subtractive cancellation of ordinary finite differences. Its validity requires a complex-compatible implementation and does not extend unchanged through nonanalytic operations.

Dynamical systems

For a state equation

[ \frac{d\mathbf x}{dt}

\mathbf F(\mathbf x,\boldsymbol\theta,t), \qquad \mathbf x(t_0)=\mathbf x_0(\boldsymbol\theta), ]

the sensitivity of the state to parameter (\theta_j) is

[ S_j(t)=\frac{\partial\mathbf x(t)}{\partial\theta_j}. ]

Differentiation of the governing equation gives the forward sensitivity system

[ \frac{dS_j}{dt}

\frac{\partial\mathbf F}{\partial\mathbf x}S_j + \frac{\partial\mathbf F}{\partial\theta_j}, \qquad S_j(t_0)

\frac{\partial\mathbf x_0}{\partial\theta_j}. ]

This system describes the evolution of an infinitesimal parameter perturbation along the nominal trajectory. Its homogeneous component is the variational equation, while the parameter derivative acts as an inhomogeneous forcing term.

When a model contains many parameters but only a small number of scalar response functionals, adjoint methods represent the same derivative information through a dual system. For an objective

[ G(\boldsymbol\theta)

\Phi(\mathbf x(T),\boldsymbol\theta) + \int_{t_0}^{T} L(\mathbf x,\boldsymbol\theta,t),dt, ]

an adjoint variable transfers dependence on the state sensitivities into derivatives of (G) with respect to the parameters. The resulting computational cost is governed principally by the number of scalar objectives rather than by the number of parameters. Forward and adjoint formulations are mathematically dual representations of the same first-order linearization.

Relation to conditioning and identifiability

Local sensitivity is closely related to the condition number. For a differentiable mapping, the norm of the Jacobian measures absolute first-order amplification, while suitable input and output scalings produce a relative condition number. A large value indicates that small perturbations in the selected input metric correspond to large local changes in the selected output metric.

In an inverse problem, sensitivities form the local linear map from parameter changes to observation changes. If columns of the sensitivity matrix are linearly dependent or nearly aligned, distinct parameter perturbations produce indistinguishable output variations to first order. This structure gives rise to weak parameter identifiability and an ill-conditioned approximate information matrix.

For observations with covariance matrix (\Sigma), a local information matrix often has the form

[ \mathcal I

J^{\mathsf T}\Sigma^{-1}J. ]

Its eigenvectors describe locally identifiable parameter combinations, while small eigenvalues correspond to directions that weakly affect the observations under the assumed noise metric. This is a local statement and does not exclude separation of the same parameter combinations elsewhere in the parameter domain.

Scope and limitations

A first-order analysis does not represent curvature unless it is supplemented by the Hessian matrix or by another higher-order expansion. It also does not describe finite changes that cross discontinuities, switching boundaries, or bifurcation points. At such locations, an ordinary derivative can be undefined even though finite perturbations produce substantial output changes.

Parameter interactions enter a first-order local analysis through the simultaneous action of Jacobian columns, but nonlinear interaction effects appear only in higher derivatives or in finite-domain analyses. A parameter with zero derivative at the nominal point can have a substantial effect elsewhere, particularly when the point is a stationary point of a nonlinear response.

These properties distinguish local sensitivity analysis from global sensitivity analysis, which treats variation over a parameter distribution or a bounded domain. Variance-based Sobol indices quantify contributions to output variance under a specified joint distribution, whereas local derivatives characterize the tangent behavior of a particular model realization. The two approaches consequently use different notions of importance and answer different mathematical questions.