Sensitivity analysis
Introduction
Definition and Purpose
Sensitivity analysis (SA) is a methodological framework used to quantify how variations or uncertainties in the input parameters of a computational model influence the variability in its output.[7] It systematically evaluates the relative importance of different inputs by apportioning the uncertainty in the model output to sources of uncertainty in the inputs, thereby revealing the robustness and reliability of the model under varying conditions.[8] The primary purposes of SA include identifying the most influential parameters that drive output changes, which aids in prioritizing research efforts and resource allocation; reducing model complexity by fixing non-influential inputs; supporting risk assessment by highlighting potential vulnerabilities in system predictions; and facilitating model validation through verification of key assumptions and behaviors.[3] These applications make SA essential in fields such as engineering, environmental science, and policy analysis, where models inform critical decisions.[9] For example, consider a simple linear model $ Y = aX + b $, where varying the input $ X $ while holding $ a $ and $ b $ constant demonstrates direct proportionality: the output $ Y $ changes linearly with $ X $, with the sensitivity coefficient $ a $ indicating the magnitude of influence.[7] This illustrates how SA can straightforwardly assess input-output relationships in basic systems. SA is distinct from uncertainty analysis (UA), which focuses on propagating input uncertainties to characterize the overall distribution of output uncertainty, whereas SA specifically examines the contributions of individual or groups of inputs to that output variability.[8]Historical Development
The origins of sensitivity analysis can be traced to the mid-20th century, with foundational work in statistics and engineering through factorial designs and response surface methodology introduced in the 1950s by George E. P. Box and others, which allowed for systematic examination of how variations in inputs affect model outputs.[10] By the 1960s, the concept began appearing in decision theory contexts, where it supported robust decision-making under uncertainty, including early explorations of expected value of perfect information in finite decision problems.[11] In the 1970s, engineering applications advanced significantly, particularly in chemistry and nonlinear modeling, with the development of the Fourier Amplitude Sensitivity Test (FAST) by Robert I. Cukier and colleagues, enabling the identification of influential parameters through spectral analysis of model responses. Concurrently, C.S. Holling and C.J. Walters pioneered global perspectives in 1978 by advocating simultaneous variation of multiple parameters to capture interactions, a departure from isolated perturbations.[10] The 1980s marked a pivotal expansion with the maturation of local sensitivity methods, driven by rising computational capabilities that facilitated derivative-based techniques and random sampling for exploring model behavior around nominal points.[10] This era emphasized one-at-a-time analyses and regression approaches in engineering and systems modeling, as reviewed in subsequent works like Helton et al. (2006).[10] Influential contributions included Bradley Efron and Charles Stein's 1981 advancements in variance decomposition, providing theoretical underpinnings for partitioning output uncertainty.[10] The 1990s ushered in the rise of global variance-based techniques, transforming sensitivity analysis into a tool for comprehensive uncertainty attribution. Ilya M. Sobol' introduced his seminal indices in 1993, formalizing the decomposition of model output variance into contributions from individual inputs and their interactions, applicable to nonlinear and high-dimensional systems. Andrey Saltelli emerged as a key figure in the late 1990s, developing efficient Monte Carlo-based estimators and ANOVA-like decompositions that made global methods computationally feasible, as detailed in his collaborative works around 2000.[10] In the 2000s, sensitivity analysis integrated closely with uncertainty quantification frameworks, particularly in environmental science and risk assessment, where Saltelli's 2004 handbook standardized variance-based practices and highlighted their role in model validation. This period saw widespread adoption in fields like climate modeling, emphasizing robust propagation of input uncertainties.[10] By the 2010s, the field advanced with moment-independent measures and open-source software, broadening accessibility.[10] Entering the 2020s, evolution accelerated toward AI-enhanced sensitivity analysis for high-dimensional data, leveraging machine learning to optimize sampling and interpret complex interactions, as evidenced in 2024 reviews of AI-driven methodologies that improve efficiency in black-box models.[12][10] In 2025, the field continued to progress, highlighted by the Eleventh International Conference on Sensitivity Analysis of Model Output (SAMO 2025) held in April in Grenoble, France, focusing on advancements from theory to applications in uncertainty quantification.[13]Theoretical Foundations
Mathematical Formulation
Sensitivity analysis is grounded in the study of input-output relationships within mathematical models. Consider a general model where the output $ Y $ is a function of a vector of input factors $ \mathbf{X} = (X_1, X_2, \dots, X_k) $, expressed as $ Y = f(\mathbf{X}) $. Here, $ \mathbf{X} $ represents the inputs, which may be deterministic or stochastic, and $ f $ denotes either an analytical function or a black-box model evaluated computationally.[14][15] For local sensitivity analysis, the impact of individual inputs is assessed near a nominal point $ \mathbf{x}^* $. The local sensitivity of the output with respect to the $ i $-th input is given by the partial derivative:
This measure quantifies the rate of change in $ Y $ for infinitesimal variations in $ X_i $, holding other inputs fixed, and is particularly useful for models where $ f $ is differentiable.[16][17]
In preliminary screening approaches, such as the elementary effects method, a finite perturbation is used to approximate local effects. The elementary effect for the $ i $-th factor is defined as:
where $ \Delta $ is a small finite increment within the input domain, and $ \mathbf{e}_i $ is the $ i $-th unit vector. Multiple such effects are computed along randomized trajectories to estimate mean and variance, identifying influential factors.[18]
Global sensitivity analysis extends this framework by considering the full range of input variability, assuming $ \mathbf{X} $ follows a joint probability density function $ p(\mathbf{x}) $ over a probability space. The inputs are typically modeled as independent random variables, and the output $ Y $ is analyzed through integrals over the input domain. A foundational approach decomposes the unconditional variance of $ Y $ as:
This decomposition isolates main effects and interactions, with the first-order term $ \mathrm{Var}(E(Y \mid X_i)) $ representing the expected variance attributable to $ X_i $ alone, averaged over other inputs. Higher-order terms capture interactions, providing a basis for variance-based indices.[19][15] Such indices, including Sobol' indices, quantify the fractional contributions to total variance and are explored in subsequent method-specific sections.
Key Concepts and Terminology
In sensitivity analysis, input factors, also referred to as parameters or variables, represent the uncertain components of a computational model that drive its outputs. These factors, such as physical constants, environmental variables, or economic parameters, are characterized by probability distributions or specified ranges to account for their inherent uncertainty.[14] The output quantities of interest (QoI) are the specific scalar or functional results from the model that analysts seek to evaluate, such as risk levels, system performance metrics, or predicted doses in environmental simulations; the goal of sensitivity analysis is to apportion the uncertainty in these QoI to the input factors.[14] Sensitivity indices provide normalized quantitative measures, typically ranging from 0 to 1, of an input factor's contribution to the variance or uncertainty in the QoI, enabling the ranking of factors by their relative importance.[14] A key distinction exists between first-order effects and total-order effects. First-order effects quantify the direct, isolated influence of a single input factor on the QoI, assuming all other factors are fixed, thereby capturing only the main effect without interactions.[14] In contrast, total-order effects measure the comprehensive impact of an input factor, incorporating its main effect plus all higher-order interactions with other factors, which is essential for understanding the full propagation of uncertainty in complex systems.[14] The concepts of linearity and nonlinearity are fundamental to selecting appropriate sensitivity methods. In linear models, the relationship between inputs and outputs is additive and proportional, resulting in constant sensitivities that can be assessed locally without considering interactions.[14] Nonlinear models, however, feature varying sensitivities across the input space, often with significant interactions, requiring global methods to fully characterize the effects and avoid underestimation of factor importance.[14] Efficient exploration of the input space relies on sampling techniques like Latin Hypercube Sampling (LHS), a stratified method that divides each input factor's probability distribution into equal intervals and selects one sample from each to ensure even coverage, thereby reducing the number of model evaluations needed compared to random sampling while maintaining representativeness. Assessing convergence ensures the reliability of sensitivity results, typically through criteria such as the stabilization of sensitivity indices or the narrowing of their confidence intervals (e.g., at 95% level) as sample size increases, confirming that further sampling yields negligible changes.[20]Classification of Sensitivity Analysis Methods
Local versus Global Approaches
Local sensitivity analysis (LSA) examines the impact of input parameters on model outputs by perturbing them individually or in small combinations around a specific nominal or baseline point in the parameter space.[21] This approach assumes that the model behaves linearly or nearly so near the reference point and that parameters act independently without significant interactions.[22] It is particularly suited for scenarios where the system operates close to known operating conditions, such as in preliminary model calibration or when computational resources are limited.[23] In contrast, global sensitivity analysis (GSA) evaluates the effects of inputs across their entire feasible range or probability distributions, capturing nonlinearities, parameter interactions, and the propagation of uncertainties throughout the model. Unlike LSA, GSA does not rely on a single reference point but instead explores the full input domain, making it ideal for complex systems with high uncertainty or where inputs may vary widely.[22] Seminal works emphasize that GSA provides a more robust assessment of parameter importance by accounting for the joint effects of all inputs, which LSA often overlooks. The primary distinction lies in their scope and assumptions: LSA is computationally inexpensive and offers quick insights into local behavior but can miss global interactions and yield misleading results for nonlinear models, while GSA is more comprehensive yet demands substantially higher computational effort due to extensive sampling.[23] For instance, LSA might suffice for low-uncertainty engineering designs near nominal values, whereas GSA is preferred for risk assessment in environmental or biological models with distributed inputs.[22] Decision criteria for selection include model linearity, parameter variability, and analysis goals—opt for LSA in resource-constrained, near-nominal cases; use GSA for thorough uncertainty quantification in nonlinear or interactive systems. Hybrid approaches bridge these methods by leveraging LSA for initial screening around multiple points in the parameter space, followed by GSA to confirm and explore interactions, thereby balancing efficiency and completeness. Examples include the Distributed Evaluation of Local Sensitivity Analysis (DELSA), which distributes LSA computations to approximate global effects,[24] and variation-based hybrids that integrate local derivatives with global variance measures.[25] These are particularly useful in high-dimensional models where pure GSA is prohibitive, allowing practitioners to refine focus before full global exploration.Deterministic versus Probabilistic Frameworks
In deterministic sensitivity analysis, input parameters are treated as fixed values that are systematically varied, often through methods like grid searches or one-at-a-time perturbations, to evaluate the impact on model outputs without incorporating probability distributions.[14] This framework focuses on nominal or worst-case scenarios, such as assessing how outputs change when inputs are set to upper and lower bounds, thereby identifying critical parameters for scenario testing in deterministic models.[26] It assumes parameters have true or representative values and ignores input correlations or variability, making it suitable for exploring local effects around a baseline point but potentially overlooking broader interactions.[14] In contrast, probabilistic sensitivity analysis models inputs as random variables following specified probability distributions, such as uniform or normal, to propagate uncertainty through the model and quantify its effects on outputs via statistical measures like expectations or variances.[14] Techniques like Monte Carlo sampling are commonly used to generate ensembles of input combinations, enabling the assessment of risk under input variability and providing probabilistic outputs, such as confidence intervals for model predictions.[26] This approach requires knowledge of input distributions and accounts for correlations if specified, offering a more comprehensive view of uncertainty compared to deterministic methods.[14] The key differences between these frameworks lie in their treatment of uncertainty: deterministic analysis excels in targeted scenario exploration and is computationally simpler, ideal for preliminary assessments or when distributions are unknown, while probabilistic analysis better captures overall risk and variability, though it demands more data and resources for distribution specification.[26] A natural transition occurs from deterministic one-at-a-time analysis to probabilistic extensions, such as randomized versions that sample from distributions to incorporate variability while retaining the focus on individual parameter effects.[14] Deterministic frameworks often assume independence and linearity, potentially underestimating complex behaviors, whereas probabilistic ones necessitate accurate distributional assumptions to avoid biased uncertainty propagation.[26]Local Sensitivity Methods
One-at-a-Time (OAT) Analysis
One-at-a-time (OAT) analysis is a fundamental local sensitivity method that assesses the impact of individual input parameters on model outputs by varying one parameter while holding all others fixed at their nominal values.[22] This approach operates within the framework of local sensitivity analysis, focusing on small perturbations around a chosen reference point to approximate partial effects.[27] It is particularly suited for initial screening in deterministic models where interactions between parameters are assumed negligible or secondary.[28] The procedure begins with selecting a nominal or baseline point for all input parameters, representing the expected or central values of the system. One input is then perturbed by fixed increments, such as ±5% or ±10% of its nominal value, or across a discrete set of levels within its range, while keeping the remaining inputs constant. For each perturbation, the model is re-evaluated to compute the corresponding change in output. This process is repeated sequentially for each input parameter in turn.[22] Sensitivity is quantified using measures like the elementary effect, often expressed as the slope of the output change relative to the input perturbation, , or the percentage change in output , where is the output and is the -th input.[29] These metrics provide a direct indicator of how responsive the output is to variations in each input near the nominal point.[22] A practical example of OAT analysis appears in financial budgeting models, such as capital investment projects, where the net present value (NPV) serves as the output. To evaluate sensitivity to costs, the nominal cost estimate is perturbed by 5% (e.g., increased or decreased) while holding revenue projections, discount rates, and other expenses fixed; the resulting NPV reveals the cost's influence on project viability. Similar applications occur in engineering models, like varying a single material property in a structural simulation to observe effects on load-bearing capacity.[30] OAT analysis offers key advantages, including its intuitive simplicity and low computational demand, as it requires only model evaluations for inputs in the basic form, making it accessible for quick assessments in resource-constrained settings.[27] However, it has notable limitations: by fixing other inputs, it overlooks interactions and nonlinear effects across the parameter space, potentially underestimating overall uncertainty and providing misleading results in complex, high-dimensional systems.[22] Its strictly local nature also restricts validity to the vicinity of the nominal point, rendering it unsuitable for exploring global behaviors.[27] Extensions of OAT incorporate limited multi-way interactions through factorial designs, where small subsets of inputs (e.g., two or three) are varied simultaneously in a structured grid, such as a full factorial for , to capture pairwise effects without full global exploration.[28] This hybrid approach balances computational efficiency with improved detection of dependencies, often used as a bridge to more advanced methods.[29]Derivative-Based Techniques
Derivative-based techniques in local sensitivity analysis compute the partial derivatives of the model output $ Y $ with respect to each input parameter $ X_i $ at a specific nominal point in the parameter space, quantifying how small changes in $ X_i $ affect $ Y $. These methods assume the model is locally differentiable and focus on the gradient of the response surface near the base case.[16] Analytical derivatives involve direct computation from the model's mathematical equations, providing exact sensitivities without additional evaluations if the model is symbolic or differentiable by hand. For instance, consider a simple model $ Y = X_1^2 X_2 $; the partial derivative with respect to $ X_1 $ is $ \frac{\partial Y}{\partial X_1} = 2 X_1 X_2 $, evaluated at nominal values to yield the local sensitivity. This approach is efficient for low-dimensional, explicit models in fields like engineering design.[16][31] When analytical forms are unavailable or complex, numerical approximations estimate derivatives using finite differences, which perturb the input slightly and compute the difference quotient. The forward difference formula is $ \frac{\partial Y}{\partial X_i} \approx \frac{f(\mathbf{X} + h \mathbf{e}_i) - f(\mathbf{X})}{h} $, where $ h $ is a small step size and $ \mathbf{e}_i $ is the unit vector in the $ i $-th direction; backward differences use $ -h $ for improved accuracy in some cases. These methods require multiple model evaluations per parameter but are versatile for black-box simulations. Step size selection is critical to balance truncation and round-off errors, often using $ h = \sqrt{\epsilon} X_i $ where $ \epsilon $ is machine precision.[16][32] Elasticity normalizes the derivative to a dimensionless measure, defined as $ \eta_i = \frac{\partial Y}{\partial X_i} \cdot \frac{X_i}{Y} $, representing the percentage change in $ Y $ per percentage change in $ X_i $ at the nominal point. This scaling facilitates comparison across parameters with different units or scales, commonly applied in economic and ecological models.[33] These techniques find applications in optimization, where gradients guide parameter tuning via methods like steepest descent, and in stability analysis of dynamical systems, such as assessing eigenvalue sensitivities in control theory.[16][34] Limitations include the requirement for model differentiability, which fails for discontinuous or non-smooth functions, and sensitivity to the choice of nominal point, potentially missing nonlinear behaviors away from it. Numerical methods also amplify errors in high dimensions due to the curse of dimensionality in evaluations.[16][35]Regression Analysis
Regression analysis serves as a local sensitivity method that infers the influence of input variables on model outputs by fitting a statistical regression model to input-output data generated from simulations or observations. In this approach, a multiple linear regression model is constructed in the form $ Y = \beta_0 + \sum \beta_i X_i + \epsilon $, where $ Y $ represents the model output, $ X_i $ are the input variables, $ \beta_i $ are the regression coefficients quantifying the sensitivity of $ Y $ to each $ X_i $, $ \beta_0 $ is the intercept, and $ \epsilon $ is the error term.[3] The absolute values of the $ \beta_i $ coefficients indicate the relative importance of each input, with larger magnitudes signifying greater sensitivity. To enable comparability across inputs with different scales or units, standardized regression coefficients (SRCs) are employed by scaling the inputs and output to zero mean and unit variance before fitting the model. These SRCs, which range from -1 to 1, measure the change in output standard deviation per standard deviation change in an input, while holding other inputs constant, and their squared values approximate the fraction of output variance attributable to each input. For instance, in an ecological predator-prey model based on the Lotka-Volterra equations, regression analysis was applied to assess species abundances as outputs regressed against environmental parameters such as growth rates ($ r \alpha m, \delta $); SRCs revealed that the predator mortality rate $ \delta $ explained up to 82.4% of variance in prey abundance at equilibrium. Stepwise regression selection enhances this method by iteratively adding or removing inputs based on statistical criteria, such as improvements in the coefficient of determination $ R^2 $ or t-tests for coefficient significance, to identify the subset of inputs most influential on the output.[3] This process prioritizes inputs that significantly contribute to explaining output variance, often achieving high $ R^2 $ values (e.g., over 0.7) indicative of adequate model fit. The validity of regression-based sensitivity analysis relies on key assumptions, including linearity between inputs and output, independence among inputs, normally distributed residuals, and no severe multicollinearity. Violations of these, such as nonlinear relationships or correlated inputs, can bias coefficient estimates and suggest the limitations of local methods, necessitating a shift to global sensitivity approaches that explore the full input space.[3]Global Sensitivity Methods
Screening Methods (e.g., Morris)
Screening methods in sensitivity analysis aim to efficiently identify the most influential input factors among many in complex models, particularly when computational resources are limited. These techniques prioritize qualitative or semi-quantitative assessment over precise quantification, making them suitable for initial factor prioritization in global sensitivity contexts. The Morris method, introduced in 1991, is a prominent screening approach that evaluates elementary effects through randomized one-factor-at-a-time perturbations along multiple trajectories in the input space. For a model $ y = f(\mathbf{x}) $ with $ p $ input factors, the elementary effect for the $ i $-th factor at a point $ \mathbf{x} $ in the input space is defined as
where $ \Delta $ is a finite increment (typically $ \Delta = 1/(p+1) $ for a standardized unit hypercube), and $ \mathbf{e}i $ is the $ i $-th unit vector. This measures the local sensitivity at various points, capturing both main effects and potential nonlinearities or interactions by varying the base point $ \mathbf{x} $. The mean absolute elementary effect, $ \mu_i^* = \frac{1}{r} \sum{j=1}^r |EE_{i,j}| $, quantifies the overall influence of factor $ i $ across $ r $ replicates, while the standard deviation $ \sigma_i = \sqrt{\frac{1}{r} \sum_{j=1}^r (EE_{i,j} - \frac{1}{r} \sum_{k=1}^r EE_{i,k})^2} $, indicates nonlinearity or interactions if $ \sigma_i / \mu_i^* > 0.1 $.
The procedure involves generating $ r $ random trajectories, each consisting of $ p $ steps (for a total of $ r(p+1) $ model evaluations), where each step perturbs one randomly selected factor while keeping others fixed. The input space is discretized into a grid to ensure perturbations align with grid points, and trajectories are designed to cover the space efficiently without overlap. Factors with high $ \mu_i^* $ and low $ \sigma_i $ are deemed influential but additive, whereas high $ \sigma_i $ suggests higher-order effects. This distribution of effects allows ranking factors for further analysis.
One key advantage of the Morris method is its low computational demand, requiring only 10 to 40 evaluations per factor for $ p $ up to 20, compared to thousands for more exhaustive global methods, enabling its use in high-dimensional problems. It effectively detects both main effects and interactions, providing a profile for each factor that guides model simplification.
As an alternative for deterministic screening, sequential bifurcation iteratively halves the set of candidate factors by evaluating group averages and eliminating those with negligible impact, assuming additivity and no interactions.[36] This method is particularly efficient for models with many factors, requiring $ O(p \log p) $ evaluations in the best case.
Variance-Based Decomposition (e.g., Sobol')
Variance-based decomposition methods quantify the contribution of input variables to the output uncertainty of a model by partitioning the total variance of the output into components attributable to individual inputs and their interactions. These approaches, rooted in the functional ANOVA (analysis of variance) framework, assume that the model's output $ Y = f(\mathbf{X}) $ can be expressed as a sum of functions depending on subsets of the input vector $ \mathbf{X} = (X_1, \dots, X_p) $, where the inputs are independent and uniformly distributed over the unit hypercube. This decomposition enables a complete attribution of variance, making it suitable for global sensitivity analysis in complex systems such as engineering simulations and environmental modeling.[37] The Sobol' decomposition, named after Ilya M. Sobol', expands the output as:
where $ f_0 = E(Y) $ is the mean, and each $ f_\mathbf{u} $ for a subset $ \mathbf{u} \subseteq {1, \dots, p} $ has zero conditional expectation over any proper subset of its variables. The total variance $ \text{Var}(Y) $ is then additively decomposed as:
with $ V_\mathbf{u} = \text{Var}(E(Y \mid \mathbf{X}\mathbf{u})) - \sum{\mathbf{v} \subsetneq \mathbf{u}} V_\mathbf{v} $ representing the variance due to interactions among the variables in $ \mathbf{u} $. This formulation ensures orthogonality and completeness, allowing all variance to be accounted for without overlap.[37][38]
From this decomposition, sensitivity indices are derived to measure the relative importance of inputs. The first-order index for input $ X_i $, denoted $ S_i $, captures the main effect:
indicating the fraction of total output variance explained solely by $ X_i $. The total-order index $ S_{T_i} $, which includes all interactions involving $ X_i $, is:
where $ \mathbf{X}{\sim i} $ denotes all inputs except $ X_i $. Higher-order interaction indices, such as the second-order $ S{ij} = V_{ij} / \text{Var}(Y) $, quantify pairwise effects with $ V_{ij} = D_{ij} / \text{Var}(Y) $ and $ D_{ij} = V(E(Y \mid X_i, X_j)) - V_i - V_j $. These indices provide a hierarchical view of sensitivity, with $ \sum_i S_i + \sum_{i<j} S_{ij} + \cdots = 1 $, facilitating identification of dominant factors and interaction strengths in nonlinear models.[37]
Estimation of Sobol' indices typically relies on Monte Carlo integration due to the intractability of analytical solutions for most models. A widely adopted scheme is the Saltelli sampler, which generates $ N(p+2) $ samples using two base matrices $ \mathbf{A} $ and $ \mathbf{B} $ of size $ N \times p $, along with auxiliary matrices $ \mathbf{A}{\mathbf{B}} $ (columns of $ \mathbf{A} $ with $ B_i $ in the $ i $-th column) and $ \mathbf{B}{\mathbf{A}} $. Conditional expectations are approximated as, for example, $ E(Y \mid X_i) \approx \frac{1}{N} \sum_{j=1}^N f_{\mathbf{A}{\mathbf{B}}}(j) f{\mathbf{B}}(j) $ for the numerator of $ S_i $, with variances estimated via sample means. This method achieves improved convergence rates compared to naive Monte Carlo, requiring on the order of $ 10^3 $ to $ 10^4 $ model evaluations for reliable indices in moderate dimensions. Extensions like the Jansen or Sobol' estimators further refine accuracy by alternative pairings of the matrices.
For higher-order terms, interactions are estimated similarly by conditioning on pairs or groups, such as $ V_{ij} $ via $ E(Y \mid X_i, X_j) \approx \frac{1}{N} \sum_{k=1}^N f_{\mathbf{A}{\mathbf{B}{-ij}}}(k) [f_{\mathbf{A}}(k) - f_0] $ where $ \mathbf{B}_{-ij} $ replaces non-$ i,j $ columns. These computations scale poorly with dimension $ p $, often limiting full decomposition to low-order terms, but total indices remain feasible as they avoid explicit higher-order evaluations. In practice, variance-based methods excel in revealing nonlinear and interaction effects, outperforming local measures in capturing full input ranges.[38]
Classical Sobol' indices assume input independence; extensions for correlated inputs, such as those using copula representations or conditional variance adjustments, have been developed to preserve additivity while handling dependence.[39]
Moment-Independent and Correlation Measures
Moment-independent measures provide a global sensitivity assessment by evaluating the influence of input uncertainty on the full probability distribution of the model output, without relying on assumptions about moments such as mean or variance. These methods are particularly valuable for models with non-normal or heavy-tailed outputs, where traditional variance-based approaches may fail or provide misleading results. A prominent example is the delta measure, introduced by Borgonovo, which quantifies the shift in the output distribution induced by varying a single input factor while integrating over the others. The delta sensitivity index for an input $ X_i $ is formally defined as
where $ f_Y(y) $ denotes the unconditional probability density function of the output $ Y = f(\mathbf{X}) $, and $ f_{Y \mid X_i}(y) $ is the conditional density given a fixed value of $ X_i $. This index represents the average L1 distance between the unconditional output density and the conditional density across all possible values of $ X_i $, normalized to lie between 0 and 1, with $ \delta_i = 0 $ indicating independence between $ X_i $ and $ Y $, and $ \sum \delta_i = 1 $ for additive models. Computationally, the integral and expectations are estimated using Monte Carlo sampling, often employing kernel density estimation (KDE) for non-parametric reconstruction of the densities from simulated data points; KDE uses a kernel function (e.g., Gaussian) to smooth the empirical distribution, enabling flexible estimation without distributional assumptions.
Correlation-based measures offer a complementary, computationally lighter approach to quantify sensitivity through the strength and direction of associations between inputs and outputs, focusing on monotonic relationships. The Spearman's rank correlation coefficient $ \rho_i $ measures the monotonic dependence between $ X_i $ and $ Y $ by correlating their ranks:
where $ d_j $ is the difference between the ranks of the $ j $-th paired observations of $ X_i $ and $ Y $, and $ n $ is the sample size; values range from -1 to 1, indicating perfect negative to positive monotonicity. To account for confounding effects from other inputs, the partial rank correlation coefficient (PRCC) extends this by partialling out the influence of remaining factors via rank-transformed linear regression residuals before computing the correlation, providing a more isolated assessment of individual sensitivities in multi-input models. These coefficients are estimated from Latin hypercube or quasi-random samples and are robust to outliers and non-linear monotonicities.
Unlike variance-based decompositions, which apportion output variance and assume finite second moments, moment-independent and correlation measures like delta and PRCC handle broader output types, including non-monotonic functions and distributions with infinite variance, thus offering greater applicability in risk analysis and complex simulations. For instance, delta captures distributional shifts holistically, while correlations emphasize directional influences, making them suitable for screening influential factors in preliminary analyses. However, these methods incur higher computational demands for delta due to the need for density estimation via KDE, which requires larger sample sizes (often $ n > 10^4 $) compared to the modest $ n \approx 500-1000 $ sufficient for correlation coefficients, potentially limiting their use in very high-dimensional problems.
Fourier Amplitude Sensitivity Test (FAST)
The Fourier Amplitude Sensitivity Test (FAST) is a variance-based global sensitivity analysis method that quantifies the contribution of input factors to the output variance by exploring the model's parameter space through periodic sampling and spectral decomposition. Developed originally for assessing sensitivities in coupled reaction systems, FAST transforms each input variable into a periodic function using distinct integer frequencies , generating a one-dimensional search curve that sweeps the input space uniformly. The model output is then expressed as a Fourier series expansion along this curve: , where parameterizes the curve. The first-order sensitivity index for input is computed as , capturing the portion of output variance attributable solely to , excluding interactions.[40] In classical FAST, a single search curve is used, with frequencies chosen as integers to prevent spectral overlap (e.g., for inputs), and sampling points generated via for , where is the cumulative distribution function of and are phase shifts. This resampling scheme ensures efficient exploration with model evaluations, where must satisfy the Nyquist criterion () to resolve frequencies up to a cutoff. An analytical form for the first-order index, assuming rescaled output with zero mean and unit variance, is , derived from the Fourier coefficients' squared amplitudes. While classical FAST excels at main effects, it overlooks interactions due to the single-curve limitation.[40][41] The extended FAST (eFAST) addresses this by employing multiple search curves, each assigning a high base frequency to one input while low frequencies to others, enabling separation of main effects from total effects (including interactions). For total sensitivity, is set large (e.g., , with as the interference order), and the index is , where is the sensitivity excluding . Sampling follows a similar integer-frequency scheme but iterates over curves (typically 5–10 per input), with total evaluations scaling as , where is the number of inputs and the number of curves. eFAST aligns with Sobol' variance decomposition by estimating indices via spectral analysis.[42] FAST and eFAST are particularly efficient for models with up to 20 input factors, requiring fewer evaluations than Monte Carlo-based methods while providing robust variance apportionment, and have been applied in fields like chemical kinetics, environmental modeling, and engineering systems.[42][43]Polynomial Chaos Expansion
Polynomial chaos expansion (PCE) provides a spectral representation of a model's output as a series of orthogonal polynomials in the input random variables, enabling efficient quantification of uncertainty and derivation of sensitivity measures. Originally developed by Wiener in 1938 for Gaussian processes using Hermite polynomials, the framework was generalized by Xiu and Karniadakis in 2002 to incorporate a wider class of orthogonal bases from the Askey scheme, matching the input distributions (e.g., Hermite for Gaussian, Legendre for uniform).[44] This generalized PCE approximates the model output $ Y = \mathcal{M}(\mathbf{X}) $, where is the vector of random inputs, as
with denoting multivariate orthogonal polynomials of multi-index and coefficients .[44] Due to the orthogonality of the basis, statistical moments of can be computed directly from the coefficients, facilitating variance-based sensitivity analysis without extensive resampling.
Sensitivity indices are derived analytically from the PCE coefficients, leveraging the decomposition of the output variance. The total variance is , where denotes the expectation (often normalized to 1). The first-order Sobol' index for input , measuring its individual contribution, is
while the total-order index, capturing all effects including interactions, follows from inclusion-exclusion as , where sums over terms excluding . This approach, formalized by Sudret in 2008, allows computation of Sobol' indices and higher-order interaction terms directly from the expansion, offering computational efficiency over traditional Monte Carlo estimation for variance-based methods.
PCE constructions are categorized as intrusive or non-intrusive. Intrusive methods, such as Galerkin projection, integrate the expansion into the model's governing equations, solving a system of deterministic equations for the coefficients; this is particularly suited for differential equation models but requires code modifications.[44] Non-intrusive approaches treat the model as a black box, estimating coefficients via regression (e.g., least-squares) or orthogonal projection on a set of model evaluations at random input samples, making them applicable to legacy codes. PCE excels in handling nonlinear responses and correlated inputs, while also yielding a surrogate model for rapid subsequent evaluations and sensitivity assessments.
For high-dimensional problems, where the full expansion grows exponentially with the number of inputs, sparse PCE variants have emerged since the 2010s to mitigate the curse of dimensionality. These methods adaptively select a subset of basis terms using techniques like least angle regression or Bayesian inference, retaining only those with significant coefficients while truncating others. Developed by Blatman and Sudret in 2011, sparse PCE reduces computational demands and improves accuracy in dimensions exceeding 10–20 variables, as demonstrated in structural reliability applications.
Game-Theoretic Approaches (e.g., Shapley Effects)
Game-theoretic approaches to sensitivity analysis draw from cooperative game theory to fairly attribute the output variability of a model to its input factors, treating inputs as "players" in a game where the "payout" is the model's variance. The Shapley effect, a prominent method in this framework, assigns to each input a value that represents its average marginal contribution to the variance across all possible coalitions of inputs. This approach ensures equitable distribution of importance, accounting for interactions without bias toward any particular order of inclusion. Introduced in the context of global sensitivity analysis by Song, Nelson, and Staum (2016), the Shapley effect for input in a model with inputs is given by
where denotes the variance of the model output explained by the subset of inputs , typically under the assumption of finite variance.[45]
The Shapley effects satisfy key axiomatic properties derived from the original Shapley value in game theory: efficiency (the sum of effects equals the total output variance), symmetry (inputs with identical marginal contributions receive equal effects), dummy (an input with no contribution gets zero effect), and additivity (effects for a sum of games add up). These properties make the method robust and interpretable, as they guarantee a complete and fair decomposition of variance even when inputs interact or are dependent. Unlike variance-based methods such as Sobol' indices, which partition variance into main and interaction effects, Shapley effects provide a unified attribution by averaging over all coalition sizes and compositions.[45]
Computing exact Shapley effects is combinatorially expensive, requiring evaluation over subsets, so Monte Carlo estimation is essential, particularly Owen sampling schemes that efficiently approximate the multilinear extension of the value function. For high-dimensional problems (), approximations such as truncated sampling or metamodel-assisted methods reduce the burden while maintaining accuracy, as detailed in algorithms by Plischke, Rabitti, and Borgonovo (2021). These computational strategies allocate model evaluations to estimate marginal contributions across random permutations of inputs.
Shapley effects offer advantages in handling all input interactions equally, providing a moment-independent measure of importance that does not rely on distributional assumptions beyond finite variance, thus applicable to non-normal outputs. This fairness in attribution contrasts with hierarchical decompositions in other global methods, making it suitable for complex systems. Shapley effects have gained traction in machine learning interpretability, where they underpin tools like SHAP for feature attribution in black-box models.[45]
Advanced and Complementary Techniques
Surrogate Modeling and Metamodels
Surrogate models, also known as metamodels, serve as computationally efficient approximations of expensive simulation models, enabling sensitivity analysis by minimizing the number of direct evaluations required from the original system.[46] These approximations capture the input-output relationship of the underlying model, allowing global sensitivity measures, such as variance-based indices, to be computed rapidly on the surrogate instead of the full model. By leveraging a limited set of training data, surrogates facilitate exploration of high-dimensional parameter spaces that would otherwise be infeasible due to computational constraints. Common types of surrogate models include polynomial response surfaces, kriging (or Gaussian processes), and radial basis functions. Polynomial response surfaces approximate the model response using low-order polynomials, often quadratic forms fitted via least-squares regression, making them suitable for smooth functions in low to moderate dimensions. Kriging models the output as a Gaussian process, providing an interpolating predictor with associated uncertainty estimates derived from the covariance structure. Radial basis function models express the response as a linear combination of basis functions centered at training points, offering flexibility for irregular data and effective performance in higher dimensions. To construct a surrogate, a design of experiments (DOE) such as Latin hypercube sampling (LHS) is used to select input points across the parameter space, the expensive model is evaluated at these points to generate training data, and the surrogate is then fitted to this dataset.[47] Sensitivity analysis is subsequently performed on the surrogate, which supports efficient Monte Carlo sampling or analytical derivations for indices like Sobol' sensitivities. For example, in a Gaussian process surrogate for the output Y(X), the posterior mean provides the predicted response, while the posterior variance quantifies uncertainty, aiding in the computation of sensitivity metrics that account for prediction error.[48] The primary benefit of surrogate modeling is its ability to enable global sensitivity analysis for computationally intensive models, such as those in computational fluid dynamics (CFD), where each evaluation may require hours or days; surrogates can reduce overall computational time by orders of magnitude while maintaining acceptable accuracy.[46] Surrogate accuracy is validated through cross-validation methods, including leave-one-out or k-fold procedures, which estimate out-of-sample prediction error and confirm the surrogate's reliability for sensitivity inference.[47] Polynomial chaos expansions may also serve as surrogates in this context, linking to variance-based decomposition techniques in global sensitivity methods.[49]High-Dimensional Model Representations (HDMR)
High-dimensional model representations (HDMR) provide a functional decomposition technique to approximate multivariate functions in sensitivity analysis, particularly addressing the curse of dimensionality in models with many input factors. The core idea is to expand the model output , where represents input variables, into a sum of component functions that capture individual and interactive effects of the inputs. This expansion is typically truncated at a low order to focus on dominant low-dimensional contributions, enabling efficient analysis even for .[50] The general HDMR expansion takes the form:
where is the constant term (mean output), are univariate components, are bivariate interactions, and higher-order terms represent multi-way interactions up to the full -way term. In practice, the series is cut off at order , assuming higher-order interactions are negligible, which holds for many physical and engineering systems. This decomposition reveals the effective dimensionality of the model by identifying which low-order terms contribute most to output variability.[50][51]
A common variant is the cut-HDMR, which anchors the expansion at a reference point (often the mean or a nominal value of the inputs). The component functions are defined such that when lower-order variables are fixed at , the higher-order terms vanish, ensuring a unique hierarchical decomposition. For instance, the first-order term is , where denotes the reference point with varying. Cut-HDMR is particularly useful for deterministic models where sampling can be controlled along "cuts" through the reference point.[51][52]
In contrast, ANOVA-HDMR employs an orthogonal decomposition over the input probability space, assuming independent inputs, which directly links to variance-based sensitivity measures. The component functions are mutually orthogonal, and their variances correspond to Sobol' indices, quantifying the contribution of each input subset to the total output variance. This makes ANOVA-HDMR suitable for global sensitivity analysis, as the zeroth-order term is the unconditional mean, and higher-order terms capture conditional expectations excluding lower interactions.[52]
Component functions in cut-HDMR are computed using the inclusion-exclusion principle, evaluating the model at specific points along the cuts and subtracting contributions from lower-order subsets. For a second-order term, , with higher orders following recursively. This method requires model evaluations for order , but truncation at low (e.g., 2 or 3) keeps costs manageable for high . Random sampling variants like RS-HDMR further reduce evaluations by projecting onto a basis over the input domain.[51]
HDMR has been applied to reduce effective dimensionality in sensitivity analysis of complex models with over 100 factors, such as chemical kinetics mechanisms and atmospheric simulations, where low-order terms often explain over 90% of output variance. For example, in combustion modeling, cut-HDMR identifies key reaction pathways influencing ignition delay, enabling targeted parameter studies without full explorations of the input space.[50][53]