Simulation-Based Sensitivity Analysis

• Simulation-based sensitivity analysis quantifies how input variations affect outputs in complex systems using numerical or stochastic simulations, especially when analytical methods are infeasible.
• This approach is vital for design optimization, uncertainty quantification, and decision-making across fields like engineering, finance, and epidemiology.
• Methods include efficient sampling, surrogate modeling for expensive simulations, various sensitivity indices (variance-based, moment-independent), and strategies for stochastic models.

Simulation-based sensitivity analysis is a family of methodologies that leverage numerical or stochastic simulations to quantify how variations in model inputs propagate through complex systems and affect outputs. This approach is critical for models where analytical sensitivity calculations are intractable or where model evaluations are expensive, nonlinear, or high-dimensional. Practitioners commonly employ simulation-based sensitivity analysis in domains such as engineering, the physical sciences, epidemiology, and finance to guide design optimization, uncertainty quantification, robustness assessment, and decision-making.

1. Methodological Foundations

Simulation-based sensitivity analysis centers on quantifying the impact of uncertain or variable inputs on model outputs by repeatedly evaluating the model (or a surrogate) at different points in the input space. Key methodological principles include:

• Defining the Input-Output Mapping: The model is formulated as a mapping $f: X \rightarrow Y$, where $X$ is the space of input variables (parameters) and $Y$ the space of model outputs.
• Sampling the Input Space: Since direct exploration of the full input space is unfeasible for high-dimensional or expensive models, strategies such as Latin Hypercube Sampling, quasi-Monte Carlo sampling, or sequential/active sampling are used to select input configurations.
• Sensitivity Index Computation: Once simulation outputs are available, sensitivity indices are computed using variance decomposition (Sobol' indices), derivative-based metrics, moment-independent indices (e.g., Borgonovo, kernel-based), or statistical dependence measures such as HSIC or MMD.
• Treatment of Intrinsic Randomness: For stochastic simulators (outputs random even for fixed input), specialized frameworks model or marginalize over latent random variables, extend sensitivity indices to capture statistical dependence, or utilize surrogate models that represent distributions instead of point estimates.

2. Surrogate Modeling and Dimension Reduction

For simulators that are expensive to evaluate or present prohibitive dimensionality, surrogate (meta) models are integral to simulation-based sensitivity workflows. Common surrogate models include:

• Kriging (Gaussian Process Regression): Provides a probabilistic predictor with analytic mean and variance expressions, facilitating active sampling and uncertainty quantification. For a new point $z$ given $N$ training samples, the GP predictor and its variance are given by:

$$\widehat{F}_N(z) = k(z)^\top K_x^{-1} y, \quad \mathrm{Var}[(z)] = K(z, z) - k(z)^\top K_x^{-1} k(z)$$

• Polynomial Regression and Moving Least Squares (MLS): MLS adds local weighting to polynomial regression to enhance flexibility for nonlinear models.
• Generalized Lambda Models (GLaM): Used for stochastic simulators, GLaMs emulate not just the mean but the full conditional output distribution as a function of inputs, parameterized via a generalized lambda distribution with coefficients fitted by polynomial chaos expansions.
• Sequential/Active Design: Algorithms such as FLOLA-Voronoi or variance-based exploration with Gaussian processes prioritize next sample points based on model uncertainty or gradient information.
• Dimension Reduction: Feature selection algorithms (e.g., FOCI) identify a relevant subset of variables by quantifying conditional independence, drastically reducing problem dimensionality and enabling tractable surrogate modeling and sensitivity analysis even in settings with thousands of variables.

3. Sensitivity Index Types and Computation

Simulation-based sensitivity analysis encompasses several types of indices and computational techniques, reflecting different notions of "influence" and suited for various model and data properties:

• Variance-Based Indices (Sobol' Indices): Decompose model output variance into contributions from main and interaction effects:

$$S_{X_i} = \frac{\mathrm{Var}(\mathbb{E}[Y|X_i])}{\mathrm{Var}(Y)}, \quad S^T_{X_i} = 1 - \frac{\mathrm{Var}(\mathbb{E}[Y|X_{-i}])}{\mathrm{Var}(Y)}$$

• Derivative-Based Sensitivity Measures (DGSM):

$$\nu_i = \mathbb{E}\left[\left( \frac{\partial f}{\partial x_i} \right)^2 \right]$$

• Moment-Independent and Distributional Indices: MMD and HSIC quantify sensitivity without reference to output variance, employing kernel methods in reproducing kernel Hilbert spaces (RKHS):

$$S_l^\mathrm{MMD} = \mathbb{E}_{X_l}\left[ \mathrm{MMD}^2(P_Y, P_{Y|X_l}) \right], \quad S_l^\mathrm{HSIC} = \mathrm{HSIC}(X_l, Y)$$

• Shapley Effects: These generalize sensitivity allocation in the presence of input dependencies or for non-variance-based indices by integrating contributions across all possible input subsets.
• Simple Binning Methods: For computationally intensive models or large simulations, efficient "binning" approaches compute first- and second-order effects via variance estimates of output means within binned input intervals.

4. Approaches for Specific Challenges

Stochastic Simulators

For simulators where the output distribution for a fixed input is random:

• Statistical Dependence Indices: Recommended methods estimate, for each input, how much the (potentially heteroskedastic) distribution of outputs changes as input varies.
• QoI-based Indices: Compute Sobol' indices on summary statistics (e.g., conditional mean, variance, quantiles) of the output, not only on point estimates.
• Surrogate Distribution Models: GLaMs emulate the full output distribution at each input, enabling efficient computation of sensitivity indices for a range of output functionals.

Expensive or Black-Box Simulations

For cases where simulation runs are costly:

• Metamodel-Based Sensitivity Analysis: Surrogate models are built with as few simulator evaluations as needed, using sequential design and stopping criteria based on the convergence of sensitivity indices rather than model fit alone.
• Automatic Model Selection and Variable Filtering: Frameworks such as the Metamodel of Optimal Prognosis (MOP) systematically test variable subsets and surrogate types, selecting the optimal combination by cross-validated predictive accuracy.

5. Sensitivity Analysis for Design and Validation

Simulation-based sensitivity analysis informs decisions and ensures robust model-based inference by:

• Identifying Key Drivers: Quantifying the main and total effects of input variables highlights the most influential design, environmental, or process factors.
• Supporting Model Simplification: Non-influential variables can be fixed or omitted, reducing model complexity and simulation cost without sacrificing predictive fidelity.
• Interaction Effects and Visualization: Modern approaches (e.g., SimDec, kernel-based indices) extend classic analyses by quantifying and visualizing high-order interactions, which are especially important in highly nonlinear or interdependent models.
• Uncertainty Quantification: Sensitivity indices guide uncertainty reduction efforts by identifying which inputs’ distributions most affect model predictions (see GSA2 methodologies).
• Feedback for Experimental Design: Sensitivity measures can aid in the selection of optimal experimental designs, targeting regions of greatest expected information gain.

6. Computational Efficiency and Practical Implications

Simulation-based approaches achieve computational tractability in high-dimensional or expensive settings through:

• Efficient Sampling: Active learning, quasi-Monte Carlo, and optimal design strategies minimize simulation runs for accurate sensitivity estimation.
• Single-Loop Weighted Estimators: For second-level analyses (uncertainty in the distributions of inputs), single-loop design with importance weighting (e.g., weighted HSIC) enables many "what-if" analyses from a single simulation batch.
• Real-Time and Incremental Workflow Integration: Frameworks such as "Simulation As You Operate" (SAYO) facilitate live, incremental computation of both simulation results and sensitivity indices as models are constructed or inputs specified.

These strategies expand the applicability of sensitivity analysis to domains where direct or classical approaches would be otherwise infeasible, such as design of analog circuits with thousands of process parameters or virtual validation of complex sensor models in autonomous vehicles.

7. Applications and Strategic Impact

Simulation-based sensitivity analysis underpins practical advances across multiple research and engineering fields:

• Epidemiology and Systems Biology: Enables quantification of the relative influence of biological or social parameters and stochasticity on outcomes such as epidemic peak timing or extinction probabilities, with explicit decomposition of variance between parameter and intrinsic noise components.
• Materials and Catalysis: Guides atomic-level catalyst design by identifying which microscopic transition rates most strongly affect macroscopic reaction rates, using tailored multi-stage approaches for computational efficiency.
• Finance: Accurately attributes risk to portfolio components using variance-based sensitivity analysis integrated with advanced (nonparametric, real-time) simulation.
• Robust Engineering Design: Facilitates component-wise and interaction sensitivity assessment, supporting both optimization and reliability-based design for complex machinery.
• Power Systems: Efficiently targets rare-event risk and identifies sensitive operational parameters in environment with high uncertainty, incorporating real-world constraints of simulation runtime.

This breadth of application is enabled by modular, efficient analysis frameworks that combine surrogate-assisted computation, modern sampling, and statistical dependence metrics, providing actionable insights for design, validation, and policy.