Multivariate Global Sensitivity Analysis

• Multivariate global sensitivity analysis (MGSA) is a framework that quantifies how uncertainties in multiple inputs affect vector-valued outputs, capturing main and interactive effects in high-dimensional systems.
• It generalizes classical ANOVA and Sobol indices by employing surrogate models, dependence measures, and information-theoretic approaches to tackle complex, correlated, and functional data scenarios.
• MGSA enhances model reduction and feature selection by enabling holistic ranking of parameter importance, guiding resource allocation in uncertainty quantification for deterministic and stochastic models.

Multivariate global sensitivity analysis (MGSA) encompasses a set of theoretical frameworks and computational methodologies for quantifying the influence of multiple uncertain input variables on vector-valued, functional, or distribution-valued outputs of deterministic or stochastic models. Unlike univariate sensitivity analysis—which targets the variance contribution of input variables to a scalar output—MGSA addresses the full joint/input-output structure, enabling holistic assessment of input importance, including main and interaction effects, in high-dimensional and correlation-rich systems. MGSA methods generalize classical ANOVA and variance-based indices, and incorporate modern dependence, information-theoretic, and surrogate-modeling approaches to handle dependent inputs, functional data, distributional outputs, and computational scaling.

1. Foundations: Decomposition Principles and Multivariate Indices

The cornerstone of most MGSA approaches is a generalization of the Hoeffding–Sobol (functional ANOVA) decomposition to vector-valued outputs. For a vector-valued response $\mathbf{Y} = f(\mathbf{X}) \in \mathbb{R}^m$ depending on $d$ independent variables $\mathbf{X} = (X_1, \ldots, X_d)$, the covariance decomposition reads

$$\operatorname{Cov}(\mathbf{Y}) = \sum_{n=1}^d \mathbf{C}_n + \sum_{n<k} \mathbf{C}_{nk} + \cdots + \mathbf{C}_{1 \ldots d},$$

where $\mathbf{C}_n$ collects the variance component attributable to $X_n$ alone, and higher-order $\mathbf{C}_{nk}$ represent interactions. Noting that $$\operatorname{tr}(\operatorname{Cov}(\mathbf{Y}))=\sum_{m=1}^M \operatorname{Var}(Y_m)$$, a general multivariate (trace-based) sensitivity index for $X_n$ is

$$G_n = \frac{\operatorname{tr}(\mathbf{C}_n)}{\operatorname{tr}(\operatorname{Cov}(\mathbf{Y}))},$$

with an analogous total-effect index $G_{T, n}$ defined via conditional covariances (e.g., fixing $X_n$) (Partovizadeh et al., 21 Nov 2025).

This structure recovers the classical scalar Sobol indices in the $m=1$ case and admits rigorous extension to the multivariate regime (Oyebamiji et al., 2022, Gratiet et al., 2013). For functional-valued responses $Y(t)$ (e.g., time series, fields), functional ANOVA and time-integrated Sobol indices generalize the variance ratio concept to entire domains (Fontana et al., 2020). For outputs in $\mathbb{W}_2$ Wasserstein space (e.g., when $f(\mathbf{X})$ is a random CDF), analogous decomposition and index construction are available via Fréchet means or indicator-based metrics (Fort et al., 2020).

2. Methodological Families: Variance-Based, Dependence-Based, and Information-Theoretic Indices

Variance-based indices: These derive from the multivariate ANOVA framework above. The first-order index $S_n$ and total-effect $S_{T,n}$ for each input, and their generalization to traces or projections (e.g., $S_n = \operatorname{tr}(C_n)/\operatorname{tr}(C)$), constitute the mainstream of MGSA (Partovizadeh et al., 21 Nov 2025, Oyebamiji et al., 2022, Huet et al., 2017). Shapley effects, grounded in cooperative game theory, allocate the variance $\sigma^2$ among inputs such that $\sum_j \phi_j = \sigma^2$, with favorable properties for interpretability and invariance (Goda, 2020).

Dependence-based and information-theoretic indices: MGSA can also be realized by measuring the change in the output (joint) law upon knowledge of (sets of) input variables, generalizing beyond variance. Csiszár $f$-divergence, distance correlation ($\mathrm{dCor}$), and Hilbert–Schmidt independence criterion (HSIC) provide sensitivity indices that vanish iff input and output are independent, capturing both linear and nonlinear dependence beyond mean/variance contributions (Veiga, 2013, Rahman, 2015). The general $f$-sensitivity index is

$$H_S^f = \mathbb{E}_{X_S}\!\left[D_f\left(P_U \| P_{U|X_S}\right)\right],$$

encompassing mutual information, squared-loss mutual information, and Borgonovo's total variation measure as cases (Rahman, 2015).

Distributional outputs: For models outputting distributions, such as stochastic codes or predictive CDFs, variance in Wasserstein space is used to define output variability, and indices assess sensitivity via Fréchet means or test-based decompositions (Fort et al., 2020).

3. Computational Algorithms and Surrogate Modeling

Direct computation of multivariate sensitivity indices is often prohibitive for expensive or high-dimensional simulators. Surrogate and emulation approaches are therefore central.

• Multivariate Gaussian Process (GP) surrogates: These provide joint modeling of vector-valued responses, allowing rigorous posterior uncertainty assessment and closed-form evaluation of ANOVA or variance-based indices via posterior predictive draws. Conventional and multi-fidelity surrogates are supported, with multi-output GPs using separable or coregionalized kernels (Oyebamiji et al., 2022, Gratiet et al., 2013).
• Polynomial Chaos Expansion (PCE): PCE enables efficient computation of MGSA indices once the PCE representation is constructed, as the variance decomposition becomes a sum over known orthogonal polynomial terms (Partovizadeh et al., 21 Nov 2025). Adaptive and sparse PCE allow scaling to larger parameter sets.
• Block coordinate and RKHS ANOVA (kernel-based) approaches: RKHS-ANOVA methodologies estimate metamodels and associated sensitivity structure via penalized least squares/minimization in structured reproducing kernel Hilbert spaces, with support recovery for both main and interaction effects (Huet et al., 2017).

Variance reduction, sparsity exploitation (e.g., via compactly supported kernels and simplex constraints), adaptive sampling, and parallelization are standard computational techniques (Oyebamiji et al., 2022).

4. Extensions: Dependent Inputs, High-Order Interactions, and Stochastic/Functional Outputs

Dependence among input variables, prevalent in engineered or empirical systems, invalidates classical orthogonality assumptions of ANOVA and renders Sobol indices uninformative or ill-defined. Nonparametric, entropy-based methods—such as minimal spanning tree estimates of Rényi entropy—and direct dependence measures such as distance correlation provide robust sensitivity metrics for dependent inputs (Eggels et al., 2018, Rahman, 2015).

Functional- and distribution-valued outputs—common in climate, econometric, and uncertainty quantification applications—necessitate dedicated functional ANOVA (domain-integrated or domain-selective) (Fontana et al., 2020) and Wasserstein-based indices (Fort et al., 2020). For stochastic simulators, the framework is extended by mapping inputs to output laws (empirical or parametric), then applying the same variability quantification at the distributional level, including so-called "second-level" sensitivity concerning the input distributions (Fort et al., 2020).

5. Interaction, Ranking, and Model Reduction

True MGSA enables holistic ranking of parameter importance, including interactions not detectable via one-at-a-time (OAT) or partial-derivative-based screening (Ballester-Ripoll et al., 9 Jun 2024). Covariance-trace and Shapley-based indices partition total output variability across inputs and their combinations, allowing groupings and rational variable elimination. Fixing non-influential factors based on those indices results in reduced models demonstrably maintaining predictive performance and uncertainty characterization (Partovizadeh et al., 21 Nov 2025, Hart et al., 2017).

For high-dimensional input spaces and multivariate outputs, feature selection (e.g., mRMR or HSIC-Lasso) and kernel/embedding-based screening can precede full computation of sensitivity indices, especially when only a few dimensions exert dominant influence (Veiga, 2013).

6. Analytical and Proxy Techniques

Recent work employs Poincaré-type and integral-equality-based proxy indices, yielding tight, computationally cheap upper bounds for Sobol-type sensitivity structure, constructed directly from gradients/cross-partials and distributional weights (Lamboni, 2019). These proxies facilitate rapid prescreening, are especially advantageous when model evaluations are expensive, and converge rapidly via Monte Carlo integration.

Analytic Taylor-series decompositions of the model response enable exact sensitivity propagation for smooth systems with known input moments, providing closed-form indices that coincide with Sobol’s indices in linear/small-variance regimes but extend seamlessly to nonlinear and higher-moment sensitivity (Zhu et al., 2016).

7. Challenges, Limitations, and Best Practices

MGSA is challenged by the curse of dimensionality, correlation structure, and computational cost in large models. Surrogate modeling, efficient estimator design (e.g., unbiased, variance-reduced, permutation-based Monte Carlo), and intelligent parameterization (e.g., dimension reduction, output projection, domain selection) are essential for scalability (Oyebamiji et al., 2022, Gratiet et al., 2013). For functional outputs, domain-selective testing and simultaneous inference are needed for localized sensitivity quantification (Fontana et al., 2020). For statistical models, careful choice of uncertainty measures (loss functions) and induced densities is critical to capture practical predictive relevance (Hart et al., 2017).

A main limitation of variance-based indices is their focus on second-moment effects; dependence- and information-based indices are essential in settings where higher-order features, distributions, or nonvariance structure are relevant (Rahman, 2015, Veiga, 2013). For correlated inputs and outputs, classical Sobol indices may be ill-posed; nonparametric entropy estimators and kernel-based measures offer alternatives (Eggels et al., 2018, Veiga, 2013).

Best practices dictate:

• Prestandardizing inputs/outputs when building GPs/PCEs (Oyebamiji et al., 2022).
• Using cross-validation to select surrogate model hyperparameters and sparsity level.
• Checking that the sum of computed indices matches total variance (for interpretability).
• Adapting methodology (e.g., surrogate, dependence measure) to output structure (functional, vector, distributional) and computational budget.
• Validating reduced models on held-out or out-of-sample data (Hart et al., 2017, Partovizadeh et al., 21 Nov 2025).

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Key Reference Papers Supporting This Article

• "Quantifying dependencies for sensitivity analysis with multivariate input sample data" (Eggels et al., 2018)
• "The $f$-Sensitivity Index" (Rahman, 2015)
• "Multivariate Sensitivity Analysis of Electric Machine Efficiency Maps and Profiles Under Design Uncertainty" (Partovizadeh et al., 21 Nov 2025)
• "Multivariate sensitivity analysis for a large-scale climate impact and adaptation model" (Oyebamiji et al., 2022)
• "Global Sensitivity Analysis with Dependence Measures" (Veiga, 2013)
• "Global sensitivity analysis and Wasserstein spaces" (Fort et al., 2020)
• "Integral equalities and inequalities: a proxy-measure for multivariate sensitivity analysis" (Lamboni, 2019)
• "A Bayesian approach for global sensitivity analysis of (multi-fidelity) computer codes" (Gratiet et al., 2013)
• "Global sensitivity analysis for statistical model parameters" (Hart et al., 2017)
• "Metamodel construction for sensitivity analysis" (Huet et al., 2017)
• "A simple algorithm for global sensitivity analysis with Shapley effects" (Goda, 2020)
• "Global Sensitivity and Domain-Selective Testing for Functional-Valued Responses: An Application to Climate Economy Models" (Fontana et al., 2020)
• "Global Sensitivity Analysis of Uncertain Parameters in Bayesian Networks" (Ballester-Ripoll et al., 9 Jun 2024)
• "An analytic method for sensitivity analysis of complex systems" (Zhu et al., 2016)