---
title: Multivariate Global Sensitivity Analysis
url: https://www.emergentmind.com/topics/multivariate-global-sensitivity-analysis
type: topic
---

# Multivariate Global Sensitivity Analysis

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 $X_n$) [2511.17099].

This structure recovers the classical scalar Sobol indices in the $m=1$ case and admits rigorous extension to the multivariate regime [2201.09681, 1307.2223]. 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 [2006.13850]. 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 [2007.12378].

## 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 [2511.17099, 2201.09681, 1701.04671]. 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 [2009.00874].

**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 [1311.2483, 1512.02303]. 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 [1512.02303].

**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 [2007.12378].

## 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 [2201.09681, 1307.2223].
- **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 [2511.17099]. 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 [1701.04671].

Variance reduction, sparsity exploitation (e.g., via compactly supported kernels and simplex constraints), adaptive sampling, and parallelization are standard computational techniques [2201.09681].

## 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 [1802.01841, 1512.02303].

Functional- and distribution-valued outputs—common in climate, econometric, and uncertainty quantification applications—necessitate dedicated functional ANOVA (domain-integrated or domain-selective) [2006.13850] and Wasserstein-based indices [2007.12378]. 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 [2007.12378].

## 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 [2406.05764]. 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 [2511.17099, 1708.07441].

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 [1311.2483].

## 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 [1911.12444]. 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 [1608.07794].

## 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 [2201.09681, 1307.2223]. For functional outputs, domain-selective testing and simultaneous inference are needed for localized sensitivity quantification [2006.13850]. For statistical models, careful choice of uncertainty measures (loss functions) and induced densities is critical to capture practical predictive relevance [1708.07441].

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 [1512.02303, 1311.2483]. For correlated inputs and outputs, classical Sobol indices may be ill-posed; nonparametric entropy estimators and kernel-based measures offer alternatives [1802.01841, 1311.2483].

Best practices dictate:
- Prestandardizing inputs/outputs when building GPs/PCEs [2201.09681].
- 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 [1708.07441, 2511.17099].

---

**Key Reference Papers Supporting This Article**
- "Quantifying dependencies for sensitivity analysis with multivariate input sample data" [1802.01841]
- "The $f$-Sensitivity Index" [1512.02303]
- "Multivariate Sensitivity Analysis of Electric Machine Efficiency Maps and Profiles Under Design Uncertainty" [2511.17099]
- "Multivariate sensitivity analysis for a large-scale climate impact and adaptation model" [2201.09681]
- "Global Sensitivity Analysis with Dependence Measures" [1311.2483]
- "Global sensitivity analysis and Wasserstein spaces" [2007.12378]
- "Integral equalities and inequalities: a proxy-measure for multivariate sensitivity analysis" [1911.12444]
- "A Bayesian approach for global sensitivity analysis of (multi-fidelity) computer codes" [1307.2223]
- "Global sensitivity analysis for statistical model parameters" [1708.07441]
- "Metamodel construction for sensitivity analysis" [1701.04671]
- "A simple algorithm for global sensitivity analysis with Shapley effects" [2009.00874]
- "Global Sensitivity and Domain-Selective Testing for Functional-Valued Responses: An Application to Climate Economy Models" [2006.13850]
- "Global Sensitivity Analysis of Uncertain Parameters in Bayesian Networks" [2406.05764]
- "An analytic method for sensitivity analysis of complex systems" [1608.07794]

Source: https://www.emergentmind.com/topics/multivariate-global-sensitivity-analysis