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Global sensitivity metrics from active subspaces
Paul G. Constantine, Paul Diaz
TL;DR
Global sensitivity analysis seeks important parameters in complex models, while active subspaces identify important input directions for dimension reduction. The paper develops activity scores from active subspaces, relates them to standard metrics, and evaluates them on two engineering models, finding consistent parameter rankings. Its scope excludes claims of computational superiority and includes limitations from Monte Carlo eigenpair estimation and from cases without a distinguished active subspace.
Problem
Complex models make it difficult to identify important parameters, while active-subspace directions do not directly provide parameter-level insight.
Method
The paper defines activity scores from active subspaces, relates them mathematically to Sobol’ total indices and derivative-based measures, and studies quadrature and Monte Carlo estimators.
Results
In two simplified engineering models, activity scores and standard metrics identify the same important and unimportant variables.
Takeaways & Limitations
Activity scores provide parameter rankings that behave similarly to standard sensitivity metrics while deriving from active-subspace structure.
Takeaways & Limitations
The paper does not claim computational advantage, and Monte Carlo activity-score estimates require bootstrap standard errors because CLT-based errors are unavailable for estimated eigenpairs.
Abstract
from arXiv · showhide
Predictions from science and engineering models depend on several input parameters. Global sensitivity analysis quantifies the importance of each input parameter, which can lead to insight into the model and reduced computational cost; commonly used sensitivity metrics include Sobol' total sensitivity indices and derivative-based global sensitivity measures. Active subspaces are an emerging set of tools for identifying important directions in a model's input parameter space; these directions can be exploited to reduce the model's dimension enabling otherwise infeasible parameter studies. In this paper, we develop global sensitivity metrics called activity scores from the active subspace, which yield insight into the important model parameters. We mathematically relate the activity scores to established sensitivity metrics, and we discuss computational methods to estimate the activity scores. We show two numerical examples with algebraic functions taken from simplified engineering models. For each model, we analyze the active subspace and discuss how to exploit the low-dimensional structure. We then show that input rankings produced by the activity scores are consistent with rankings produced by the standard metrics.
1. Introduction
The paper addresses how to extract parameter-level insight from active-subspace dimension reduction by proposing activity scores and relating them to established sensitivity metrics. It develops estimation procedures and compares the metrics on simplified engineering models.
- Motivation: Global sensitivity analysis identifies important parameters, potentially reducing the cost of optimization and uncertainty-quantification studies.The paper distinguishes global metrics from local metrics that measure response near a nominal parameter value.
- Active-subspace context: Active subspaces extend parameter-space reduction from individual coordinates to important directions formed by linear combinations of inputs.These directions define active variables and can support reduced-dimensional parameter studies.
- Contribution: Activity scores are proposed to recover insight into original parameters after identifying an active subspace.The paper mathematically relates them to Sobol’ total sensitivity indices and derivative-based global sensitivity measures.
- Contribution: The paper discusses tensor-product Gauss quadrature and Monte Carlo procedures for estimating activity scores.Bootstrap-based standard errors are proposed for Monte Carlo estimates.
- Evaluation: Two simplified engineering models—a seven-parameter piston-cycle model and a six-parameter transformerless-circuit model—are used for comparisons.The paper does not claim that activity scores are superior to existing metrics in sensitivity characterization or computational cost.
2. Global sensitivity metrics
The paper reviews variance-based, derivative-based, and regression-based global sensitivity metrics, including their definitions, computation, and error estimation. These metrics capture different aspects of model behavior and use quadrature or Monte Carlo methods under different conditions.
- Variance-based metrics: Variance-based total sensitivity indices decompose output variance and attribute contributions to subsets containing each input variable.The paper uses τ_i for the total sensitivity index and notes alternative notations such as ST_i.
- Variance-based metrics: Tensor-product Gauss-Legendre quadrature provides accurate total-index estimates when dimension is low, the function is smooth, and evaluations are inexpensive.Otherwise, the reviewed Jansen Monte Carlo method uses sampled function evaluations, with bootstrap standard errors for general nonlinear functions.
- Derivative-based metrics: Derivative-based global sensitivity measures average local derivative information over the input domain to obtain a global metric.They can be estimated by high-order quadrature for sufficiently smooth low-dimensional functions or by Monte Carlo sampling.
- Regression-based metrics: Standardized regression coefficients provide signed sensitivity measures from a least-squares linear approximation.Under smoothness and monotonicity, their signs indicate whether the output increases or decreases with an input, unlike the unsigned total and derivative-based measures.
- Regression-based metrics: Bootstrap standard errors quantify Monte Carlo uncertainty for regression coefficients when reference values are too expensive to compute.The randomness comes from sampling rather than from evaluating the deterministic function.
3. Active subspaces
An active subspace is obtained from the eigenvectors of a gradient-based, symmetric positive semidefinite matrix, with eigenvalue gaps indicating useful dimension reduction. The resulting structure can approximate the model with fewer active directions, but such structure need not exist for every function.
- Definition: The active subspace is defined by the eigenvectors of an m × m symmetric positive semidefinite matrix.Its eigenvalues and eigenvectors are functionals of the model, analogous to established sensitivity metrics.
- Structure: If λ_i = 0, the function is constant along eigenvector direction w_i; a gap λ_n > λ_n+1 supports partitioning active and inactive directions.The n-dimensional active subspace is the column span of the first n eigenvectors.
- Limitation: Not every square-integrable differentiable function has a useful active subspace; for f(x) = ∥x∥2 under a uniform density, all eigenvalues are equal.Equal eigenvalues provide no distinguished active direction for selecting a lower-dimensional subspace.
- Dimension reduction: When trailing eigenvalues are sufficiently small, the model admits a low-dimensional approximation through a function of the leading active variables.The approximation error can be bounded in terms of the neglected eigenvalues.
- Computation: Monte Carlo estimation of the active-subspace matrix samples inputs, evaluates gradients, and computes the estimated eigenpairs.The estimated eigenvalues and subspace distance are used to assess approximation quality.
- Dimension selection: A large eigenvalue gap λ_n − λ_n+1 supports accurate Monte Carlo estimation and motivates choosing n at the largest eigenvalue gap.This is presented as a practical heuristic for selecting active-subspace dimension.
4. Sensitivity metrics from active subspaces
The paper proposes two sensitivity metrics from active-subspace eigenvectors and eigenvalues, including activity scores that rank input importance. These metrics relate to established derivative-based and variance-based measures, but their agreement depends on an appropriate active-subspace dimension and eigenvalue gap.
- Proposed metrics: The paper proposes two metrics from active-subspace eigenvectors and eigenvalues: first-eigenvector components and eigenvalue-weighted activity scores.The activity scores combine squared eigenvector components with eigenvalues.
- Activity scores: Activity scores rank model inputs by combining contributions from the active-subspace eigenpairs.The paper calls αi(n) the activity score for parameter i and uses these values to rank input importance.
- Relations to existing metrics: The activity score is a truncated derivative-based metric, and it is bounded above by the corresponding derivative-based global sensitivity measure.The bound becomes an equality when n = m.
- Comparison and scope: Activity scores are most appropriate when an eigenvalue gap supports a genuinely low-dimensional active subspace.Without such a gap, selecting a lower-dimensional active subspace can misidentify parameter importance.
- Relations to existing metrics: If λn+1 and αi(n) are small, the corresponding total sensitivity index is also bounded small.The bound is τi ≤ 1/(4π^2) V (αi(n) + λn+1).
- Comparison and scope: Activity-score rankings can differ from derivative-based or total-sensitivity rankings because the metrics measure different characteristics of the model.The paper reports consistency for many practical engineering examples but also constructs functions with different rankings.
5. Numerical experiments
Numerical experiments on piston and circuit models evaluate active-subspace sensitivity metrics against standard metrics, examining low-dimensional structure, estimation error, and parameter rankings. The active-subspace metrics show consistent rankings across both models, while convergence and model-specific summary plots indicate how the reduced structure may be exploited.
- Experimental design: The experiments compare Sobol’ total indices, derivative-based measures, linear coefficients, first eigenvector components, and activity scores for piston and circuit models.The piston model has 7 physical inputs, while the circuit model has 6.
- Experimental design: The parameter spaces are normalized to the [−1, 1] hypercube, so sensitivity metrics and active-subspace computations depend on the specified parameter ranges.Partial derivatives are scaled appropriately for the normalized model.
- Experimental design: Active subspaces were computed using 7-point-per-dimension Gauss-Legendre quadrature and eigenpairs of the resulting matrix estimate.The quadrature-based eigenvalues were found accurate to 10 digits, using 7^m total points.
- Piston model: The piston model shows active-subspace evidence for dimensions 1 to 6; its first active variable has a gross monotonic trend but substantial spread, while two variables capture curvature better.The spread likely reflects output changes from perturbations orthogonal to the first eigenvector.
- Estimation error: Monte Carlo errors for both models decrease at the expected M −1/2 rate, while total-index errors are larger because the fixed evaluation budget allocates fewer effective samples.For the piston model, errors were averaged over 10 independent trials; the circuit model shows the same convergence pattern.
- Circuit model: The circuit model shows active-subspace evidence for dimensions 1 to 5, with a tight, nearly linear relationship between circuit voltage and the first active variable.Its two-dimensional summary plot has nearly linear contours, supporting a univariate linear approximation more strongly than in the piston model.
- Ranking comparison: The five metrics produce consistent input-importance rankings for both models, although piston parameters M and k swap rankings as active-subspace dimension increases from 1 to 2.Activity scores converge quickly as spectral decay weights higher-dimensional eigenvector components.
6. Summary and remarks
The paper introduces active-subspace sensitivity metrics and finds that, in two engineering models, they rank important and unimportant variables like standard metrics without claiming computational advantage.
- The study proposes eigenvector components and activity scores as global sensitivity metrics derived from active-subspace eigenvectors and eigenvalues.Activity scores combine squared eigenvector components with eigenvalues.
- Across two test models, all metrics identify the same important and unimportant variables.The authors note that different rankings can occur because the metrics measure different model characteristics.
- The paper does not claim that the proposed metrics reduce computational cost.For a fixed number of samples, Monte Carlo errors are comparable, particularly between activity scores and derivative-based measures.
- Estimating active subspaces, like derivative-based measures, requires access to gradients unless alternative gradient-free approaches are used.Gradient-free sensitivity metrics remain a proposed direction for future work.