Overcoming global sensitivity limitations: using active subspaces to explore discrepancies between global and local parameter sensitivities

TL;DR

The paper addresses the discrepancy between global and local parameter sensitivities in complex models by introducing an active-subspace framework that uses the gradient-based matrix $C = ∫ ∇_x f(x) ∇_x f(x)^T ρ(x) dx$ and its decomposition $C = W Λ W^T$ to define active directions. It demonstrates how to compute and compare global versus local sensitivity information through activity scores and a subspace-distance metric, enabling identification of regions where global rankings are trustworthy. Through two-dimensional examples and a six-parameter Lotka-Volterra tumor model, the work reveals substantial misalignment between global and local sensitivities, with hundreds of local rankings diverging from the global one and directly impacting calibration and surrogate modeling. The findings provide a practical framework for region-aware sensitivity analysis and suggest using local subspaces to guide robust calibration and efficient, accurate surrogate construction in high-dimensional settings.

Abstract

Global sensitivity metrics are essential tools for assessing parameter importance in complex models, particularly when precise information about parameter values is unavailable. In many cases, such metrics are used to provide parameter rankings that allow for necessary dimension reduction in moderate-to-high dimensional systems. However, globally-derived sensitivity results may obscure localized variability in parameter sensitivities, resulting in misleading conclusions about parameter importance and ensuing consequences for subsequent tasks such as model calibration and surrogate model construction. In this study, we illustrate how discrepancies between globally- and locally-based sensitivity information may arise for an emerging sensitivity metric based on active subspace methodology, as well as for other commonly used sensitivity techniques. In response, we outline a framework that exploits the active subspace to evaluate the stability of parameter sensitivities over the admissible parameter space. This analysis allows one to determine the subregions of the parameter space in which a globally-derived sensitivity metric may be trustworthy. The proposed framework is illustrated on a collection of simple examples for ease of visualization, as well as the highly-applicable Lotka-Volterra model, for which we demonstrate how these issues may be exacerbated in higher dimensions. Our findings suggest that globally-derived sensitivity information should be treated with caution, and that incorporating analysis on local subregions may improve robustness and accuracy in downstream modeling tasks.

Figures (13)

• Figure 1: Visualization of locally-computed gradient vectors (white) versus a gradient vector computed from global sampling (black).
• Figure 2: Subspace distance visualization. For function $f_3$, the dashed curve indicates the separation of regions where $x_1$ is ranked first (above) and where $x_2$ is prioritized (below), according to locally-computed activity scores.
• Figure 3: Competitive Lotka–Volterra model trajectories with a 9:1 initial Type-$S$ : Type-$R$ ratio. Trajectories for Type-$S$ (blue, solid) and Type-$R$ (orange, dashed) cell volumes illustrate how different choices of growth rates, carrying capacities, and interaction strengths can impact the behavior of the system.
• Figure 4: Percentage of local regions in which each parameter is ranked among the top 1, top 2, or top 3 of the ranking list.
• Figure 5: Eigenvalues of the globally-computed matrix $\hat{\mathbf{C}}$ for the Lotka-Volterra model.