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Global Activity Scores

Ruilong Yue, Giray Ökten

TL;DR

This work introduces global activity scores (GA scores), a derivative-free global sensitivity measure based on first-order finite differences, to address limitations of gradient-dependent active-subspace methods in noisy or non-smooth models. The authors establish theoretical links between GA scores and Sobol' indices, and compare GA scores with Sobol' indices, DGSM, and traditional activity scores across several examples, showing that GA scores are more robust to noise and nonlinearity, and often align with upper Sobol' indices. In noiseless settings, GA scores recover results similar to existing measures, but in challenging scenarios they offer improved interpretability and reliability. The study also discusses well-posedness, computational costs, and potential extensions to dependent inputs through mapping approaches or Shapley-value frameworks, highlighting GA scores as a practical addition to global sensitivity analysis toolbox.

Abstract

We introduce a new global sensitivity measure, the global activity scores. We establish its theoretical connection with Sobol' sensitivity indices and demonstrate its performance through numerical examples. In these examples, we compare global activity scores with Sobol' sensitivity indices, derivative-based sensitivity measures, and activity scores. The results show that in the presence of noise or high variability, global activity scores outperform derivative-based measures and activity scores, while in noiseless settings the three approaches yield similar results.

Global Activity Scores

TL;DR

This work introduces global activity scores (GA scores), a derivative-free global sensitivity measure based on first-order finite differences, to address limitations of gradient-dependent active-subspace methods in noisy or non-smooth models. The authors establish theoretical links between GA scores and Sobol' indices, and compare GA scores with Sobol' indices, DGSM, and traditional activity scores across several examples, showing that GA scores are more robust to noise and nonlinearity, and often align with upper Sobol' indices. In noiseless settings, GA scores recover results similar to existing measures, but in challenging scenarios they offer improved interpretability and reliability. The study also discusses well-posedness, computational costs, and potential extensions to dependent inputs through mapping approaches or Shapley-value frameworks, highlighting GA scores as a practical addition to global sensitivity analysis toolbox.

Abstract

We introduce a new global sensitivity measure, the global activity scores. We establish its theoretical connection with Sobol' sensitivity indices and demonstrate its performance through numerical examples. In these examples, we compare global activity scores with Sobol' sensitivity indices, derivative-based sensitivity measures, and activity scores. The results show that in the presence of noise or high variability, global activity scores outperform derivative-based measures and activity scores, while in noiseless settings the three approaches yield similar results.
Paper Structure (17 sections, 7 theorems, 70 equations, 10 figures, 4 tables)

This paper contains 17 sections, 7 theorems, 70 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Global Sensitivity Measures
  3. Sobol' Sensitivity Indices
  4. Derivative-based sensitivity measures
  5. Activity scores
  6. Global Activity Scores
  7. Numerical Results
  8. Example 1: A test function with noise
  9. No noise ($k=0$)
  10. Low noise ($k=0.01$)
  11. Moderate noise ($k=0.1$)
  12. High noise ($k=1$)
  13. Example 2: IEEE 14-bus power system
  14. Example 3: A counterexample from Sobol' and Kucherenko
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 3.2

\newlabelgas10 If $\Omega =(0,1)^d$, endowed with the uniform probability measure, then for $i=1,\ldots,d$.

Figures (10)

  • Figure 1: Normalized cumulative sum of eigenvalues (left) and the first eigenvector (right) of $\pmb C_\text{as}$ and $\pmb C_\text{gas}$, when $k=0$, in Example 1
  • Figure 2: Sensitivity indices when $k=0$, in Example 1
  • Figure 3: Normalized cumulative sum of eigenvalues (left) and the first eigenvector (right) of $\pmb C_\text{as}$ and $\pmb C_\text{gas}$, when $k=0.01$, in Example 1
  • Figure 4: Sensitivity indices when $k=0.01$, in Example 1
  • Figure 5: Normalized cumulative sum of eigenvalues (left) and the first eigenvector (right) of $\pmb C_\text{as}$ and $\pmb C_\text{gas}$, when $k=0.1$, in Example 1
  • ...and 5 more figures

Theorems & Definitions (19)

  • Definition 3.1
  • Theorem 3.2
  • Proof 1
  • Corollary 3.3
  • Proof 2
  • Theorem 3.4
  • Proof 3
  • Remark 3.5
  • Corollary 3.6
  • Proof 4
  • ...and 9 more