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arxiv: 2505.00711 · v5 · submitted 2025-03-30 · 🧮 math.ST · stat.TH

Global Activity Scores

Ruilong Yue , Giray \"Okten This is my paper

Pith reviewed 2026-05-22 21:48 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords global sensitivity analysisSobol indicesfinite differencesactivity scoresnoise robustnessvariable importancesensitivity measures
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The pith

Global activity scores based on finite differences identify influential variables more stably than derivative-based measures when additive noise is present.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces global activity scores as a new global sensitivity measure that relies on finite differences of the model function. It establishes a theoretical link between these scores and the well-known Sobol' sensitivity indices. Numerical comparisons show that the new scores maintain consistent identification of important variables even when the function outputs contain additive noise or exhibit high variability. Derivative-based measures and standard activity scores fluctuate more under the same conditions. This distinction matters for applications where data or model evaluations include unavoidable noise, as it supports more dependable ranking of input importance.

Core claim

Global activity scores provide a finite-difference based alternative to derivative-based sensitivity measures, with a theoretical link to Sobol' indices, and demonstrate greater stability in noisy conditions for ranking variable importance.

What carries the argument

Global activity scores, computed by aggregating finite differences of the function to quantify each input's overall influence.

If this is right

  • Global activity scores can serve as a direct substitute for derivative-based methods when model outputs contain additive noise.
  • In the absence of noise the three compared approaches produce similar variable rankings.
  • The finite-difference construction avoids the noise amplification that affects derivative estimates.
  • The established connection to Sobol' indices allows global activity scores to inherit some of their interpretability properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite-difference foundation may allow direct use of the scores on black-box simulators where analytic derivatives are unavailable.
  • Hybrid procedures that combine global activity scores with partial Sobol' computations could reduce overall cost while preserving robustness.
  • Extension to time-dependent or stochastic models would test whether the stability advantage persists beyond static functions.

Load-bearing premise

The numerical examples used to demonstrate superiority are representative of real-world functions with additive noise.

What would settle it

Run global activity scores alongside derivative-based measures on a standard test function with known influential variables, add independent Gaussian noise at several signal-to-noise ratios, and check whether the variable ranking produced by global activity scores stays consistent across repeated noise realizations while the derivative-based ranking changes.

Figures

Figures reproduced from arXiv: 2505.00711 by Giray \"Okten, Ruilong Yue.

Figure 1
Figure 1. Figure 1: Normalized cumulative sum of eigenvalues (left) and the first eigenvector [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sensitivity indices when k = 0, in Example 1 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Normalized cumulative sum of eigenvalues (left) and the first eigenvector [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sensitivity indices when k = 0.01, in Example 1 4.1.3. Moderate noise (k = 0.1) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normalized cumulative sum of eigenvalues (left) and the first eigenvector [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sensitivity indices when k = 0.1, in Example 1 DGSMs, and activity scores to analyze the model. Here we will apply the global activity score method to the power system problem and compare it with the other sensitivity measures. The output of interest in the sensitivity analysis is the steady state solution for the voltage at busbar 9. The inputs are the active power at each node i. We set up This manuscrip… view at source ↗
Figure 7
Figure 7. Figure 7: Normalized cumulative sum of eigenvalues (left) and the first eigenvector [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Sensitivity indices when k = 1, in Example 1 the differential equations as in Yue et al. [30], and then find the output by computing the load flow solution with Newton-Raphson method (Tinney and Hart [29]). We assume the ith input (i = 1, . . . , 14) follows the distribution N [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Sensitivity indices for power system report some of these results - specifically the no-noise case in Example 1 - in the paper). These findings highlight the robustness of global activity scores in challenging settings where other sensitivity measures may be misleading. However, we emphasize that global activity scores are not immune to noise, as demonstrated by the high noise setting of Example 1. Beyond … view at source ↗
Figure 10
Figure 10. Figure 10: Sensitivity indices for the example from Sobol’ and Kucherenko [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
read the original abstract

We introduce a new global sensitivity measure, the global activity scores. The measure is based on finite differences of the underlying function, in contrast to several sensitivity measures in the literature that are based on derivatives of the function. 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 additive noise or high variability, global activity scores provide more stable and reliable identification of influential variables than derivative-based measures and activity scores, which are more sensitive to noise. In noiseless settings, however, all three approaches yield comparable results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces global activity scores, a sensitivity measure based on finite differences of the model function. It claims a theoretical connection to Sobol' indices (in the noiseless case) and, via numerical examples, asserts that these scores yield more stable identification of influential inputs than derivative-based measures or activity scores when additive noise or high variability is present, while all methods perform comparably without noise.

Significance. If the claimed noise robustness generalizes beyond the specific examples, the measure could provide a practical finite-difference alternative for global sensitivity analysis in noisy settings, complementing variance-based indices.

major comments (2)
  1. [Numerical examples] Numerical examples (section containing the comparisons): the superiority claim under additive noise rests entirely on unspecified test functions, noise amplitudes, finite-difference step-size choices, and absence of error bars or statistical tests. Without these details it is impossible to judge whether the observed stability is general or an artifact of the chosen setups.
  2. [Theoretical connection] Theoretical connection (section establishing link to Sobol' indices): the connection is stated to hold only in the noiseless case, yet the central practical claim concerns noisy regimes where no supporting analysis (e.g., bias-variance trade-off for finite differences) is provided.
minor comments (1)
  1. [Abstract] The abstract should explicitly note that the noise-robustness result is empirical and limited to the presented examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below.

read point-by-point responses

  1. Referee: [Numerical examples] Numerical examples (section containing the comparisons): the superiority claim under additive noise rests entirely on unspecified test functions, noise amplitudes, finite-difference step-size choices, and absence of error bars or statistical tests. Without these details it is impossible to judge whether the observed stability is general or an artifact of the chosen setups.

    Authors: We agree that the numerical examples require more explicit detail for reproducibility and assessment. In the revised manuscript we will specify the test functions, noise amplitudes, finite-difference step sizes, and include error bars from repeated runs together with statistical tests. revision: yes

  2. Referee: [Theoretical connection] Theoretical connection (section establishing link to Sobol' indices): the connection is stated to hold only in the noiseless case, yet the central practical claim concerns noisy regimes where no supporting analysis (e.g., bias-variance trade-off for finite differences) is provided.

    Authors: The manuscript states that the theoretical connection holds only in the noiseless case. Claims of improved stability under noise are supported exclusively by the numerical examples. A bias-variance analysis for finite differences in noisy settings is a natural extension but is outside the scope of the present work, which focuses on introducing the measure and its empirical behavior. revision: no

Circularity Check

0 steps flagged

No circularity: new finite-difference measure defined independently with external theoretical link to Sobol' indices

full rationale

The paper defines global activity scores directly from finite differences of the function. It states a theoretical connection to Sobol' indices is established (abstract), which is presented as a derived property rather than an input. Performance comparisons under noise are shown via numerical examples, not by renaming fitted quantities or self-referential definitions. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems from the same authors are invoked as load-bearing steps. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the measure itself is presented as newly defined without listed assumptions beyond standard finite-difference and Sobol' background.

pith-pipeline@v0.9.0 · 5634 in / 1249 out tokens · 21533 ms · 2026-05-22T21:48:12.416693+00:00 · methodology

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Reference graph

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