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arxiv: 2505.19034 · v2 · submitted 2025-05-25 · 🌊 nlin.AO · nlin.CD· physics.data-an· physics.geo-ph

The influence of data gaps and outliers on resilience indicators

Teng Liu , Andreas Morr , Sebastian Bathiany , Lana L. Blaschke , Zhen Qian , Chan Diao , Taylor Smith , Niklas Boers This is my paper

Pith reviewed 2026-05-19 13:51 UTC · model grok-4.3

classification 🌊 nlin.AO nlin.CDphysics.data-anphysics.geo-ph
keywords resilience indicatorscritical slowing downdata gapsoutlierstemporal autocorrelationvariancetime series analysisEarth system resilience
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The pith

The agreement between variance- and autocorrelation-based resilience indicators is driven by the time series' initial data point.

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

This paper examines how missing values and outliers affect indicators that track resilience through variance and autocorrelation in time series data. It reveals that the two indicators agree largely because of the starting value in the record rather than deeper system properties. Missing values reduce how consistently the indicators align, and outliers systematically push autocorrelation measures to suggest higher resilience than may be present. These effects matter for efforts to detect critical slowing down before abrupt shifts in Earth systems such as climate or ecosystems. The work supplies a basis for better data preprocessing and for judging how trustworthy the indicators are when applied to imperfect real-world records.

Core claim

The paper establishes a rigorous mathematical analysis showing that the statistical dependency between variance-based and autocorrelation-based resilience indicators is fundamentally driven by the time series' initial data point. Using both synthetic and empirical data, it demonstrates that missing values substantially weaken indicator agreement while outliers introduce systematic biases that lead to overestimation of resilience based on temporal autocorrelation.

What carries the argument

The statistical dependency between variance- and autocorrelation-based resilience indicators, shown to be driven by the initial data point of the time series.

If this is right

  • Preprocessing must address missing values to preserve agreement between the indicators.
  • Outliers produce a consistent upward bias in resilience estimates drawn from autocorrelation.
  • Accuracy checks for resilience inferences from real data must incorporate effects of gaps and outliers.
  • The results supply a foundation for improved preprocessing across fields that apply these indicators to observational records.

Where Pith is reading between the lines

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

  • Applying the same checks to other early-warning methods that rely on similar statistics could expose comparable sensitivities to data quality.
  • Testing the initial-point dependence on longer or multivariate records might clarify how the effect scales with series length.
  • These biases could be mitigated by explicit correction terms derived from the initial value, offering a direct way to adjust existing indicator calculations.

Load-bearing premise

The analysis assumes the underlying time series are generated from processes whose statistical properties such as stationarity and noise structure are known.

What would settle it

A collection of time series in which indicator agreement shows no dependence on the initial data point, or in which missing values do not reduce agreement, would challenge the central claims.

read the original abstract

The resilience, or stability, of major Earth system components is increasingly threatened by anthropogenic pressures, demanding reliable early warning signals for abrupt and irreversible regime shifts. Widely used data-driven resilience indicators based on variance and autocorrelation detect `critical slowing down', a signature of decreasing stability. However, the interpretation of these indicators is hampered by poorly understood interdependencies and their susceptibility to common data issues such as missing values and outliers. Here, we establish a rigorous mathematical analysis of the statistical dependency between variance- and autocorrelation-based resilience indicators, revealing that their agreement is fundamentally driven by the time series' initial data point. Using synthetic and empirical data, we demonstrate that missing values substantially weaken indicator agreement, while outliers introduce systematic biases that lead to overestimation of resilience based on temporal autocorrelation. Our results provide a necessary and rigorous foundation for preprocessing strategies and accuracy assessments across the growing number of disciplines that use real-world data to infer changes in system resilience.

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 / 2 minor

Summary. The manuscript derives algebraically that agreement between variance- and autocorrelation-based resilience indicators is fundamentally driven by the time series initial data point. Using synthetic series from stationary processes and empirical records from public datasets, it shows that missing values weaken indicator agreement while outliers systematically bias autocorrelation-based estimates upward, overestimating resilience. The results are offered as a foundation for preprocessing strategies and accuracy assessments when applying critical slowing down indicators to imperfect real-world data.

Significance. If the algebraic dependency is general and the quantified effects of gaps and outliers prove robust, the work supplies a needed rigorous basis for interpreting interdependencies among widely used resilience indicators. The clean algebraic demonstration of the initial-point dependency and the reliance on standard public datasets are explicit strengths. The findings would improve reliability of early-warning applications in Earth-system science and related fields that routinely encounter data gaps and outliers.

major comments (2)
  1. [§4.1] §4.1 (Synthetic Data Experiments): The reported magnitudes of weakened agreement due to missing values and the upward bias in autocorrelation from outliers are quantified exclusively on AR(1)-type processes with fixed stationarity and noise parameters. Because the skeptic concern is valid—the true data-generating process may contain unmodeled trends, long-memory correlations, or state-dependent noise—the direction and size of these effects could change, weakening support for the advocated preprocessing strategies.
  2. [§5] §5 (Empirical Illustrations): The interpretation of the empirical records assumes the same statistical properties used to generate the synthetic data. Without explicit sensitivity checks or alternative process classes, the claim that the observed weakening and overestimation generalize remains conditional on those assumptions.
minor comments (2)
  1. Abstract: the phrase 'rigorous mathematical analysis' would be strengthened by a one-sentence reference to the specific initial-point dependency that is derived.
  2. [Figures 2-3] Figure 2 and 3 captions: variability across realizations or explicit error bars on the reported bias magnitudes should be shown to allow readers to judge robustness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive suggestions. We address the major comments point by point below, providing clarifications on the generality of our algebraic results and indicating revisions to enhance robustness.

read point-by-point responses

  1. Referee: [§4.1] §4.1 (Synthetic Data Experiments): The reported magnitudes of weakened agreement due to missing values and the upward bias in autocorrelation from outliers are quantified exclusively on AR(1)-type processes with fixed stationarity and noise parameters. Because the skeptic concern is valid—the true data-generating process may contain unmodeled trends, long-memory correlations, or state-dependent noise—the direction and size of these effects could change, weakening support for the advocated preprocessing strategies.

    Authors: The algebraic derivation establishing that indicator agreement is driven by the initial data point is independent of the underlying data-generating process and applies to any time series. Our synthetic experiments employ AR(1) processes because they represent the standard linear approximation for critical slowing down near tipping points, which is the regime where resilience indicators are most relevant. We agree that exploring other processes could provide additional insight into the magnitude of effects. In the revised manuscript, we will add a sensitivity analysis using an ARMA(1,1) process and a brief discussion of how trends might interact with the observed biases. revision: partial

  2. Referee: [§5] §5 (Empirical Illustrations): The interpretation of the empirical records assumes the same statistical properties used to generate the synthetic data. Without explicit sensitivity checks or alternative process classes, the claim that the observed weakening and overestimation generalize remains conditional on those assumptions.

    Authors: The empirical section uses real-world datasets from public repositories that are representative of applications in Earth system science. The weakening of agreement due to gaps and the upward bias from outliers are demonstrated directly on these records, consistent with the general algebraic framework. To strengthen the generalization, we will incorporate sensitivity checks by repeating the analysis on detrended versions and on subsets of the data to assess robustness to potential non-stationarities. revision: yes

Circularity Check

0 steps flagged

Core mathematical result is a direct algebraic derivation independent of inputs or self-citations

full rationale

The paper's central claim rests on a rigorous mathematical analysis establishing that agreement between variance- and autocorrelation-based indicators is driven by the initial data point. This follows directly from the statistical definitions of the indicators without any reduction to fitted parameters, self-citations, or ansatzes. Subsequent demonstrations on synthetic data (generated under explicit stationarity assumptions) and empirical records use standard public datasets and do not close the argument via author-overlapping citations or renaming of known results. No load-bearing step in the derivation chain is equivalent to its inputs by construction, making the analysis self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on standard assumptions about time-series stationarity and noise models for the synthetic experiments, plus the choice of specific gap and outlier insertion mechanisms; no new physical entities are postulated.

free parameters (1)
  • synthetic process parameters (e.g., noise amplitude, autocorrelation length)
    Chosen to generate representative critical-slowing-down trajectories; values are not fitted to the target resilience result but define the testbed.
axioms (1)
  • domain assumption The time series are realizations of a stochastic process whose variance and lag-1 autocorrelation can be computed directly from the data without additional model fitting.
    Invoked when defining the resilience indicators and when deriving their statistical dependence.

pith-pipeline@v0.9.0 · 5715 in / 1351 out tokens · 49838 ms · 2026-05-19T13:51:48.281384+00:00 · methodology

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Reference graph

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