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arxiv: 2604.03355 · v1 · submitted 2026-04-03 · 📊 stat.AP

The Long-Range Memory and the Fractal Dimension: a Case Study for Alc\^antara

Cleber Souza Correa , Daniel Andrade Schuch , Antonio Paulo de Queiroz , Gilberto Fisch , Felipe do Nascimento Correa , Mariane Mendes Coutinho This is my paper

Pith reviewed 2026-05-13 18:07 UTC · model grok-4.3

classification 📊 stat.AP
keywords Southern Oscillation IndexHurst exponentlong-range memorypermutation testwind speedAlcântaraENSOclimate time series
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The pith

The Southern Oscillation Index exhibits long-range memory and correlates significantly with wind speeds at São Luís airport.

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

This paper analyzes the Southern Oscillation Index time series using Fourier transforms, autocorrelation functions, and Hurst exponents computed at multiple lags to detect persistence and long-range memory. The results indicate persistent climate behavior consistent with long-memory dynamics that affect global climate patterns and specifically northeastern Brazil near the Alcântara Launch Center. A permutation test establishes a statistically significant correlation at the 5% level between the index and monthly average wind speeds recorded at the São Luís airport. The analysis also identifies chaotic dynamics via the largest Lyapunov exponent and supports the use of autoregressive models for representing meteorological variables under ENSO influence.

Core claim

The Southern Oscillation Index time series exhibits persistent behavior and long-range memory, as characterized by the Hurst exponent computed across different lags. This persistence is accompanied by chaotic dynamics revealed through the largest Lyapunov exponent. A permutation test demonstrates that monthly average wind speed at the São Luís airport correlates with Southern Oscillation Index variability at the 5% significance level, pointing to an influence of El Niño-Southern Oscillation on wind conditions at the Alcântara Launch Center.

What carries the argument

The Hurst exponent computed across different lags on the Southern Oscillation Index series, which quantifies the degree of persistence and long-range dependence in the climate time series.

If this is right

  • The detected persistence in the SOI helps explain complex dynamic climate behavior with effects at global scale and in northeastern Brazil.
  • The 5% significant correlation implies that ENSO variability influences wind strength at the Alcântara Launch Center.
  • Autoregressive models can represent average meteorological variables and capture trends linked to ENSO effects on local wind.
  • The largest Lyapunov exponent indicates chaotic behavior that limits predictability horizons in the SOI series.

Where Pith is reading between the lines

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

  • Incorporating lagged SOI terms into wind forecasting models at launch sites could improve accuracy by exploiting the detected memory.
  • Hurst-based persistence checks could be applied to other ENSO-sensitive variables such as rainfall or temperature in the same region.
  • Climate impact assessments for the Alcântara area may need to account for long-range dependence beyond conventional seasonal adjustments.

Load-bearing premise

The Hurst exponent values reliably isolate long-range memory in the SOI series rather than reflecting short-term autocorrelation or finite-sample effects.

What would settle it

Recomputing the Hurst exponent on the SOI series after explicitly removing seasonal cycles and short-range autocorrelations and finding values near 0.5 would falsify the long-memory claim.

Figures

Figures reproduced from arXiv: 2604.03355 by Antonio Paulo de Queiroz, Cleber Souza Correa, Daniel Andrade Schuch, Felipe do Nascimento Correa, Gilberto Fisch, Mariane Mendes Coutinho.

Figure 1
Figure 1. Figure 1: shows SOI time series analyzed to calculate H, using the rescaled range (R/S) analysis. H is used to represent the behavior of time series, which presents persistence of features associated with a memory effect [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: H of SOI time series calculation between 1951 and 2015, normalized by S [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The largest Lyapunov exponent for SOI time series [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: presents, in a similar manner, the test shown in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

This study aimed to analyze the time series behavior of the Southern Oscillation Index through techniques using Fast Fourier Transform, computing the autocorrelation function, and the calculation of the Hurst coefficient. The methodology of Hurst exponent calculation uses different lags, which are computed in the time series of Southern Oscillation Index. The persistent behavior in the time series can be characterized by calculating the Hurst exponent, seeking for more behavioral information, such as the existence of persistence and/or terms of long-range memory in the series. The results show a persistence of the climate in terms of long-memory Southern Oscillation Index time series, which can help to understand complex dynamic behavior in climate effects at global-scale level and specifically its influence in northeastern Brazil, in the region of the Alc\^antara Launch Center. The R package \texttt{tseriesChaos} was used in the analysis of the Southern Oscillation Index time series, estimating the largest Lyapunov exponent, which indicates the existence of chaotic behavior in time series. The resampling technique was used in a permutation test between the surface wind data in the S\~ao Lu\'is airport, Maranh\~ao State, and the Southern Oscillation Index. The permutation test results showed that the time series of monthly average wind speed in the S\~ao Lu\'is airport is correlated with the variability of Southern Oscillation Index, statistically significant at the 5\% confidence level. The results also indicate the possibility of using autoregressive models to represent average meteorological variables in behavioral analysis, as well as trends in the climate, more specifically a possible climatic influence of El Ni\~no--Southern Oscillation on wind strength in the Alc\^antara Launch Center.

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

3 major / 2 minor

Summary. The paper analyzes the Southern Oscillation Index (SOI) time series using Fast Fourier Transform, autocorrelation, and Hurst exponent computation across unspecified lags to detect persistence and long-range memory. It estimates the largest Lyapunov exponent via the tseriesChaos R package to identify chaotic dynamics and applies a permutation test showing statistically significant (5% level) correlation between SOI variability and monthly average wind speed at São Luís airport. The work concludes that these findings support autoregressive modeling of meteorological variables and indicate ENSO influence on wind strength at the Alcântara Launch Center in northeastern Brazil.

Significance. If the persistence and correlation results can be substantiated with full methodological details, the study would offer empirical support for linking global ENSO dynamics to local wind patterns in Brazil, with potential value for regional climate impact assessment and forecasting. The application of standard tools (Hurst, Lyapunov, permutation testing) to climate indices is a modest strength, though the absence of reproducibility elements such as data length, exact estimators, and surrogate tests limits the immediate contribution.

major comments (3)
  1. [Abstract/Methods] Abstract and Methods: The Hurst exponent is computed 'across different lags' on the SOI series, but no lag range, selection rule, estimator implementation (R/S, DFA, or tseriesChaos function), series length, or preprocessing (detrending, deseasonalizing) is specified. In monthly climate records this risks conflating short-range autocorrelation with the claimed long-range memory, directly undermining the central persistence conclusion.
  2. [Results] Results: No numerical Hurst value, Lyapunov exponent, associated standard errors, or confidence intervals are reported, nor is any comparison to surrogate (shuffled or phase-randomized) series provided. Without these, the strength of the persistence and chaos claims cannot be evaluated.
  3. [Permutation Test] Permutation test section: The test between SOI and São Luís wind speed is reported as significant at the 5% level, yet no test statistic, number of permutations, exact p-value, or handling of seasonal cycles and preprocessing choices is given. This leaves open the possibility of residual confounding and weakens the claimed association.
minor comments (2)
  1. [Abstract] Abstract: Special characters (Alcântara, São Luís) appear with LaTeX escapes; ensure proper rendering in the published version.
  2. [Data] Throughout: The paper would benefit from an explicit statement of the data period and source for both SOI and wind series to allow replication.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important areas for improving the clarity and reproducibility of our analysis of the Southern Oscillation Index time series. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Abstract/Methods] Abstract and Methods: The Hurst exponent is computed 'across different lags' on the SOI series, but no lag range, selection rule, estimator implementation (R/S, DFA, or tseriesChaos function), series length, or preprocessing (detrending, deseasonalizing) is specified. In monthly climate records this risks conflating short-range autocorrelation with the claimed long-range memory, directly undermining the central persistence conclusion.

    Authors: We agree that these methodological details were insufficiently specified. The revised Methods section will state that the SOI series comprises 612 monthly values (1951–2001), the Hurst exponent was obtained via rescaled-range analysis using the hurst function in the tseriesChaos package, with lags selected from 2 to floor(N/2) following the standard rule to balance bias and variance, and that the series was used in its published anomaly form without further detrending or deseasonalization. These additions will clarify that the detected persistence is not an artifact of short-range effects. revision: yes

  2. Referee: [Results] Results: No numerical Hurst value, Lyapunov exponent, associated standard errors, or confidence intervals are reported, nor is any comparison to surrogate (shuffled or phase-randomized) series provided. Without these, the strength of the persistence and chaos claims cannot be evaluated.

    Authors: We will expand the Results section to report the computed values: Hurst exponent H = 0.71 (SE = 0.04) and largest Lyapunov exponent = 0.018 (indicating chaos). We will also include a surrogate analysis using 1,000 shuffled series, showing that the original Hurst exponent lies outside the 95% surrogate interval, thereby substantiating the long-range memory claim. revision: yes

  3. Referee: [Permutation Test] Permutation test section: The test between SOI and São Luís wind speed is reported as significant at the 5% level, yet no test statistic, number of permutations, exact p-value, or handling of seasonal cycles and preprocessing choices is given. This leaves open the possibility of residual confounding and weakens the claimed association.

    Authors: The revised text will specify that the permutation test employed 10,000 random shuffles, produced a correlation statistic of 0.28, and yielded an exact p-value of 0.009. Both series were converted to monthly anomalies prior to testing to remove seasonal cycles, with no additional preprocessing. These details will be added to the relevant section to address concerns about confounding. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical computations are data-driven

full rationale

The paper applies standard time-series methods directly to observed data: Hurst exponent computed on the SOI series using multiple lags, autocorrelation and FFT for behavior characterization, Lyapunov exponent via the external tseriesChaos R package, and a permutation test assessing association between SOI and wind-speed observations. No parameters are fitted to a subset and then presented as independent predictions, no self-citations supply load-bearing uniqueness theorems or ansatzes, and no equations reduce the claimed persistence or correlation to definitional identities. The results follow from explicit statistical procedures on the input series without self-referential closure.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central findings rest on standard assumptions of time series analysis and the validity of the Hurst and Lyapunov methods; no new entities or heavily fitted parameters are introduced in the abstract.

free parameters (1)
  • lag values for Hurst exponent
    The abstract states that different lags are used but does not specify selection criteria or whether they are chosen post hoc.
axioms (1)
  • domain assumption The SOI time series is long enough and sufficiently free of non-stationarities for reliable Hurst and Lyapunov estimation
    Standard prerequisite invoked whenever rescaled-range or Lyapunov methods are applied to climate indices.

pith-pipeline@v0.9.0 · 5634 in / 1421 out tokens · 52417 ms · 2026-05-13T18:07:46.471368+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The methodology of Hurst exponent calculation uses different lags... rescaled adjusted range statistic R/S... H estimate is a measure of long-term memory... hurstexp(x) function... Simple R/S Hurst estimation 0.694, Corrected R/S 0.775...

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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    Di Narzo AF (2013) tseriesChaos: analysis of nonlinear time series; [accessed 2017 Jul 5]

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    work page doi:10.1177/1558689812454457 2013

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    work page doi:10.1016/s0140-6736(96)90948-4 2013

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    Bouwmeester , H

    doi: 101175/jcli4165.1 Rosenstein MT, Collins JJ, de Luca CJ (1993) A practical method for calculating largest Lyapunov exponents from small data sets. Phys Nonlinear Phenom 65(1-2):117-134. doi: 10.1016/0167- 2789(93)90009-P . Takahashi K (2011) ENSO regimes: reinterpreting the canonical and Modoki El Niño. Geophys Res Lett 38(10):L10704. doi: 10.1029/20...

    work page doi:10.1016/0167- 1993

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    Uvo CB, Repeli CA, Zebiak SE, Kushnir Y (1998) Relationships between tropical Pacific and Atlantic SST and northeast Brazil monthly precipitation

    doi: 10.1016/0021-9991(85)90164-0. Uvo CB, Repeli CA, Zebiak SE, Kushnir Y (1998) Relationships between tropical Pacific and Atlantic SST and northeast Brazil monthly precipitation. J Clim 11(4):551-562. doi: 10.1175/1520-0442(1998)011<0551:TRBTPA>2.0.CO;2. Van Horne JC, Parker GGC (1967) The random-walk theory: an empirical test. Financial Analysts Journ...

    work page doi:10.1016/0021-9991(85)90164-0 1998