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Noise-Induced Predictability Redistribution Across Forecast Horizons of Extreme Events in Chaotic Dynamics

Andrei Velichko, Viet-Thanh Pham

arXiv:2609.27877v1nlin.CDcs.LGphysics.data-an

TL;DR

This study examines how dynamical noise changes both broad extreme-event occurrence and forecast skill across lead times. Using paired noise realizations and a positive-count-matched control, it finds short-horizon gains at intermediate noise but reduced skill at strong noise farther ahead, a pattern termed noise-induced predictability redistribution.

  • Problem

    For broad extreme events, it has not been systematically resolved whether noise changes event occurrence as well as the distribution of forecast skill across lead times.

    Source: §1 · p. 3

  • Method

    The study evaluates broad-event forecasts across noise amplitudes and forecast gaps using ten paired realizations and a positive-training-count-matched control.

    Source: §1 · p. 4, §2.7 · p. 8

  • Results

    The short-horizon score rises from 0.456 in paired clean data to 0.546 at σ = 7 × 10−3, a paired gain of 0.0895 (95% CI 0.0494–0.1295; Holm-adjusted p = 0.0234), while strong noise reduces matched intermediate-horizon skill.

    Source: §5 · p. 21

  • Takeaways & Limitations

    The authors describe the nonuniform, horizon-dependent change in forecast skill as noise-induced predictability redistribution (NIPR).

    Source: §2.7 · p. 9, §4 · p. 19

    The analysis uses ten realizations and is restricted to one chaotic system and one observable; the representative noise subset followed an exploratory scan rather than independent preregistration.

    Source: §4 · p. 20

Abstract

from arXiv · show

Extreme events (EEs) in chaotic dynamics are rare broad excursions whose forecastability can be altered by dynamical noise. We investigate how noise changes EE occurrence and prediction skill across forecast horizons in a third-order autonomous chaotic flow. A single clean-data threshold is frozen for all realizations, broad events are defined by one maximum per excursion, and a future window W=15 is predicted from a 15-time-unit history using HistGradientBoosting with chronological data separation. As the forecast gap G between the observed history and the future event window increases, the clean Matthews correlation coefficient (MCC) decreases from 0.641 at G=0 to 0.165 at G=15. Noise dependence is evaluated with ten paired realizations at eight amplitudes. The mean short-horizon score increases from 0.456 in clean data to 0.546 at sigma=0.007; the paired gain is 0.0895 (95% CI 0.0494-0.1295; Holm-adjusted p=0.0234). Noise strongly increases EE occurrence while event amplitude and width remain comparatively stable. Equalizing positive training counts across noise levels substantially attenuates the short-horizon gain, whereas strong noise reduces intermediate-horizon skill. We term this horizon-dependent, nonuniform change in forecast skill noise-induced predictability redistribution (NIPR).

1. Introduction

Forecasting broad extreme events depends on the event definition, available precursor history, and forecast lead time. This study asks how dynamical noise changes both event occurrence and forecast skill across horizons, introducing NIPR for a nonuniform horizon-dependent response.

1. Introduction

  • Forecastability depends on the event definition, observed precursor information, and required lead time, while similar-amplitude events may follow dynamically different routes.Prior work motivates resolving skill by lead time rather than relying on one aggregate accuracy.Source: §1 · p. 2
  • Noise may alter event frequency, waiting times, and positive-example counts, so forecast-skill changes need not mean deterministic precursors became intrinsically sharper.The study therefore includes a positive-training-count-matched control.Source: §1 · pp. 3–4
  • Noise can change both broad-event occurrence and the distribution of forecast skill across lead times, a distinction not systematically resolved in prior work.The paper names this horizon-dependent, nonuniform response noise-induced predictability redistribution (NIPR).Source: §1 · p. 3
  • The study uses one broad-event marker per excursion and retains the same event definition and threshold under noise.Its forecast target asks whether the broad maximum occurs within a future window.Source: §1 · p. 3
  • Ten paired realizations compare noise amplitudes with clean trajectories sharing the same initial-condition perturbation and rescaled Gaussian-noise sequence.The analysis also tests how much apparent gain is attributable to having more positive training examples.Source: §1 · pp. 3–4

2. Methods

The methods define broad extreme events with a frozen clean-data threshold, predict whether a broad maximum falls within a future window, and assess skill using chronological test data. Noise responses are evaluated with paired realizations and a control that equalizes positive training counts.

2.1. Chaotic system, integration, and stochastic forcing

  • The system is a third-order autonomous flow integrated with RK4 at Δt=10^-3 for 50,000 time units, with the first 100 discarded.Dynamical noise is applied only to the y coordinate after each deterministic RK4 step.Source: §2.1 · p. 4

2.2. Single working extreme-event threshold; 2.3. Broad-peak extraction and event geometry

  • One threshold is computed from clean local maxima and frozen across noise and initial-condition changes; broad maxima define one event coordinate per excursion.Broad maxima are found after Gaussian smoothing, then the original-signal maximum is recovered within a ±8-time-unit neighborhood; width is defined independently of the threshold.Source: §2.2 · p. 5, §2.3 · p. 5

2.4. Reference dense noise scan

  • The reference dense noise scan applies the same frozen threshold across a 16-level grid and records event counts, rates, amplitudes, widths, and inter-event intervals.The scan localizes response regimes and selects amplitudes for the paired ensemble; its single-realization statistics are not uncertainty estimates.Source: §2.4 · p. 7, Table 1 · p. 7

2.5. Broad-event finite-window forecasting

  • For anchor time t, the binary target is positive when a catalogued broad EE peak occurs in [t + G, t + G + W), and zero otherwise.The window is slightly larger than the reference clean median W1/2 = 11.553; threshold crossings within one excursion are not separate positive targets.Source: §2.5 · p. 7

2.6. Classifier, chronological validation, and clean history selection

  • The classifier uses 60 fixed feature positions to encode a 15-time-unit history, with longer candidate histories adding older samples while preserving the immediate history representation.A HistGradientBoostingClassifier is evaluated with chronological train, validation, and test regions; validation MCC selects the probability threshold, which is then fixed for testing.Source: §2.5 · p. 7, §2.6 · p. 8

2.7. Paired multi-realization noise validation and count-matched control

  • Ten paired realizations compare selected noise amplitudes with clean trajectories sharing initial-condition perturbations and Gaussian-noise sequences; positive-training-count matching tests the role of extra positive examples.Validation and test sets retain natural event prevalence, and paired comparisons use Wilcoxon signed-rank tests with Holm correction.Source: §2.7 · pp. 8–10
  • Noise effects are summarized as paired changes in MCC across forecast gaps and amplitudes, with the replicate-averaged response forming the operational NIPR map.Positive regions indicate noise-assisted skill relative to paired clean dynamics, negative regions indicate degradation, and no separate resonance mechanism is assumed.Source: §2.7 · p. 9

3. Results

Dynamical noise changes broad-event occurrence more than morphology, while its effect on forecast skill varies across noise levels and forecast gaps. The short-horizon gain is substantially attenuated by matching positive training counts, and strong noise degrades intermediate-horizon skill.

3.1. The single threshold isolates broad extreme excursions

  • One broad maximum per excursion marks the event, whose clean median peak amplitude is 13.993 and median half-height width is 11.553.The broad excursion spans roughly a dozen time units, rather than being a sub-unit threshold-occupancy episode.Source: §3.1 · p. 10

3.2. The dense reference scan shows stronger changes in occurrence than morphology

  • The dense scan’s EE count rose from 45 in clean data to 101 at σ = 3×10−2, while median inter-event interval shortened from 501.7 to 305.4 and amplitude and width varied comparatively little.Noise was small relative to deterministic excursions and did not cause obvious visual roughening; its main effects appeared in long-run occurrence and timing statistics.Source: §3.2 · p. 10, Figure 2 · p. 11, Figure 3 · p. 12

3.3. The recent 15-time-unit history contains the useful clean precursor information

  • Validation scores averaged over G = 0–15 were similar across history lengths 15–90, so Thist = 15 was selected and frozen for the noise experiments.The strongest validation dependence was on forecast gap, not older context.Source: §3.3 · p. 14, Figure 5 · p. 14

3.4. The clean broad-event predictability horizon is short

  • With Thist = 15, clean test MCC declined from 0.641 at G = 0 to 0.165 at G = 15; prediction skill fell sharply over the first six time units, then left a weak residual tail.Source: §3.4 · p. 14, Figure 6 · p. 15

3.5. Paired ensemble reveals noise-induced predictability redistribution

  • For intermediate noise (σ = 0.002–0.01), mean Snear rose to 0.524–0.546.The reported paired difference from clean was 0.0895 (95% CI 0.0494–0.1295); 9 of 10 replicates improved, with Holm-adjusted p = 0.0234.Source: §3.5 · p. 15
  • The paired ensemble supports a short-horizon noise gain but does not establish a unique statistically confirmed optimum for intermediate-horizon Smid.Several intermediate-noise levels had positive paired differences, but none remained significant after Holm correction.Source: §3.5 · pp. 15, 17
  • Noise responses vary with forecast gap: at σ = 3 × 10−2, Snear remains above clean while Smid falls to 0.205 from 0.232.Intermediate noise gives a positive band at shorter and moderate gaps, whereas strong noise suppresses skill at larger gaps.Source: §3.5 · pp. 15, 17, Figure 8 · p. 17

3.6. Count matching decomposes the NIPR response and exposes strong-noise degradation

  • After positive-count matching, the short-horizon gain at σ = 0.007 shrank from about 0.090 to 0.028 and was not significant after Holm correction.The authors attribute a substantial part of the natural-data gain to the larger number of positive training examples; strong-noise intermediate-horizon degradation remained significant.Source: §3.6 · pp. 17–18

4. Discussion: noise-induced predictability redistribution

The discussion interprets noise as reorganizing extreme-event occurrence and redistributing forecast skill across lead times, rather than shifting skill uniformly. It links part of the short-horizon gain to additional positive training examples, while count matching leaves strong-noise intermediate-horizon degradation.

4. Discussion: noise-induced predictability redistribution

  • A frozen clean-data threshold and one broad-peak maximum per excursion separate event selection from internal oscillatory micropeaks.Event width is measured relative to the realization mean rather than the selection threshold.Source: §4 · p. 19
  • Clean-data skill is concentrated near the event: test MCC stays above 0.5 through G = 2 and above 0.3 through G = 5.Histories longer than 15 time units do not improve validation MCC.Source: §4 · p. 19
  • Noise redistributes forecast skill across horizons, with positive short-horizon responses but matched intermediate-horizon losses at strong noise.The pattern is qualitatively reminiscent of resonance-like behavior, but the results do not establish a stochastic- or coherence-resonance mechanism.Source: §4 · pp. 19–20
  • Noise increases broad-event counts and shortens typical inter-event separation, while peak amplitude and width change less.The increase in event density is specific to this dynamical regime; noise can suppress extreme events in other systems.Source: §4 · p. 19
  • Matching positive training counts substantially reduces the short-horizon gain, and the remaining positive differences are not statistically significant.The authors therefore do not claim that intermediate noise intrinsically sharpens deterministic precursors; matched intermediate-horizon degradation remains significant at strong noise.Source: §4 · pp. 19–20
  • The analysis is limited to one chaotic system and observable, with ten realizations; the authors call for tests using other systems, classifiers, and experimentally measured noise.The validation-selected operating threshold means reported MCC reflects optimized alarm operation, not calibration transfer across noise regimes.Source: §4 · pp. 20–21

5. Conclusions

Using a fixed-threshold broad-event framework, the study finds that noise changes forecast skill nonuniformly across horizons. Near-event skill improves at some noise levels, while strong noise weakens intermediate-horizon skill; the authors call this pattern NIPR.

5. Conclusions

  • The framework fixes the threshold at HEE = 13.052073 from the reference clean peak distribution and assigns one coordinate to each broad excursion.Its future window is W = 15, motivated by the reference clean median half-height width W1/2 = 11.553.Source: §5 · p. 21
  • The clean test MCC falls from 0.641 at G = 0 to 0.369 at G = 5 and 0.165 at G = 15, indicating a short useful horizon and weak residual tail.Chronological validation selected a 15-time-unit history; older context up to 90 time units provided no systematic benefit.Source: §5 · p. 21
  • At σ = 7 × 10−3, the natural-data short-horizon score rises from 0.456 in paired clean data to 0.546, while strong noise reduces matched intermediate-horizon skill by 0.0518.The short-horizon paired gain is 0.0895 (95% CI 0.0494–0.1295; Holm-adjusted p = 0.0234); at σ = 3 × 10−2, all ten replicates decrease (Holm-adjusted p = 0.0137).Source: §5 · p. 21
  • Noise strongly increases extreme-event occurrence while broad-event amplitude and width remain comparatively stable, and matching positive training counts attenuates most short-horizon gain.The remaining positive differences after matching are not statistically significant.Source: §5 · p. 21
  • NIPR names the nonuniform change in extreme-event forecast skill across lead times under noise, not a universal increase or decrease in predictability.The near-event positive component is partly associated with increased event density and positive-example availability, while strong noise produces a negative component at longer horizons.Source: §5 · pp. 21–22

CRediT authorship contribution statement

The statement assigns conceptualization, investigation, and manuscript responsibilities across the two authors, with Velichko listed for a broader set of project roles.

CRediT authorship contribution statement

Data and code availability

The numerical workflow is organized into six reproducible supplementary programs covering trajectory generation, event analysis, forecasting, and ensemble validation.

Data and code availability

  • Programs 01–03 generate clean and noisy trajectories, construct broad-event catalogs, and produce event-statistics figures.Source: Data and code availability · p. 22
  • Program 04 selects clean-data history length and analyzes the predictability horizon, while Program 05 runs an exploratory dense noise-prediction scan.Source: Data and code availability · p. 22
  • Program 06 performs paired ten-realization ensemble validation, including the positive-training-count-matched control and final mean/CI and paired-test tables.It regenerates trajectories in memory and stores compact event, prediction, and statistical outputs.Source: Data and code availability · p. 22
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