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

Andrei Velichko; Viet-Thanh Pham

nlin.CDAug 20, 2026

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

Authors: Andrei Velichko, Viet-Thanh Pham

Organizations: Institute of Physics and Technology, Petrozavodsk State University, 33 Lenin Ave., Petrozavodsk, 185910, Russia · Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Abstract

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).

Figures & tables

Figure 1: Single-threshold broad-event definition. (a) Reference clean dynamics with the fixed threshold $H_{\mathrm{EE}}=13.0521$ and the interval enlarged in panel (b). (b) Broad maxima below the threshold and EE maxima above the same fixed threshold. (c) Raw and Gaussian-smoothed signals around one representative broad EE. The half-height level is measured relative to the mean of the complete realization, so $W_{1/2}$ is independent of the EE threshold.

$\sigma$	$N_{\rm EE}$	rate/1000	median $A_{p}$	median $W_{1/2}$
$0$	45	0.902	13.993	11.553
$1.3\times 10^{-3}$	62	1.242	14.361	11.733
$3\times 10^{-3}$	74	1.483	14.438	11.669
$10^{-2}$	82	1.643	14.459	11.555
$3\times 10^{-2}$	101	2.024	14.168	11.299

Table 1: Selected event statistics from the reference dense noise scan with the same frozen threshold

H_{\mathrm{EE}}=13.052073

Figure 2: Representative $y(t)$ segments for four noise intensities, plotted over the same nominal time interval with the same frozen $H_{\mathrm{EE}}$ . Filled markers indicate selected broad EE maxima. The dynamical noise is small relative to the deterministic excursion scale and therefore does not produce obvious visual roughening; its main effect is revealed by long-time EE occurrence statistics.

Figure 3: Reference dense-scan EE occurrence statistics. (a) Event rate per $1000$ time units. (b) Median interval between consecutive EE maxima; shading shows the interquartile range.

Figure 4: Reference dense-scan broad-event morphology. (a) Median EE peak amplitude and interquartile range. (b) Median half-height width $W_{1/2}$ and interquartile range. The morphology varies much less than the occurrence statistics.

Figure 5: Validation MCC used for clean history selection. Extending the history from $15$ to $90$ time units produces no systematic improvement, whereas forecast skill decreases strongly with increasing $G$ .

Figure 6: Clean-system broad-event predictability horizon for $T_{\rm hist}=15$ and $W=15$ . Prediction skill decreases sharply over the first six time units and then forms only a weak residual tail.

Panel of Figure 7 — Figure 7: Paired ensemble summary of horizon-dependent prediction skill. Top: natural training data. Bottom: positive-training-count-matched control. Curves show ensemble mean $S_{\rm near}$ and $S_{\rm mid}$ ; shading denotes 95% confidence intervals over ten paired realizations. Intermediate noise reproducibly improves the natural short-horizon score, whereas matching the positive training count substantially attenuates the gain.

Figure 8: Noise-induced predictability redistribution (NIPR) map: mean paired change in test MCC relative to the corresponding clean replicate. Positive values indicate noise-assisted prediction and negative values indicate degradation. The positive region is concentrated at shorter forecast gaps, whereas strong noise increasingly suppresses longer-horizon skill.

Figure 9: Ensemble number of selected broad EEs as a function of dynamical-noise amplitude. Points are means over ten paired realizations and shading denotes 95% confidence intervals. Noise strongly increases event occurrence while broad-event amplitude and width remain comparatively stable.

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Rare-Event Estimation Chaotic Time-Series Prediction Chaotic Dynamical Systems

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Extreme events in chaotic systems are difficult to learn from short trajectories because they are controlled by transient finite-time instability rather than by frequently observed bulk dynamics. We propose a mechanism-aware conditioning plug-in framework that turns a nudged coarse ensemble into a non-intrusive sensor of local instability geometry. In the small-noise regime, the ensemble covariance aggregates the same finite-time deformation kernels that govern local instability, providing a Jacobian-free proxy for the local amplification structure around a synchronized coarse trajectory. A small FiLM module injects statistics of this ensemble geometry into an otherwise unchanged backbone while leaving the coarse simulator unchanged. We demonstrate this interface in two distinct pipelines: a Transformer-style residual-attention corrector for a controlled low-dimensional chaotic system and a probabilistic recurrent STORN corrector for topographic two-layer quasi-geostrophic (QG) flow. In the low-dimensional benchmark, ensemble covariance directions co-activate with OTD modes and FiLM conditioning improves 99th-percentile exceedance-frequency errors over an identical no-context Transformer baseline. In QG, a fixed ensemble-conditioned FiLM-STORN model trained on only

50

time units substantially improves long-horizon rare-event statistics in the data-limited regime, including density-tail errors, exceedance frequencies, and spatial exceedance-area distributions relative to an unconditioned STORN trained on the same data; on averaged high-threshold exceedance diagnostics, it also outperforms the baseline STORN trained with

20

times more high-resolution data. These results show that local instability geometry is not merely interpretable post hoc, but an actionable conditioning signal for data-efficient rare-event emulation.

Abstract

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