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arxiv: 2605.17282 · v1 · pith:RHTH2AXGnew · submitted 2026-05-17 · 🌊 nlin.CD · cs.LG· math.DS

FEG-Pro: Forecast-Error Growth Profiling for Finite-Horizon Instability Analysis of Nonlinear Time Series

Andrei Velichko , N'Gbo N'Gbo , Bruno Carpentieri , Mudassir Shams This is my paper

Pith reviewed 2026-05-19 23:10 UTC · model grok-4.3

classification 🌊 nlin.CD cs.LGmath.DS
keywords forecast error growthLyapunov exponent estimationnonlinear time seriesscalar observationsnearest-neighbor predictionchaos detectioninstability analysiserror profile
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The pith

A forecast-error growth slope from scalar time series estimates the largest Lyapunov exponent when the growth curve shows a quasi-linear regime.

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

The paper proposes a method to quantify dominant instability rates in nonlinear dynamics using only scalar observations, without access to governing equations or full state vectors. It builds autocorrelation-guided sparse histories and applies distance-weighted nearest-neighbor forecasting over multiple horizons to track how prediction errors grow. The resulting logarithmic error-growth curve yields a slope, lambda_FEG, that can be compared directly to reference largest Lyapunov exponents in cases where the growth appears quasi-linear. This matters because it offers a data-driven route to instability analysis for systems where traditional tangent-space methods are unavailable. The framework also produces additional descriptors from the full error profile, such as curvature and error-distribution entropy, to characterize signal behavior beyond a single rate.

Core claim

By constructing autocorrelation-guided sparse histories and performing distance-weighted k-nearest-neighbor multi-horizon forecasting on scalar time series, the method extracts a finite-horizon forecast-error growth slope lambda_FEG that approximates the dominant instability rate whenever the error-growth curve supports a quasi-linear regime, as demonstrated through agreement with known exponents on chaotic maps, Mackey-Glass delay dynamics, and Lorenz-63 observables.

What carries the argument

The forecast-error growth profile and its quasi-linear-regime slope lambda_FEG, obtained from geometrically averaged logarithmic errors after distance-weighted k-nearest-neighbor prediction.

If this is right

  • In quasi-linear regimes the extracted slope provides a usable numerical estimate of the largest Lyapunov exponent from scalar data alone.
  • Secondary profile descriptors including curvature, residual roughness after quadratic fit, monotonicity, and forecast-error distribution entropy serve as built-in diagnostics for the slope reliability.
  • These same descriptors function as candidate features for machine-learning tasks in nonlinear signal classification.
  • The method remains interpretable under progressive record-length reduction, with roughness and mean error entropy often changing monotonically even when the slope itself becomes variable.

Where Pith is reading between the lines

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

  • The profile approach could extend instability analysis to experimental time series from partially observed physical systems where only one variable is recorded.
  • Combining the slope with the entropy and roughness descriptors might improve automated detection of transitions between regular and chaotic regimes in long records.
  • Applying the pipeline to non-stationary data with slow parameter drifts could reveal how instability rates evolve over time without requiring model retraining.

Load-bearing premise

The distance-weighted nearest-neighbor forecasting step accurately captures the local expansion rates present in the underlying dynamics.

What would settle it

A systematic comparison in which lambda_FEG deviates markedly from the known largest Lyapunov exponent on a system whose error-growth curve displays a clear quasi-linear segment over several decades of horizon length.

Figures

Figures reproduced from arXiv: 2605.17282 by Andrei Velichko, Bruno Carpentieri, Mudassir Shams, N'Gbo N'Gbo.

Figure 1
Figure 1. Figure 1: Typical forecast-error growth profiles considered by FEG-Pro. A clean quasi-linear region supports [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reference Lyapunov exponent versus FEG-Pro [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mackey-Glass delay benchmark: reference largest Lyapunov exponent versus FEG-Pro estimate [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative Mackey-Glass diagnostic curve for [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Representative Mackey-Glass diagnostic curve for [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Lorenz-63 scalar-module benchmark: full-state variational reference exponent versus FEG-Pro [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Primary λFEG response under dyadic record shortening. Open circles show individual fragment estimates. Blue markers and bars show the mean and one standard deviation across valid fragments at the same length. The dotted black line is the full-record FEG-Pro value, whereas the dashed red line is the known/reference Lyapunov value used in the benchmark tables. The label n = · above each mean indicates the nu… view at source ↗
Figure 8
Figure 8. Figure 8: Residual roughness after quadratic detrending under dyadic shortening. Open circles show indi [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mean normalized FEDE under dyadic shortening. Open circles show individual fragment values; [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
read the original abstract

Estimating the largest Lyapunov exponent from a scalar time series is difficult when the governing equations, tangent dynamics, and full state vector are unavailable. We propose FEG-Pro, a forecast-error growth profiling framework for nonlinear scalar time series. The method constructs autocorrelation-guided sparse histories, performs distance-weighted k-nearest-neighbor multi-horizon forecasting, and analyzes the logarithmic growth of geometrically averaged forecast errors. Its primary output is the finite-horizon forecast-error growth slope, lambda_FEG. When the error-growth curve supports a quasi-linear regime, this slope can be compared with reference largest Lyapunov exponents as an estimate of the dominant instability rate. The same pipeline also extracts the formal fit-selection regime, curvature, residual roughness after quadratic detrending, monotonicity, and forecast-error distribution entropy (FEDE) from signed multi-horizon errors. These secondary descriptors are intended not only as diagnostic controls for the slope, but also as candidate machine-learning features for nonlinear signal analysis, because they encode profile geometry and distributional uncertainty not captured by lambda_FEG alone. We evaluate the method on chaotic maps, Mackey-Glass delay dynamics, and scalar Lorenz-63 observables with known or reference exponents. Full-record experiments show good agreement in quasi-linear cases and meaningful curve-shape information in curved or weak profiles. A dyadic length-halving experiment on representative logistic, Mackey-Glass, and Lorenz records shows that residual roughness and mean FEDE often change monotonically and remain interpretable as record length decreases, even when the slope becomes biased or highly variable. The results support treating forecast-error growth as a structured profile and feature-generation framework rather than a single-number estimator.

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 paper introduces FEG-Pro, a forecast-error growth profiling framework for nonlinear scalar time series. It constructs autocorrelation-guided sparse histories, applies distance-weighted k-nearest-neighbor multi-horizon forecasting, and extracts the finite-horizon forecast-error growth slope lambda_FEG from the logarithmic growth of geometrically averaged forecast errors. When a quasi-linear regime is present, lambda_FEG is proposed as an estimate of the largest Lyapunov exponent. Secondary descriptors including curvature, residual roughness, monotonicity, and forecast-error distribution entropy (FEDE) are also derived for diagnostics and as machine-learning features. Evaluations on chaotic maps, Mackey-Glass delay dynamics, and scalar Lorenz-63 observables report good agreement in quasi-linear cases and interpretable changes under dyadic length-halving.

Significance. If rigorously validated, the approach could supply a practical data-driven route to dominant instability rates from scalar observations when equations and full-state vectors are unavailable. Treating the error-growth curve as a structured profile rather than a single-number estimator, together with the dyadic length-halving experiments that track monotonicity of roughness and FEDE, adds diagnostic value beyond conventional Lyapunov estimation. The secondary descriptors could serve as useful features for nonlinear signal classification.

major comments (2)
  1. [Abstract / Evaluation] Abstract and evaluation section: the statement that 'full-record experiments show good agreement in quasi-linear cases' supplies no quantitative metrics (e.g., mean absolute deviation, correlation, or tabulated lambda_FEG versus reference LLE values with error bars) for the logistic, Mackey-Glass, or Lorenz-63 examples. This absence is load-bearing for the central claim that lambda_FEG can be compared with reference largest Lyapunov exponents as an estimate of the dominant instability rate.
  2. [Method (forecasting procedure)] Method description of the forecasting procedure: distance-weighted kNN multi-horizon prediction forms a weighted average over neighbors selected by Euclidean distance in the autocorrelation-guided history. This averaging can damp the observed growth rate relative to the true local stretching factor whenever the neighbors exhibit curvature or separation. No analytic bound on the resulting bias in lambda_FEG nor an ablation that replaces kNN with exact local linearization (Jacobian or tangent map) on identical histories is provided, which directly affects the validity of equating the slope to the largest Lyapunov exponent.
minor comments (2)
  1. [Notation] The definition of lambda_FEG as the slope of the log-averaged forecast-error curve should be tied to an explicit equation number for reproducibility.
  2. [Figures] Error-growth figures would be clearer if they indicated the identified quasi-linear regime and included variability across realizations or parameter choices for k and the autocorrelation threshold.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback and for acknowledging the potential utility of treating forecast-error growth as a structured profile. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract / Evaluation] Abstract and evaluation section: the statement that 'full-record experiments show good agreement in quasi-linear cases' supplies no quantitative metrics (e.g., mean absolute deviation, correlation, or tabulated lambda_FEG versus reference LLE values with error bars) for the logistic, Mackey-Glass, or Lorenz-63 examples. This absence is load-bearing for the central claim that lambda_FEG can be compared with reference largest Lyapunov exponents as an estimate of the dominant instability rate.

    Authors: We agree that quantitative support is needed to substantiate the claim of good agreement. In the revised manuscript we will add a dedicated table in the evaluation section reporting lambda_FEG values next to reference LLEs for the logistic map, Mackey-Glass, and Lorenz-63 cases. The table will include mean absolute deviations, Pearson correlations across multiple realizations, and error bars obtained from repeated runs with varied random seeds for neighbor selection. revision: yes

  2. Referee: [Method (forecasting procedure)] Method description of the forecasting procedure: distance-weighted kNN multi-horizon prediction forms a weighted average over neighbors selected by Euclidean distance in the autocorrelation-guided history. This averaging can damp the observed growth rate relative to the true local stretching factor whenever the neighbors exhibit curvature or separation. No analytic bound on the resulting bias in lambda_FEG nor an ablation that replaces kNN with exact local linearization (Jacobian or tangent map) on identical histories is provided, which directly affects the validity of equating the slope to the largest Lyapunov exponent.

    Authors: The concern about possible damping bias from neighbor averaging is well taken. Because FEG-Pro targets settings where governing equations and full-state Jacobians are unavailable, exact local linearization is not feasible in general. For the benchmark systems with known dynamics we will add an ablation that recomputes the growth slopes using local linear fits on the identical autocorrelation-guided histories and directly compares the resulting slopes to the kNN-based lambda_FEG. We will also expand the discussion to clarify that lambda_FEG is offered as a practical proxy whose validity is conditioned on the presence of a quasi-linear regime, rather than an exact dynamical equivalent. revision: partial

standing simulated objections not resolved
  • Deriving a general analytic bound on the bias in lambda_FEG that would hold for arbitrary nonlinear maps and reconstruction parameters.

Circularity Check

0 steps flagged

No circularity: lambda_FEG defined directly from observable forecast-error slope without reduction to inputs or self-citations

full rationale

The paper introduces FEG-Pro as a data-driven pipeline that builds autocorrelation-guided histories, applies distance-weighted kNN multi-horizon forecasting, computes geometrically averaged forecast errors, and extracts lambda_FEG as the slope of their logarithmic growth in quasi-linear regimes. This slope is then compared empirically to reference Lyapunov exponents on known systems (logistic map, Mackey-Glass, Lorenz-63). No equation or step equates lambda_FEG to the largest Lyapunov exponent by algebraic construction, fitted-parameter renaming, or load-bearing self-citation; the comparison is presented as an empirical approximation supported by experiments rather than a tautological identity. Secondary descriptors (curvature, FEDE, roughness) are likewise extracted directly from the same error profiles without circular re-use of the target quantity. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The method depends on several procedural choices that function as free parameters and on the domain assumption that quasi-linear error-growth regimes exist and are meaningful for Lyapunov comparison.

free parameters (2)
  • k (number of neighbors)
    The number of neighbors and distance-weighting scheme in the kNN forecaster; exact value or selection rule not stated in the abstract.
  • autocorrelation threshold or lag set for sparse histories
    Parameters that determine which past observations are retained; specific criteria not detailed.
axioms (1)
  • domain assumption A quasi-linear regime exists in the forecast-error growth curve for the systems under study and its slope approximates the largest Lyapunov exponent.
    Invoked when the abstract states that lambda_FEG 'can be compared with reference largest Lyapunov exponents' in quasi-linear cases.

pith-pipeline@v0.9.0 · 5853 in / 1553 out tokens · 79754 ms · 2026-05-19T23:10:49.359078+00:00 · methodology

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