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

2026-05-17

AI Summary

Accurately estimating the maximal Lyapunov exponent from scalar time series is highly challenging in the absence of governing equations, tangent-space dynamics, and full state information. This work proposes the FEG-Pro framework, which leverages autocorrelation-guided sparse embedding and distance-weighted k-nearest neighbor multi-step prediction to analyze the finite-horizon slope of the logarithmic growth of geometric mean prediction errors. For the first time, error growth is treated as a structured profile, incorporating multidimensional diagnostic features such as curvature, residual roughness, monotonicity, and entropy of the error distribution. The method demonstrates strong performance on scalar observations from chaotic maps, the Mackey–Glass system, and the Lorenz-63 attractor, achieving close agreement with true Lyapunov exponents in near-linear regimes and retaining interpretable characteristics even under short-data conditions, thereby offering a novel paradigm for instability rate estimation and machine learning.

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.

Problem

• Lyapunov exponent
• nonlinear time series
• forecast-error growth
• instability analysis
• scalar observables

Innovation

• forecast-error growth profiling
• finite-horizon instability
• Lyapunov exponent estimation
• nonlinear time series analysis
• feature generation