Mittag-Leffler-Type Forecast-Error Growth as a Diagnostic Indicator of Fractional Dynamics

Abstract

Fractional calculus is a powerful framework for modeling nonlocal behavior in complex systems. However, the identification of fractional dynamics from measured time series remains challenging, as most existing approaches require knowledge of the underlying governing equations. In this work, we propose a data-driven diagnostic pipeline that detects fractional signatures directly from scalar observations using a multi-horizon k-nearest neighbors (kNN) forecast-error growth framework. The central idea is that fractional systems exhibit power-law or Mittag-Leffler error growth, in contrast to the exponential divergence characteristic of chaotic integer-order systems. By comparing the empirical error-growth curve against exponential and Mittag-Leffler models, and by examining the local slope of the logarithmic curve, we construct a preliminary fractionality indicator. The method is evaluated on a fractional chaotic system and in a controlled stable fractional relaxation setting, including a kNN-based contraction test. On a fractional chaotic system the Mittag-Leffler model achieved a 58% reduction in RMSE over the exponential model, with $\Delta>0$ in 100% of bootstrap replicates. In the stable relaxation setting, Mittag-Leffler decay strongly outperformed the exponential alternative; in the kNN contraction test, the free-order Mittag-Leffler model reduced the RMSE from $4.810\times 10^{-3}$ to $5.14\times10^{-4}$. The fitted Mittag-Leffler order should be interpreted as an effective shape parameter of the error-growth curve rather than as a direct estimate of the true system order, the recovery of which remains a more difficult inverse problem. Our results demonstrate that multi-horizon forecast-error geometry can serve not only for forecasting and chaos detection, but also for dynamical characterization in fractional systems.