发表机构
International University of Grand-Bassam; Laboratory for Intelligence and Mathematics (LIMAs); Petrozavodsk State University; Free University of Bozen-Bolzano（大巴萨姆国际学校; 智能与数学实验室; 彼得罗扎沃茨克国立大学; 博尔扎诺自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究如何从标量观测中检测分数动力学，提出用多步kNN预测误差增长框架构建分数指标。通过与指数和米塔格-莱夫勒模型比较及检查对数曲线斜率来评估，在分数混沌系统等上测试，结果显示该方法可用于分数系统动力学表征。
AI 中文摘要
分数微积分是对复杂系统中的非局部行为进行建模的有力框架。然而，从测量时间序列中识别分数动力学仍然具有挑战性，因为大多数现有方法需要了解潜在的控制方程。在这项工作中，我们提出了一种数据驱动的诊断流程，该流程使用多步k近邻（kNN）预测误差增长框架直接从标量观测中检测分数特征。核心思想是分数系统呈现幂律或米塔格-莱夫勒误差增长，这与混沌整数阶系统的指数发散特征形成对比。通过将经验误差增长曲线与指数和米塔格-莱夫勒模型进行比较，并检查对数曲线的局部斜率，我们构建了一个初步的分数指标。该方法在分数混沌系统和受控稳定分数弛豫设置上进行了评估，包括基于kNN的收缩测试。在分数混沌系统上，米塔格-莱夫勒模型的均方根误差（RMSE）比指数模型降低了58%，在100%的自举重复中$\Delta>0$。在稳定弛豫设置中，米塔格-莱夫勒衰减明显优于指数替代方案；在kNN收缩测试中，自由阶米塔格-莱夫勒模型将RMSE从$4.810\times10^{-3}$降低到$5.14\times10^{-4}$。拟合的米塔格-莱夫勒阶应解释为误差增长曲线的有效形状参数，而不是真实系统阶的直接估计，恢复真实系统阶仍然是一个更困难的逆问题。我们的结果表明，多步预测误差几何不仅可用于预测和混沌检测，还可用于分数系统的动力学表征。
英文摘要
Fractional models provide a natural description of systems with memory, but a noninteger derivative should not be introduced solely because a time series is curved or slowly relaxing. We develop an equation-free preliminary screening framework that asks whether a scalar time series produces a multi-horizon k-nearest-neighbor (kNN) profile more compatible with Mittag-Leffler-type behavior than with selected conventional alternatives. In an ideal matched Caputo-relaxation benchmark, the complete generation-kNN-profile-model-comparison pipeline reproduces the expected Mittag-Leffler geometry and recovers the generating order to within approximately $10^{-3}$; this is interpreted as controlled calibration rather than as general fractional-order identification. Under 3% trajectory-specific observational noise, the held-out Mittag-Leffler preference is most consistent when the generating dynamics are well separated from the integer-order limit and becomes progressively less decisive as $α\rightarrow1$. The fitted order $α_{\mathrm{fit}}$, however, shows substantially larger realization-to-realization variability. Thus, relative model compatibility is more robust than single-realization order estimation in the present noisy benchmark. Noise-free nonfractional controls show a separate limitation of specificity: a stretched exponential can generate a strongly Mittag-Leffler-compatible profile, whereas inclusion of the generating rational/Hill family recovers that family and its parameters to numerical precision in the matched setting. A positive Mittag-Leffler-versus-exponential screen therefore does not uniquely establish fractional origin. A fractional chaotic system is treated only as an exploratory extension: the Mittag-Leffler growth family gives lower finite-window RMSE than exponential and logistic/saturating alternatives over the detected pre-transition interval.
CommentsSubstantially revised version; 20 pages, 5 figures, 8 tables
