选择嵌入延迟：嵌入技术综述与一种基于持续同调的新方法
Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology
AI总结：
本文提出SToPS方法，利用持续同调构造特征时间谱来选择嵌入滞后，在快慢时间序列预测中优于现有嵌入方法。
摘要：
延迟嵌入方法是时间序列分析与预测领域的基础工具。然而，嵌入参数的选择会对最终分析结果产生重大影响，这促使大量用于优化嵌入滞后等参数选择的方法被提出。本文旨在为初次接触该领域的读者提供嵌入理论基础的全面概述。我们梳理了在均匀和非均匀延迟嵌入两种情形下现有的嵌入滞后选择方法。针对现有非均匀滞后选择方法在动力学可解释性方面的不足，我们提出了一种结合动力学论证与拓扑论证的替代性嵌入滞后选择方法。所提出的方法名为“持续链上的显著时间”（Significant Times on Persistent Strands，SToPS），该方法利用持续同调构造一个特征时间谱，用以量化每个时间滞后的相对动力学重要性。我们在周期、混沌和快慢时间序列上对该方法进行了测试，发现其性能与现有的自动化非均匀嵌入方法相当。此外，在预测快慢时间序列时，使用SToPS构建的嵌入所训练的n步预测器被证明优于其他嵌入方法。
英文摘要：
Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimise the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, {\em Significant Times on Persistent Strands} (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, $n$-step predictors trained on embeddings constructed with SToPS was found to outperform other embedding methods when predicting fast-slow time series.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
