AI 中文总结
针对复杂动力系统状态变化检测，提出结合拓扑数据分析（TDA）的0维子水平持久性与机器学习分类器的方法，从稀疏时间序列提取特征训练分类器，经实验验证该方法有效，为分析此类序列提供了新途径。
AI 中文摘要
复杂动力系统常随分岔参数变化从周期行为转变为混沌行为，及时检测这些变化至关重要。基于最大Lyapunov指数的传统方法通常需要控制方程知识或足够长的均匀采样时间序列。当可用数据稀少或有缺失观测时，其性能下降，难以进行可靠的相空间重构。我们提出一种将拓扑数据分析（TDA），特别是0维子水平持久性，与机器学习（ML）分类器相结合的方法，直接从时间序列中区分周期和混沌状态。子水平持久性通过分析最小值和最大值的演变来提取拓扑特征，揭示周期动力学的重复特征和混沌动力学的更分散模式。这些特征用于训练逻辑回归、支持向量机和k近邻分类器。使用K折交叉验证验证超参数，平均分类准确率超过90%。训练后的分类器提供二元预测，识别未见数据中的周期和混沌行为。该方法在Duffing、Rössler和Lorenz系统上进行评估，检测到的转变与使用MLE识别的转变密切一致，证明了该方法的可靠性。它进一步应用于实际心电图信号，对正常和异常心跳进行分类，在标准统计指标上产生了令人鼓舞的性能。所提出的框架为分析稀疏或不完整时间序列提供了一种有效的替代方法，在传统非线性时间序列方法受限的实验设置中特别有用。
英文摘要
Complex dynamical systems often undergo transitions from periodic to chaotic behaviour as bifurcation parameters vary, making timely detection of these changes essential. Conventional approaches based on the maximal Lyapunov exponent (MLE) generally require either knowledge of the governing equations or sufficiently long, uniformly sampled time series. Their performance degrades when the available data are scanty or contain missing observations, making reliable phase-space reconstruction difficult. We propose a methodology that combines Topological Data Analysis (TDA), specifically 0-D sublevel persistence, with Machine Learning (ML) classifiers to distinguish periodic and chaotic regimes directly from time series. Sublevel persistence extracts topological features by analysing the evolution of minima and maxima, revealing repeating signatures for periodic dynamics and more scattered patterns for chaotic dynamics. These features are used to train logistic regression, support vector machine, and k-nearest neighbour classifiers. Hyperparameters are validated using K-fold cross-validation, yielding average classification accuracies exceeding 90%. The trained classifiers provide binary predictions, identifying periodic and chaotic behaviour in previously unseen data. The proposed methodology is evaluated on the Duffing, Rössler, and Lorenz systems, where the detected transitions closely agree with those identified using the MLE, demonstrating the reliability of the approach. It is further applied to real-world ECG signals to classify normal and abnormal heartbeats, producing encouraging performance across standard statistical metrics. The proposed framework provides an effective alternative for analysing sparse or incomplete time series and is particularly useful in experimental settings where conventional nonlinear time-series methods are limited.
Comments24 pages, 20 Figures, 8 Tables
