Detecting early warning indicators for abrupt dynamical transitions in complex systems or high-dimensional observation data is essential in many real-world applications, such as brain diseases, natural disasters, financial crises, and engineering reliability. To this end, we develop a novel approach: the directed anisotropic diffusion map that captures the latent evolutionary dynamics in the low-dimensional manifold. Then three effective warning signals (Onsager-Machlup Indicator, Sample Entropy Indicator, and Transition Probability Indicator) are derived through the latent coordinates and the latent stochastic dynamical systems. To validate our framework, we apply this methodology to authentic electroencephalogram (EEG) data. We find that our early warning indicators are capable of detecting the tipping point during state transition. This framework not only bridges the latent dynamics with real-world data but also shows the potential ability for automatic labeling on complex high-dimensional time series.

翻译：在复杂系统或高维观测数据中检测突变动态转换的早期预警指标，对于脑部疾病、自然灾害、金融危机及工程可靠性等众多实际应用至关重要。为此，我们提出了一种创新方法：有向各向异性扩散映射，用于捕捉低维流形中的潜在演化动力学。进而通过隐式坐标与隐式随机动力系统推导出三种有效预警信号（Onsager-Machlup指标、样本熵指标及转移概率指标）。为验证该框架的有效性，我们将此方法应用于真实脑电图数据。研究发现，我们的早期预警指标能够成功检测状态转换过程中的临界点。该框架不仅建立了隐式动力学与现实数据之间的桥梁，还展现出对复杂高维时间序列进行自动标注的潜在能力。