Early warnings for dynamical transitions in complex systems or high-dimensional observation data are essential in many real world applications, such as gene mutation, brain diseases, natural disasters, financial crises, and engineering reliability. To effectively extract early warning signals, we develop a novel approach: the directed anisotropic diffusion map that captures the latent evolutionary dynamics in low-dimensional manifold. Applying the methodology to authentic electroencephalogram (EEG) data, we successfully find the appropriate effective coordinates, and derive early warning signals capable of detecting the tipping point during the state transition. Our method bridges the latent dynamics with the original dataset. The framework is validated to be accurate and effective through numerical experiments, in terms of density and transition probability. It is shown that the second coordinate holds meaningful information for critical transition in various evaluation metrics.

翻译：针对复杂系统或高维观测数据中的动力学相变进行早期预警，在基因突变、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中至关重要。为有效提取早期预警信号，我们开发了一种新型方法：定向各向异性扩散映射，该方法可捕捉低维流形中的潜演化动力学。将该方法应用于真实脑电图数据后，我们成功找到了合适的有效坐标，并推导出能够检测状态转换过程中临界点的早期预警信号。本研究架起了潜动力学与原始数据集之间的桥梁。通过数值实验在密度和转移概率方面的验证，该框架被证实具有准确性和有效性。研究表明，在各种评估指标下，第二坐标均包含临界转换的关键信息。