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.

通过发展新方法：有向各向异性扩散图（directed anisotropic diffusion map），我们能够有效地提取出低维流形中的潜在演化动力学，并应用于真实的脑电图（EEG）数据，找到了适当的有效坐标，推导出能够检测状态转变过程中的临界点的早期预警信号。