Early warning indicators via latent stochastic dynamical systems.

Detecting early warning indicators for abrupt dynamical transitions in complex systems or high-dimensional observation data are essential in many real-world applications, such as brain diseases, natural disasters, 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 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.