Early warning indicators via latent stochastic dynamical systems

TL;DR

This work tackles detecting tipping points in high-dimensional time-evolving data by learning latent dynamics on a low-dimensional manifold using a directed anisotropic diffusion map. It models latent dynamics with a stochastic differential equation ${dz(t)=\\mu(z(t))dt+\\eta(z(t))dB_t}$ learned via neural networks, and derives three early warning indicators—Onsager-Machlup, Sample Entropy, and Transition Probability—solely from the latent coordinates. The approach is validated on real EEG data, showing that the indicators can detect state transitions (pre-ictal to ictal) earlier than traditional measures, and enables automatic labeling of complex high-dimensional time series. This framework reduces dimensionality, enhances robustness of warnings, and holds promise for broader applicability beyond EEG data.

Abstract

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, 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.

Figures (8)

• Figure 1: A schematic diagram of our framework.
• Figure 2: An example: the EEG data of a patient.
• Figure 3: Comparison of diffusion map and directed anisotropic diffusion map. The first two columns (a) (b) represent the kernel matrices, and (c) (d) give the eigenvalues for diffusion map and directed anisotropic diffusion map respectively. For all the cases, we set $\epsilon = 1$. Note that (a) (c) are constructed without the directed term, while (b) (d) are calculated with the data driven directed term learned by Eq.(\ref{['empiricalFun']}) .
• Figure 4: Comparison of low-dimensional embedding patterns with (a) diffusion map and (b) directed anisotropic diffusion map.
• Figure 5: The validation of the neural network approximation with regard to $\Phi_1$. (a) Diffusion map. (b) Directed anisotropic diffusion map.