Maximally predictive states: From partial observations to long timescales.

Area of Science:

• Dynamical Systems and Complex Systems Analysis
• Non-linear Dynamics and Time Series Analysis
• Statistical Mechanics and Information Theory

Background:

• Analyzing dynamical systems often requires complete knowledge of underlying equations, which is frequently unavailable.
• Distinguishing slow dynamics from fast fluctuations is crucial but challenging with partial observations.
• Existing methods struggle to extract meaningful information from incomplete time-series data.

Purpose of the Study:

• To develop a novel method for analyzing dynamical systems from partial observations without prior knowledge of equations.
• To identify and characterize slow dynamics, timescale separation, and long-lived collective modes.
• To provide a new, robust estimator for Kolmogorov-Sinai entropy applicable to diverse systems.

Main Methods:

• Constructing maximally predictive states by time-concatenating measurements.
• Partitioning measurement sequences using maximum entropy principles.
• Approximating the transfer operator from state transitions to analyze system dynamics.
• Utilizing the operator spectrum to reveal timescale separation and collective modes.

Main Results:

• Successfully revealed timescale separation and long-lived collective modes in the Lorenz system and a double-well potential model.
• Developed a novel estimator for Kolmogorov-Sinai entropy.
• Demonstrated the method's applicability to discrete-time, continuous-time, and biological systems (C. elegans movement).

Conclusions:

• The developed method effectively extracts slow dynamics and collective behaviors from partial observations of complex systems.
• The transfer operator approximation provides a powerful tool for analyzing systems lacking known equations.
• The new Kolmogorov-Sinai entropy estimator offers a valuable tool for quantifying complexity across different system types.