Maximally predictive states: from partial observations to long timescales

TL;DR

This paper introduces a method to construct maximally predictive states from partial observations of dynamical systems, enabling the analysis of long timescales and revealing system properties like timescale separation and collective modes.

Contribution

It presents a novel approach to extract predictive states and approximate transfer operators from partial data, applicable to both deterministic and stochastic systems.

Findings

Successfully applied to Lorenz system and particle in double-well potential

Provides a new estimator for Kolmogorov-Sinai entropy

Reveals long-lived modes and timescale separation

Abstract

Isolating slower dynamics from fast fluctuations has proven remarkably powerful, but how do we proceed from partial observations of dynamical systems for which we lack underlying equations? Here, we construct maximally-predictive states by concatenating measurements in time, partitioning the resulting sequences using maximum entropy, and choosing the sequence length to maximize short-time predictive information. Transitions between these states yield a simple approximation of the transfer operator, which we use to reveal timescale separation and long-lived collective modes through the operator spectrum. Applicable to both deterministic and stochastic processes, we illustrate our approach through partial observations of the Lorenz system and the stochastic dynamics of a particle in a double-well potential. We use our transfer operator approach to provide a new estimator of the Kolmogorov-Sinai entropy, which we demonstrate in discrete and continuous-time systems, as well as the movement behavior of the nematode worm $C. elegans$.