Entropy-based Generating Markov Partitions for Complex Systems

TL;DR

This paper introduces a method to approximate Generating Markov Partitions for complex systems using finite data, enabling better symbolic encoding of trajectories and analysis of system complexity.

Contribution

It proposes a novel approach to construct GMPs from finite-resolution data in complex systems, facilitating invariant measure estimation and complexity analysis.

Findings

Method effectively encodes trajectories with minimal information loss.

Allows estimation of invariant probability measures from finite data.

Applicable to real-world data like EEG and climate measurements.

Abstract

Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are \emph{a priori} unknown, and, as it happens in any real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time trajectory. We critically test our method on networks of coupled maps, encoding their trajectories into symbolic sequences. We show that these sequences are optimal because they minimise the information loss and also any spurious information added. Consequently, our method allows us to approximately calculate the invariant probability measures of complex systems from observed data. Thus, we can efficiently define complexity measures that are applicable to a wide range of complex phenomena, such as the characterisation of brain activity from EEG signals measured at different brain regions or the characterisation of climate variability from temperature anomalies measured at different Earth regions.