A Tutorial on Time-Evolving Dynamical Bayesian Inference

TL;DR

This paper provides a comprehensive tutorial on dynamical Bayesian inference for time-evolving coupled systems with noise, including theory, algorithms, pseudocode, and practical MATLAB examples for phase and state domain analysis.

Contribution

It introduces a unified Bayesian inference framework for analyzing noisy, time-varying coupled systems, with detailed algorithms and implementation guidance.

Findings

Effective inference of phase dynamics in coupled oscillators

Successful state domain inference in chaotic systems

Potential application to secure communications

Abstract

In view of the current availability and variety of measured data, there is an increasing demand for powerful signal processing tools that can cope successfully with the associated problems that often arise when data are being analysed. In practice many of the data-generating systems are not only time-variable, but also influenced by neighbouring systems and subject to random fluctuations (noise) from their environments. To encompass problems of this kind, we present a tutorial about the dynamical Bayesian inference of time-evolving coupled systems in the presence of noise. It includes the necessary theoretical description and the algorithms for its implementation. For general programming purposes, a pseudocode description is also given. Examples based on coupled phase and limit-cycle oscillators illustrate the salient features of phase dynamics inference. State domain inference is illustrated with an example of coupled chaotic oscillators. The applicability of the latter example to secure communications based on the modulation of coupling functions is outlined. MatLab codes for implementation of the method, as well as for the explicit examples, accompany the tutorial.