A tutorial on time-evolving dynamical Bayesian inference

Abstract

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.

Experimental Realization of the Coupling Function Secure Communications Protocol and Analysis of Its Noise Robustness

“…Note that the chosen coupling functions q(x i , x j ) represent the encryption key. Bearing this in mind, if given a 2 × M time-series X = {x n ≡ x(t n )} (t n = nh) as input, the main task for the Bayesian dynamical inference is to reveal the unknown model parameters and the noise diffusion matrix M = {c, D}, which eventually comes down to maximization of the posterior conditional probability p X (M|X ) of observing the parameters M when given the data X [49]. The relationship of this posterior conditional probability to the prior density p prior (M) (which encompasses observation based prior knowledge of the unknown parameters), and to the likelihood function (X |M) (that is the conditional probability density to observe X given choice M), is given by Bayes' theorem:…”

“…Further details of the development, software implementation and application of the dynamic Bayesian inference approach can be found in [21], [27], [28], [49], and references therein.…”

Noise robustness of communications provided by coupling-function-encryption and dynamical Bayesian inference

“…the encrypted signals. More details about the method, its implementation and coding can be found in [28], [29].…”

Detecting Chronotaxic Systems from Single-Variable Time Series with Separable Amplitude and Phase

“…The odd parameters correspond to the coupling terms inferred for ϕ 1 in the direction 2 → 1, and the even parameters correspond to the coupling terms inferred for ϕ 2 in the direction 1 → 2. See [32] for further details and an in depth tutorial on dynamical Bayesian inference and its implementation. In summary, Bayesian inference is applied to ϕ * x and ϕ A *…”