Efficient Bayesian estimation of permutation entropy with Dirichlet priors

TL;DR

This paper introduces a Bayesian approach for estimating permutation entropy that effectively incorporates uncertainty and prior knowledge, enabling accurate estimation even with short time series by approximating the posterior with a Beta distribution.

Contribution

It presents a novel Bayesian estimation method for permutation entropy using Dirichlet priors, which simplifies computation and relaxes data length requirements compared to traditional methods.

Findings

PE posterior approximates a Beta distribution

Method works well with short time series

PE varies periodically in laser system

Abstract

Estimation of permutation entropy (PE) using Bayesian statistical methods is presented for systems where the ordinal pattern sampling follows an independent, multinomial distribution. It is demonstrated that the PE posterior distribution is closely approximated by a standard Beta distribution, whose hyperparameters can be estimated directly from moments computed analytically from observed ordinal pattern counts. Equivalence with expressions derived previously using frequentist methods is also demonstrated. Because Bayesian estimation of PE naturally incorporates uncertainty and prior information, the orthodox requirement that $N \gg D!$ is effectively circumvented, allowing PE to be estimated even for very short time series. Self-similarity tests on PE posterior distributions computed for a semiconductor laser with optical feedback (SLWOF) system show its PE to vary periodically over time.