Efficient Bayesian estimation of permutation entropy with Dirichlet priors

Abstract

Estimation of permutation entropy (PE) using Bayesian statistical methods is presented for systems where the ordinal pattern sampling follows an independent, multinomial distribution. The desired PE posterior distribution is demonstrated to closely approximate a standard Beta distribution whose hyperparameters can be computed directly from ordinal pattern counts. It is further shown that this Bayesian approach is a generalization of previously-published frequentist methods. Because Bayesian posterior distributions can be estimated for very short time series, this method enables PE analysis on data sets currently not compatible with existing methods due to their limited size. To demonstrate the power and flexibility of this technique, the PE of a semiconductor laser with optical feedback (SLWOF) is shown to be time-variant by visualizing how the PE posterior distribution varies over sequences of small (1000 point) partitions of the output time-series.