On the Deterministic Estimaton of Multiscale Permutation Entropy of High-Order Autoregressive-Moving-Average Processes as a Function of ARMA Parameters

Multiscale Permutation Entropy (MPE) is one of the most common techniques to assess the ordinal information content within a time series. In the present paper we propose an explicit, deterministic function of the MPE of a general ARMA process, as a function only of its parameters and time scale. We compare our theoretical results with the MPE of corresponding simulated signals, which further support our formulation. We also present an exploration of the effects of the ARMA parameters on the MPE curve, where we found a monotonic decrease of entropy for long-term time scales, and highly non-linear effects on short scales. With these results, we aim to provide a benchmark for the MPE of any real time series modelled as an ARMA process.