# Permutation entropy revisited

## TL;DR

This paper introduces a generalized permutation entropy measure for time-series analysis that depends on two window lengths, providing insights into the structure of invariant measures and aiding in estimating Kolmogorov-Sinai entropy.

## Contribution

It extends permutation entropy by incorporating a second window parameter, enabling a more detailed analysis of time-series dynamics and invariant measure structures.

## Key findings

- The $w$-dependence reveals invariant measure structure.
- The $L$-dependence aids in estimating Kolmogorov-Sinai entropy.
- Partition structure becomes elongated with increasing $w$.

## Abstract

Time-series analysis in terms of ordinal patterns is revisited by introducing a generalized permutation entropy $H_p(w,L)$, which depends on two different window lengths: $w$, implicitly defining the resolution of the underlying partition; $L$, playing the role of an embedding dimension, analogously to standard nonlinear time-series analysis. The $w$-dependence provides information on the structure of the corresponding invariant measure, while the $L$-dependence helps determining the Kolmogorov-Sinai entropy. We finally investigate the structure of the partition with the help of principal component analysis, finding that, upon increasing $w$, the single atoms become increasingly elongated.