# Improved Reconstruction of Chaotic Signals from Ordinal Networks

## TL;DR

This paper explores how well chaotic time series can be reconstructed from ordinal patterns, using a Markov approximation and the Hénon map as a test case.

## Contribution

The novel contribution is a method for reconstructing chaotic signals from ordinal networks, even with observational noise.

## Key findings

- Chaotic signals can be meaningfully reconstructed from ordinal patterns using ergodic Markov approximation.
- Reconstruction remains effective even in the presence of small observational noise.
- The Hénon map was used to validate the reconstruction approach.

## Abstract

Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is not clear to what extent a given ordinal pattern indeed provides a faithful reconstruction of the original signal. Here, we address this question by attempting the reconstruction of the original time series by invoking an ergodic Markov approximation of the symbolic dynamics, thereby inverting the encoding procedure. Using the Hénon map as a testbed, we show that a meaningful reconstruction can also be made in the presence of a small observational noise.