# Permutation Entropy Based on Non-Uniform Embedding

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Permutation Entropy

#### 2.2. Non-Uniform Embedding

## 3. The Proposed Visualization Scheme for PE

- Determine the optimal embedding dimension D for a given time series.
- Set L (the upper range for all time lags). Determine the set of optimal time lags ${\tau}_{1}^{\u2605},{\tau}_{2}^{\u2605},\dots ,{\tau}_{D-1}^{\u2605}$ according to Equation (5).
- Repeat in lexicographical order and construct planar images of PE:
- Average all $(D-1)(D-2)/2$ planar digital images of PE.

**Example**

**1.**

## 4. Computational Experiments

#### 4.1. The Sine Wave

#### 4.2. The Rössler Time Series

#### 4.3. Real-World Time Series

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PE | Permutation Entropy |

FNN | False Nearest Neighbors |

EEG | Electroencephalogram |

BNCI | Brain/Neural Computer Interaction |

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**Figure 1.**Permutation entropy for the sine wave: ${H}_{4}({\tau}_{1},{\tau}_{2},5)$ is depicted in (

**a**); ${H}_{4}({\tau}_{1},5,{\tau}_{3})$ in (

**b**); ${H}_{4}(5,{\tau}_{2},{\tau}_{3})$ in (

**c**); and the averaged ${\overline{H}}_{4}(\zeta ,\eta )$ in (

**d**); Numerical values of permutation entropy are indicated in color bars.

**Figure 2.**The geometric structure of a single periodic cell produced by the averaged PE reveals the three different inclination lines from three planar projections of the Permutation entropy (PE). (

**a**) The PE reconstructed by the technique presented in [44] is depicted in (

**b**).

**Figure 3.**All possible planar projections of PE for the Rössler time series ($D=6$): (

**a**) ${H}_{6}({\tau}_{1},{\tau}_{2},42,21,23)$; (

**b**) ${H}_{6}({\tau}_{1},11,{\tau}_{3},21,23)$; (

**c**) ${H}_{6}({\tau}_{1},11,42,{\tau}_{4},23)$; (

**d**) ${H}_{6}({\tau}_{1},11,42,21,{\tau}_{5})$; (

**e**) ${H}_{6}(38,{\tau}_{2},{\tau}_{3},21,23)$; (

**f**) ${H}_{6}(38,{\tau}_{2},42,{\tau}_{4},23)$; (

**g**) ${H}_{6}(38,{\tau}_{2},42,21,{\tau}_{5})$; (

**h**) ${H}_{6}(38,11,{\tau}_{3},{\tau}_{4},23)$; (

**i**) ${H}_{6}(38,11,{\tau}_{3},21,{\tau}_{5})$; (

**j**) ${H}_{6}(38,11,42,{\tau}_{4},{\tau}_{5})$.

**Figure 4.**Averaged PE for the Rössler time series: (

**a**) uniform embedding with no additive noise; (

**b**) non-uniform embedding with no additive noise; (

**c**) ${\overline{H}}_{6}$ with $10\%$ noise; (

**d**) ${\overline{H}}_{7}$ with $50\%$ noise; (

**e**) ${\overline{H}}_{7}$ with $75\%$ noise; (

**f**) ${\overline{H}}_{8}$ with $200\%$ noise.

**Figure 5.**Averaged PE for the Rössler time series with no additive noise: (

**a**) the pattern produced by a random set of time delays $\{7,18,39,27,12\}$; (

**b**) by a random set of time delays $\{30,40,16,9,29\}$.

**Figure 6.**The Electroencephalogram (EEG) signal available from the Brain/Neural Computer Interaction (BNCI) Horizon 2020 project database [49]. Insets (

**a**) and (

**b**) are used to depict the zoomed parts of the signal.

**Figure 7.**Digital images of PE reconstructed for Interval A. The proposed scheme yields the image in (

**a**). The scheme without the assessment of the optimal embedding dimension and the optimal set of time lags results in the image in (

**b**).

**Figure 8.**Digital images of PE reconstructed for Interval B. The proposed scheme yields the image in (

**a**). The scheme without the assessment of the optimal embedding dimension and the optimal set of time lags results in the image in (

**b**).

${\mathit{\tau}}_{1}$ | ${\mathit{\tau}}_{2}$ | ${\mathit{\tau}}_{3}$ | ${\mathit{\tau}}_{4}$ | ${\mathit{H}}_{5}\left(\mathit{\tau}\right)$ |
---|---|---|---|---|

$1,\dots ,L$ | $1,\dots ,L$ | ${\tau}_{3}^{\u2605}$ | ${\tau}_{4}^{\u2605}$ | ${H}_{5}\left({\tau}_{1},{\tau}_{2},{\tau}_{3}^{\u2605},{\tau}_{4}^{\u2605}\right)$ |

$1,\dots ,L$ | ${\tau}_{2}^{\u2605}$ | $1,\dots ,L$ | ${\tau}_{4}^{\u2605}$ | ${H}_{5}\left({\tau}_{1},{\tau}_{2}^{\u2605},{\tau}_{3},{\tau}_{4}^{\u2605}\right)$ |

$1,\dots ,L$ | ${\tau}_{2}^{\u2605}$ | ${\tau}_{3}^{\u2605}$ | $1,\dots ,L$ | ${H}_{5}\left({\tau}_{1},{\tau}_{2}^{\u2605},{\tau}_{3}^{\u2605},{\tau}_{4}\right)$ |

${\tau}_{1}^{\u2605}$ | $1,\dots ,L$ | $1,\dots ,L$ | ${\tau}_{4}^{\u2605}$ | ${H}_{5}\left({\tau}_{1}^{\u2605},{\tau}_{2},{\tau}_{3},{\tau}_{4}^{\u2605}\right)$ |

${\tau}_{1}^{\u2605}$ | $1,\dots ,L$ | ${\tau}_{3}^{\u2605}$ | $1,\dots ,L$ | ${H}_{5}\left({\tau}_{1}^{\u2605},{\tau}_{2},{\tau}_{3}^{\u2605},{\tau}_{4}\right)$ |

${\tau}_{1}^{\u2605}$ | ${\tau}_{2}^{\u2605}$ | $1,\dots ,L$ | $1,\dots ,L$ | ${H}_{5}\left({\tau}_{1}^{\u2605},{\tau}_{2}^{\u2605},{\tau}_{3},{\tau}_{4}\right)$ |

**Table 2.**Optimal embedding dimensions and optimal time lags for the Rössler time series with different noise levels.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tao, M.; Poskuviene, K.; Alkayem, N.F.; Cao, M.; Ragulskis, M. Permutation Entropy Based on Non-Uniform Embedding. *Entropy* **2018**, *20*, 612.
https://doi.org/10.3390/e20080612

**AMA Style**

Tao M, Poskuviene K, Alkayem NF, Cao M, Ragulskis M. Permutation Entropy Based on Non-Uniform Embedding. *Entropy*. 2018; 20(8):612.
https://doi.org/10.3390/e20080612

**Chicago/Turabian Style**

Tao, Mei, Kristina Poskuviene, Nizar Faisal Alkayem, Maosen Cao, and Minvydas Ragulskis. 2018. "Permutation Entropy Based on Non-Uniform Embedding" *Entropy* 20, no. 8: 612.
https://doi.org/10.3390/e20080612