# Permutation Entropy: New Ideas and Challenges

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Some Theoretical Background

#### 2.1. The Kolmogorov–Sinai Entropy

#### 2.2. Observables and Ordinal Partitioning

#### 2.3. No Information Loss

#### 2.4. Conditional Entropy of Ordinal Patterns

**Definition**

**2.**

#### 2.5. Permutation Entropy

**Definition**

**3.**

#### 2.6. The Practical Viewpoint

## 3. Generalizations Based on the Families of Renyi and Tsallis Entropies

#### 3.1. The Concept

**Definition**

**4.**

#### 3.2. Some Properties

#### 3.3. Demonstration

## 4. Classification on the Base of Different Entropies

**Definition**

**5.**

#### 4.1. The Data

- group A: surface EEG’s recorded from healthy subjects with open eyes,
- group B: surface EEG’s recorded from healthy subjects with closed eyes,
- group C: intracranial EEG’s recorded from subjects with epilepsy during a seizure-free period from within the epileptogenic zone,
- group D: intracranial EEG’s recorded from subjects with epilepsy during a seizure-free period from hippocampal formation of the opposite hemisphere of the brain,
- group E: intracranial EEG’s recorded from subjects with epilepsy during a seizure period.

#### 4.2. Visualization and Classification for Delay One

#### 4.3. Other Delays

## 5. Resume

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Different estimates of the KS entropy of $T(\omega )=4\omega (1-\omega );\phantom{\rule{0.166667em}{0ex}}\omega \in [0,1]$ for different orbit lengths. On the top: $h(d,2)-h(d,1)$ for $d=2,3,\dots ,7$; in the middle: $h(7,k)-h(7,k-1)$ for $k=2,3,\dots ,10$; on the bottom: $h(d,2)-h(d,1)$ versus $\frac{h(d,1)}{d}$ for $d=5,6,7$.

**Figure 4.**Comparison of empirical Renyi Permutation entropy (eRPE) for $\alpha =0.5$, 2, empirical Tsallis Permutation entropy (eTPE) for $\alpha =2$ and ePE, computed from EEG recordings before stimulator implantation (data set 1) and after stimulator implantation (data set 2) for 19 channels using a shifted time window, order $d=3$ and delay $\tau =4$. Highlighting, in particular, the channels T3 (green line) and P3 (red line) as well as the entropy over all channels by a fat black line. The sampling rate is 256 Hz.

**Figure 5.**eRPE for $\alpha =0.01$, 35 and eTPE for $\alpha =35$ computed from data set 1 for 19 channels using a shifted time window, $d=3$ and $\tau =4$ (cf. Figure 4). T3: green line, P3: red line.

**Figure 9.**Second principal component versus the first one obtained from principal component analysis on three entropy variables.

α | $0.5$ | $0.8$ | $0.9$ | $1.1$ | $1.2$ | $1.5$ | 2 | 250 |
---|---|---|---|---|---|---|---|---|

Fp2 | $98.69\phantom{\rule{0.166667em}{0ex}}\%$ | $99.52\phantom{\rule{0.166667em}{0ex}}\%$ | $99.77\phantom{\rule{0.166667em}{0ex}}\%$ | $99.78\phantom{\rule{0.166667em}{0ex}}\%$ | $99.58\phantom{\rule{0.166667em}{0ex}}\%$ | $99.05\phantom{\rule{0.166667em}{0ex}}\%$ | $98.38\phantom{\rule{0.166667em}{0ex}}\%$ | $94.85\phantom{\rule{0.166667em}{0ex}}\%$ |

T3 | $95.41\phantom{\rule{0.166667em}{0ex}}\%$ | $98.39\phantom{\rule{0.166667em}{0ex}}\%$ | $99.22\phantom{\rule{0.166667em}{0ex}}\%$ | $99.26\phantom{\rule{0.166667em}{0ex}}\%$ | $98.59\phantom{\rule{0.166667em}{0ex}}\%$ | $96.92\phantom{\rule{0.166667em}{0ex}}\%$ | $95.20\phantom{\rule{0.166667em}{0ex}}\%$ | $89.31\phantom{\rule{0.166667em}{0ex}}\%$ |

P3 | $93.18\phantom{\rule{0.166667em}{0ex}}\%$ | $97.71\phantom{\rule{0.166667em}{0ex}}\%$ | $98.93\phantom{\rule{0.166667em}{0ex}}\%$ | $99.05\phantom{\rule{0.166667em}{0ex}}\%$ | $98.21\phantom{\rule{0.166667em}{0ex}}\%$ | $96.30\phantom{\rule{0.166667em}{0ex}}\%$ | $94.33\phantom{\rule{0.166667em}{0ex}}\%$ | $83.66\phantom{\rule{0.166667em}{0ex}}\%$ |

Entropy | Classification Accuracy (In %) |
---|---|

ApEn | 31.0 |

SampEn | 37.8 |

ePE | 32.0 |

eCE | 30.0 |

Entropy | Classification Accuracy (In %) |
---|---|

ApEn & SampEn | 51.0 |

ApEn & ePE | 58.0 |

ApEn & eCE | 61.8 |

SampEn & ePE | 64.0 |

SampEn & eCE | 64.6 |

ePE & eCE | 48.2 |

Entropy | Classification Accuracy (In %) |
---|---|

ApEn & SampEn & ePE | 67.4 |

ApEn & SampEn & eCE | 66.8 |

ApEn & ePE & eCE | 65.4 |

SampEn & ePE & eCE | 71.8 |

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**MDPI and ACS Style**

Keller, K.; Mangold, T.; Stolz, I.; Werner, J. Permutation Entropy: New Ideas and Challenges. *Entropy* **2017**, *19*, 134.
https://doi.org/10.3390/e19030134

**AMA Style**

Keller K, Mangold T, Stolz I, Werner J. Permutation Entropy: New Ideas and Challenges. *Entropy*. 2017; 19(3):134.
https://doi.org/10.3390/e19030134

**Chicago/Turabian Style**

Keller, Karsten, Teresa Mangold, Inga Stolz, and Jenna Werner. 2017. "Permutation Entropy: New Ideas and Challenges" *Entropy* 19, no. 3: 134.
https://doi.org/10.3390/e19030134