# Differentiating Interictal and Ictal States in Childhood Absence Epilepsy through Permutation Rényi Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. EEG Recording

#### 2.2. Permutation Rényi Entropy

_{t}were constructed selecting m equally-spaced samples from x, starting from time point t:

_{t}are reshaped in an increasing order, and the time points are renamed yielding Xr

_{t}, a reshaped version of X

_{t}:

_{t}can be considered uniquely mapped onto a symbol vector π = [t

_{1}, t

_{2}, …, t

_{m}]. The vector π is a sequence of time points, hence a symbol. The frequency of occurrence of each possible π is indicated as p(π), which represents the frequency of occurrence of the specific vector π in the time series under analysis, normalized by N − (m − 1)L, where N is the number of samples of the time series x. PE is finally computed as:

_{i}) by log(p(π

_{i})) is more computationally intensive than calculating p(π

_{i})

^{α}, even for high α values, and that Rényi’s PE calculation is faster that standard PE.

#### 2.2.1. PEr vs. PE

#### 2.2.2. Selecting the Optimal Parameter Configuration for PEr

- Order m ranging from 3–7;
- Lag L ranging from 1–10;
- α ranging from 2–7 (for PEr only).

- the difference D = avgPE(interictal) − avgPE(ictal);
- the ratio R = avgPE(interictal)/avgPE(ictal).

## 3. Results

#### 3.1. Analysis for Patient 4: Pilot Trial

#### 3.1.1. Optimization for Patient 4

_{i}and R

_{i}were selected.

#### 3.1.2. Evaluating the Effects of Alpha

#### 3.2. Analysis of the Entire Dataset

**(avgPE(interictal))**and an average ictal entropy value (

**avgPE**(

**ictal**)), for each patient. The two values are used to compute the difference

**D**

_{PE}=

**avgPE(interictal) − avgPE(ictal)**and the ratio

**R**

_{PE}=

**avgPE(interictal)/avgPE(ictal)**. In this way, we came up with a vector

**avgPE(interictal)**, a vector

**avgPE(ictal)**, a vector

**D**and a vector

**R**, each one with N elements, where N is the number of analyzed EEG recordings (N = 23).

**R**or

**D**vectors were plotted in Figure 6.

**avgPEr(interictal)**and

**avgPEr(ictal)**vectors are plotted in Figure 7.

**avgPEr(interictal)**and

**avgPEr(ictal)**vectors for each specific area.

**R**or the

**D**perspective.

#### 3.3. Classifying the Brain States according to PE and PEr

## 4. Conclusions

## Acknowledgments

## Appendix

## A. Statistical Analysis

#### A.1. D_{PEr} is Higher than D_{PE}

_{PEr}> D

_{PE}was tested statistically considering the two

**D**

_{PEr}and

**D**

_{PE}vectors as the two populations to be compared: X1 =

**D**

_{PEr}and X2 =

**D**

_{PE}. The hypothesis to be tested is that PEr outperforms PE in discriminating interictal from ictal EEG in absence seizure EEG. The alternative hypothesis, H1, can be formulated as follows: the difference between the average entropy of the interictal EEG and the average entropy of the ictal EEG is higher with PEr than with PE. The null hypothesis, H0, thus can be formulated as: the difference between the average entropy of the interictal EEG and the average entropy of the ictal EEG is the same for PEr and PE. First of all, we tested the normality of the two populations X1 =

**D**

_{PEr}and X2 =

**D**

_{PE}by means of the Shapiro–Wilk test [29]. Both populations were estimated to be normally distributed, and hence, the one-tailed t-test was chosen. The decision rule is to reject H0 if:

_{t}= 0.05. The critical value tn1 + n2 − 2, α was 1.6808, and as the result was tâĹŠtest = 4.26, according to the decision rule, the null hypothesis was rejected.

#### A.2. R_{PEr} is Higher than R_{PE}

**R**vectors as the two populations to be compared: X1 =

**R**

_{PEr}and X2 =

**R**

_{PE}. The alternative hypothesis, H1, can be formulated as follows: the ratio between the average entropy of the interictal EEG and the average entropy of the ictal EEG is higher with PEr than with PE. The null hypothesis, H0, can be formulated as: the ratio between the average entropy of the interictal EEG and the average entropy of the ictal EEG is the same for PEr and PE. First of all, we tested the normality of the two populations X1 =

**R**

_{PEr}and X2 =

**R**

_{PE}by means of the Shapiro–Wilk test again. Both populations were estimated to be normally distributed, and hence, the one-tailed t-test was chosen again. The critical value tn1 + n2 − 2, was 1.6808, and as the result was tâĹŠtest = 4.52, according to the decision rule, the null hypothesis was rejected.

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The flowchart of the procedure. (1) The n-channel EEG is recorded and stored on a computer; (2) the EEG is partitioned into m overlapping windows and processed window by window; (3) given a window under analysis, permutation entropy (PE) is estimated channel by channel, and the n values are arranged in an n × 1 PE vector; (4) once the m PE vectors were estimated, the n × m matrix showing the PE trend of each channel can be displayed. PEr, permutation Rényi entropy.

**Figure 2.**The transient logistic map data x (top), variations of PE and PEr with r for m = 5, L = 1 (middle), and m = 5, L = 2 (bottom).

**Figure 3.**R vs. D in every possible parameter setting for either PE (blue circles) or PEr (red circles). The parameter setting that ensured the highest Di and Ri for PE was m = 4, L = 8 (bold blue circle), whereas the parameters that ensured the best performance for PEr with respect to either D or R was m = 4, L = 7 and α = 7 (red bold circle; the configuration ensured 96.5% of both the highest D and the highest R).

**Figure 4.**Effect of alpha on the behavior of PEr. The Figure shows the PE and PEr profiles of an EEG trace (from Channel Fp2), estimated in the same parameter configuration (m = 4 and L = 8, which is optimal for PE, but not for PEr). In the PEr estimation, α ranged from 2 to 7. The seizure onset is marked with a vertical dashed line.

**Figure 6.**Boxplot of R and D vectors. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.

**Figure 7.**Boxplot of

**avgPEr(interictal)**and

**avgPEr(ictal)**vectors. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.

**Figure 8.**Investigation on the effect of the EEG channel on the discrimination ability of PEr for ictal/interictal segments. Given a patient, PEr profiles are averaged over every cerebral area of interest (frontal, temporal, parietal, central, occipital), then the elements of

**R**vector are calculated as avgPEr(interictal/avgPEr(ictal). At the end of the process, we have one R vectors with n elements, where n is the number of patients, for every specific area. The boxplot of such

**R**vectors is shown. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.

**Figure 9.**Comparison of the accuracy in the cerebral state classification (interictal or ictal) provided by PE + learning vector quantization (LVQ) and by PEr + LVQ.

Pt | PE | PEr | |||
---|---|---|---|---|---|

m | L | m | L | α | |

1 | 4 | 3 | 5 | 4 | 6 |

2 | 4 | 2 | 5 | 3 | 3 |

3 | 4 | 10 | 4 | 7 | 7 |

4 | 4 | 8 | 4 | 7 | 7 |

5 | 3 | 10 | 3 | 5 | 7 |

6 | 4 | 3 | 5 | 5 | 7 |

7 | 4 | 2 | 5 | 3 | 7 |

8 | 4 | 10 | 5 | 4 | 7 |

9 | 4 | 3 | 5 | 6 | 7 |

10 | 4 | 3 | 5 | 5 | 6 |

11 | 4 | 3 | 4 | 3 | 5 |

12 | 4 | 3 | 5 | 5 | 5 |

13 | 4 | 8 | 4 | 7 | 7 |

14 | 4 | 3 | 5 | 4 | 7 |

15 | 4 | 2 | 4 | 3 | 4 |

16 | 4 | 3 | 5 | 4 | 6 |

17 | 5 | 3 | 4 | 6 | 7 |

18 | 4 | 2 | 4 | 2 | 2 |

19 | 4 | 3 | 5 | 4 | 6 |

20 | 4 | 3 | 4 | 3 | 4 |

21 | 4 | 3 | 5 | 5 | 6 |

22 | 4 | 3 | 5 | 3 | 4 |

23 | 4 | 3 | 4 | 7 | 7 |

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**MDPI and ACS Style**

Mammone, N.; Duun-Henriksen, J.; Kjaer, T.W.; Morabito, F.C. Differentiating Interictal and Ictal States in Childhood Absence Epilepsy through Permutation Rényi Entropy. *Entropy* **2015**, *17*, 4627-4643.
https://doi.org/10.3390/e17074627

**AMA Style**

Mammone N, Duun-Henriksen J, Kjaer TW, Morabito FC. Differentiating Interictal and Ictal States in Childhood Absence Epilepsy through Permutation Rényi Entropy. *Entropy*. 2015; 17(7):4627-4643.
https://doi.org/10.3390/e17074627

**Chicago/Turabian Style**

Mammone, Nadia, Jonas Duun-Henriksen, Troels W. Kjaer, and Francesco C. Morabito. 2015. "Differentiating Interictal and Ictal States in Childhood Absence Epilepsy through Permutation Rényi Entropy" *Entropy* 17, no. 7: 4627-4643.
https://doi.org/10.3390/e17074627