# Multivariate Matching Pursuit Decomposition and Normalized Gabor Entropy for Quantification of Preictal Trends in Epilepsy

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Univariate Matching Pursuit Decomposition (MP)

^{th}level iteration, the MP algorithm selects the most suitable basis function ${g}_{\gamma n}\left(t\right)$ for the previous residual signal ${R}^{n}f\left(t\right)$, leaving a new residual signal ${R}^{n+1}f\left(t\right)$:

_{n}equals the inner product ${R}^{n}f\left(t\right),{g}_{\gamma n}\left(t\right)$.

_{n}and the final residual:

#### 2.2. Multivariate Matching Pursuit Decomposition (MMP)

#### 2.3. MMP-Based Gabor Measures of Complexity

**GAD**is the number of Gabor atoms selected by MMP from the basis functions dictionary in the decomposition of the original signals under the predetermined threshold criteria. GAD appears to be a “natural” measure of complexity under the assumption that more complex activity corresponds to higher number of atoms in the decomposed EEG signal. However, GAD treats all atoms equally and does not account for any difference in their characteristics, e.g., in their energy.

**GMF**is the mean frequency that the Gabor atoms exhibit, estimated by averaging the modulation frequencies ξ

_{i}of each Gabor atom i in the decomposition.

**GE**is

_{n}**GE**) can be defined as:

**NGE**) as

## 3. Results

#### 3.1. Simulation Data and Gabor Measures of Complexity

#### 3.2. Intracranial EEG Data

#### 3.3. Estimation and Trend Analysis of Gabor Measures of Complexity from the EEG data

**Hypothesis (I)**, existence of “short-term” trend during the one 2-min EEG epoch immediately preceding seizures onset; and

**Hypothesis (II)**, existence of “long-term” trend along the six epochs over the 60-min period prior to seizures.

**Test of Hypothesis I:**The 120 values derived from each measure within the one 2-min epoch immediately prior to seizures’ onset were used in the trend analysis. The trend coefficients ${\alpha}_{i}$s that were estimated only within this epoch per patient were used in the modeling with the three models to quantify any characteristic preictal (short-term) trends per patient within this epoch, and then test the null hypothesis of no statistically significant average trend $\overline{\alpha}$ across patients in this epoch.

**Test of Hypothesis II:**We followed the following steps: (1) Estimate the mean μ and the standard deviation σ of the 120 values per epoch for each of the six pre-seizure epochs per seizure. (2) Apply the model fitting (m1, m2, or m3) on those mean and standard deviation values per seizure to derive the long-term preictal trends ${a}_{i,j}$ per seizure over time. (3) Average ${a}_{i,j}$ across seizures to derive the ${\alpha}_{i}$ for μ and σ per patient I. (4) Estimate the statistical significance of the average $\overline{\alpha}$ of ${\alpha}_{i}$ for both μ and σ across all patients as in testing Hypothesis (I). Under this framework, we sought to detect any long preictal trends (within 1 h prior to seizures) in the mean and standard deviation of the complexity values that were common across seizures and patients. In the test of this hypothesis we used an uneven sampling of epochs before a seizure, selecting more epochs closer than farther away from seizure onsets. The rationale for this non-uniform sampling is that EEG epochs closer to seizure onset are expected to contain more information relevant to transition to impending seizure than ones farther away from seizure onset. In addition, farther away from seizure onset epochs are expected to produce more spurious (i.e., irrelevant with respect to a trend in) values of complexity and, thus, the more such epochs we include in our trend analysis the more noisy and difficult to identify any trends would be. In addition, given that preictal periods cannot be accurately determined and may vary in duration per seizure in the same patient as well as across patients, uniform sampling of epochs over a long (e.g., 1 h) period prior to seizures may increase the probability of inclusion of interictal (normal) brain activity that would mask real preictal activity.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The three Lyapunov exponents of the 3-D Lorenz system (

**top panel**) and the Lyapunov dimension (

**bottom panel**) as a function of the model’s Rayleigh number $\rho $.

**Figure 2.**The mean values of the five MMP measures of complexity employed to characterize the evolution of the 3-D Lorenz system as a function of $\rho $ (logarithmic scale is used for GEn values).

**Figure 3.**Electrode montages for the analyzed intracranial EEG recordings: (

**a**) strip electrodes placed on the right and left orbitofrontal (ROF and LOF, respectively), and right and left subtemporal cortex (RST and LST, respectively) and depth electrodes on the right and left hippocampus (RTD and LTD, respectively); and (

**b**) electrodes placed in same places as in (

**a**) and additional depth electrodes placed on the right and left amygdala (RA and LA, respectively), and right and left frontal areas (RO and LO, respectively).

**Figure 4.**Diagrammatic representation of the temporal location of the six preictal EEG epochs that were analyzed from the available 1 h preictal iEEG recordings per seizure and patient.

**Figure 5.**Complexity values per Gabor measure within each of six EEG epochs prior to Seizure 6 of Patient 2. Epochs are 2-min in duration and each measure value was estimated from 1-s non-overlapping EEG segments within each epoch (120 measure values per epoch).

Patient | Gender | # Recording Electrodes | Available iEEG Duration (hours) | Number of Isolated Clinical Seizures |
---|---|---|---|---|

1 | F | 40 | 34.67 | 4 |

2 | M | 28 | 281.68 | 6 |

3 | F | 28 | 86.3 | 14 |

4 | M | 28 | 334.62 | 7 |

5 | M | 28 | 85.02 | 3 |

6 | M | 28 | 156.22 | 2 |

7 | M | 28 | 145.77 | 3 |

8 | F | 28 | 18.77 | 3 |

Measure | Model | $\overline{\mathit{\alpha}}$ | FDR Adjusted p-Value |
---|---|---|---|

GAD | m3 | 0.0092 | 1 |

GMF | m3 | 0.0032 | 1 |

GEn | m3 | $8.2920\times {10}^{-5}$ | 1 |

GE | m3 | $3.0526\times {10}^{-5}$ | 1 |

NGE | m3 | $-2.7492\times {10}^{-5}$ | 1 |

**Table 3.**The optimally selected models (m1, m2 or m3) for identification of long-term (across epochs) trends ${\alpha}_{i}$ in $\mu $ and $\sigma $ profiles per Gabor complexity measure.

Complexity Measure | GAD | GMF | GEn | GE | NGE |
---|---|---|---|---|---|

Statistic of Measure | Optimized Model for Trend Identification of Statistic across Epochs | ||||

$\mu $ | m1 | m3 | m1 | m1 | m1 |

$\sigma $ | m1 | m3 | m2 | m1 | m2 |

Patient | P1 (α_{1}) | P2 (α_{2}) | P3 (α_{3}) | P4 (α_{4}) | P5 (α_{5}) | P6 (α_{6}) | P7 (α_{7}) | P8 (α_{8}) | $\overline{\mathit{\alpha}}$ | FDR Adjusted p-Value for $\overline{\mathit{\alpha}}$ Significance |
---|---|---|---|---|---|---|---|---|---|---|

Statistic (Measure) | ||||||||||

$\mu $(GAD) | 0.0242 | −0.0102 | 0.479 | −0.0173 | −0.0097 | −0.0213 | −0.0215 | 0.1185 | 0.0138 | 0.7266 |

$\sigma $(GAD) | 0.031 | 0.0241 | 0.0248 | 0.0424 | 0.0501 | 0.0467 | 0.0237 | 0.0588 | 0.0316 | 0.0206 |

$\mu $(GMF) | 0.0177 | 0.0089 | 0.0209 | −0.0031 | −0.0047 | −0.0017 | −0.0079 | 0.052 | 0.0103 | 0.2359 |

$\sigma $(GMF) | 0.0076 | 0.0042 | 0.0025 | 0.003 | 0.0036 | 0.0013 | 0.0026 | 0.0243 | 0.0061 | 0.0195 |

$\mu $(GEn) | −0.0009 | 0.0063 | −0.0035 | 0.0088 | 0.0038 | 0.0023 | 0.0126 | −0.0018 | 0.0034 | 0.1773 |

$\sigma $(GEn) | 0.0013 | 0.0034 | 0.0014 | 0.002 | 0.0024 | 0.0005 | 0.0062 | 0.0055 | 0.0028 | 0.0195 |

$\mu $(GE) | −0.0018 | −0.0082 | 0.0141 | −0.0147 | −0.0035 | −0.019 | −0.0162 | 0.0252 | −0.003 | 0.6645 |

$\sigma $(GE) | 0.0471 | 0.0201 | 0.021 | 0.0777 | 0.0371 | 0.0652 | 0.06 | 0.1004 | 0.0536 | 0.0097 |

$\mu $(NGE) | −0.0077 | −0.0074 | −0.001 | −0.0091 | 0 | −0.0117 | −0.0098 | −0.0176 | −0.008 | 0.0195 |

$\sigma $(NGE) | 0.0017 | 0.0014 | −0.0004 | 0.0027 | 0.0024 | 0.0039 | 0.004 | −0.001 | 0.0018 | 0.0409 |

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**MDPI and ACS Style**

Liu, R.; Karumuri, B.; Adkinson, J.; Hutson, T.N.; Vlachos, I.; Iasemidis, L.
Multivariate Matching Pursuit Decomposition and Normalized Gabor Entropy for Quantification of Preictal Trends in Epilepsy. *Entropy* **2018**, *20*, 419.
https://doi.org/10.3390/e20060419

**AMA Style**

Liu R, Karumuri B, Adkinson J, Hutson TN, Vlachos I, Iasemidis L.
Multivariate Matching Pursuit Decomposition and Normalized Gabor Entropy for Quantification of Preictal Trends in Epilepsy. *Entropy*. 2018; 20(6):419.
https://doi.org/10.3390/e20060419

**Chicago/Turabian Style**

Liu, Rui, Bharat Karumuri, Joshua Adkinson, Timothy Noah Hutson, Ioannis Vlachos, and Leon Iasemidis.
2018. "Multivariate Matching Pursuit Decomposition and Normalized Gabor Entropy for Quantification of Preictal Trends in Epilepsy" *Entropy* 20, no. 6: 419.
https://doi.org/10.3390/e20060419