# Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### Data

## 3. Conditional Volatility Models

#### 3.1. Autoregressive Moving Average Models

#### 3.2. Generalized Autoregressive Conditional Heteroscedasticity Models

_{t-1}, Y

_{t-2},…}), and, thus, is not satisfactory for use in modeling time series with time varying variance [56]. Variance that changes of over time may affect the inferential validity and efficiency of the parameters of an ARMA model [58]. Models that account conditionally for non-constant volatility (non-constant conditional variance) allow for better predictions of (local) variability and better prediction intervals [59]. Time series with non-constant volatility are common in finance, and the most common family of volatility models were developed to model financial time series; their use in biomedical research has been rarely utilized [60,61,62].

## 4. Confidence Intervals for Half-Life (HL)

## 5. Analysis

## 6. Results

#### 6.1. GARCH Models

#### 6.2. Half-life and Confidence Intervals

## 7. Random Starting Points

## 8. Discussion

## 9. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**GARCH parameters and unconditional volatility for three seizure channels (RAST3, RAST4, RMST4) compared to summarized values for non-seizure channels.

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 1.64 (1.29,1.98) 5 | 1.78 (1.55,2.01) 5 | 6.36 (3.24,9.47) 32 | 12.37 6.00 69.59 |

2 | 1.83 (1.37,2.29) 5 | 1.49 (1.26,1.72) 5 | N/A | 12.24 4.03 141.25 |

3 | 5.24 (4.27,6.22) 22 | 1.93 (1.68,2.18) 3 | 6.49 (3.11,9.88) 26 | 14.09 3.24 452.16 |

4 | 1.90 (1.58,2.23) 5 | 1.79 (1.55,2.03) 5 | 6.33 (1.52,11.14) 33 | 18.61 4.44 106.59 |

5 | 1.27 (0.98,1.55) 4 | 1.50 (1.30,1.70) 5 | 6.42 (0.10,12.74) 37 | 8.27 2.16 208.17 |

6 | 1.61 (1.30,1.91) 2 | 1.98 (1.73,2.23) 2 | 268.82 (0.14,537.50) 53 | 28.11 4.82 83.82 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 1.13 (0.86,1.41) 2 | 1.47 (1.18,1.75) 2 | 6.49 (2.73,10.24) 42 | 7.64 1.16 186.05 |

2 | 1.72 (1.20,2.23) 2 | 1.88 (1.47,2.29) 2 | 70.74 (5.59,135.88) 52 | 10.94 4.89 67.61 |

3 | 1.75 (1.06,2.44) 7 | 1.71 (1.31,2.11) 5 | 13.65 (3.83,23.47) 53 | 15.25 3.06 115.7 |

4 | 1.84 (1.35,2.34) 4 | 1.35 (1.14,1.58) 1 | 39.8 (6.99,72.6) 50 | 12.83 6.13 74.15 |

5 | 5.61 (4.30,6.95) 31 | 6.18 (4.94,7.43) 31 | 16.56 (7.42,25.69) 48 | 6.61 2.58 54.82 |

6 | 1.67 (1.26,2.09) 3 | 1.55 (1.29,1.80) 3 | 8.98 (3.1,14.86) 45 | 10.68 4.67 49.25 |

7 | 5.50 (4.66,6.33) 30 | 1.71 (1.47,1.95) 4 | N/A | 6.84 2.65 187.9 |

8 | 1.47 (1.17,1.78) 3 | 1.55 (1.32,1.79) 3 | 6.46 (3.07,9.85) 44 | 6.45 5.68 73.61 |

9 | 1.41 (1.12,1.70) 2 | 1.34 (1.15,1.54) 2 | 6.49 (3.28,9.7) 32 | 13.08 4.59 92.17 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 8.43 (6.39,10.47) 3 | 14.86 (9.97,19.75) 23 | 6.26 (4.89,7.64) 0 | 36.04 12.69 96 |

2 | 7.93 (5.61,10.25) 2 | 13.28 (7.10,19.47) 18 | 14.38 (7.94,20.82) 21 | 41.09 12.95 85.08 |

3 | 7.38 (5.82,8.93) 2 | 15.51 (10.79,20.24) 16 | 7.84 (6.35,9.32) 3 | 39.87 7.64 90.62 |

4 | 10.13 (7.92,12.33) 7 | 6.44 (4.98,7.91) 0 | 13.05 (9.92,16.17) 9 | 34.97 13.22 65.6 |

5 | 18.57 (14.58,22.55) 23 | 16.33 (10.77,21.88) 22 | 22.54 (16.63,28.45) 37 | 41.25 8.21 151.69 |

6 | 10.31 (7.99,12.62) 6 | 13.41 (8.53,18.20) 14 | 13.39 (9.93,16.85) 11 | 37.05 8.69 127.36 |

7 | 13.70 (11.47,15.93) 6 | 29.42 (18.56,40.29) 55 | 41.87 (26.61,57.13) 56 | 40.41 16.16 91.14 |

8 | 28.15 (19.96,36.34) 56 | 25.51 (14.18,36.84) 56 | 32.90 (20.28,45.51) 60 | 38.86 1.26 94.01 |

9 | 12.05 (9.23,14.97) 7 | 12.26 (8.59,15.93) 13 | 22.84 (16.49,29.19) 43 | 32.98 12.27 93.52 |

10 | 12.13 (9.97,14.29) 10 | 10.73 (8.26,13.20) 7 | 20.78 (15.07,26.49) 41 | 34.07 9.67 98.84 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 8.47 (5.74,11.21) 22 | 10.00 (7.10,12.90) 23 | 37.68 (12.44,62.93) 49 | 29.9 2.82 308.76 |

2 | 17.07 (12.91,21.22) 29 | 29.08 (12.12,46.04) 43 | 23.14 (8.68,37.61) 49 | 30.64 6.32 134 |

3 | 6.32 (5.11,7.54) 24 | 8.27 (5.20,11.35) 25 | 6.23 (3.87,8.59) 27 | 21.72 0.77 126.04 |

4 | 12.14 (8.40,15.88) 38 | 181.4 (80.7,282.1) 12 | 36.23 (13.53,58.92) 27 | 9.26 5.42 342.14 |

5 | 8.41 (6.59,10.24) 36 | 8.68 (4.51,12.85) 41 | 6.33 (3.94,8.72) 33 | 18.84 6.18 178.51 |

6 | 18.86 (13.29,24.44) 41 | 22.62 (5.77,39.47) 59 | 10.51 (6.7,14.33) 40 | 20.32 4.56 136.41 |

7 | 15.83 (10.98,20.68) 28 | 72.72 (24.29,121.2) 27 | 6.43 (2.43,10.44) 39 | 11.35 3.63 188.24 |

8 | 20.30 (14.19,26.40) 36 | 380.8 (134.7,626.9) 3 | 7.92 (1.45,14.39) 41 | 24.55 4.94 98.15 |

9 | 10.76 (7.27,14.24) 37 | 5.94 (4.40,7.49) 23 | 6.42 (2.5,10.34) 32 | 25.01 6.29 106.96 |

10 | 25.37 (12.71,38.03) 32 | 44.89 (20.83,68.94) 29 | 6.39 (1.64,11.14) 40 | 14.58 3.16 164.74 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 1.64 (1.34,2.05) 0 | 1.78 (1.57,2.04) 0 | 6.36 (4.23,12.16) 29 | 14.24 6 312.3 |

2 | 1.83 (1.45,2.41) 1 | 1.49 (1.29,1.75) 0 | N/A | 12.27 4.03 165.01 |

3 | 5.25 (4.41,6.44) 21 | 1.93 (1.7,2.21) 0 | 6.49 (4.23,13.17) 27 | 18.42 3.24 452.16 |

4 | 1.9 (1.62,2.28) 0 | 1.79 (1.57,2.06) 0 | 6.33 (3.53,23.46) 38 | 19.56 4.44 152.9 |

5 | 1.27 (1.02,1.61) 0 | 1.5 (1.32,1.72) 0 | 6.42 (3.15,99.28) 50 | 8.27 2.16 413.78 |

6 | 1.61 (1.35,1.97) 0 | 1.98 (1.75,2.26) 1 | 268.82 (134.36,148185.87) 14 | 29.05 4.24 139.58 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 1.13 (0.90,1.47) 1 | 1.47 (1.22,1.80) 1 | 6.49 (4.06,14.82) 42 | 9.33 1.16 283.79 |

2 | 1.72 (1.30,2.39) 0 | 1.88 (1.53,2.37) 0 | 70.74 (36.74,850.50) 23 | 11.43 4.89 166.48 |

3 | 1.75 (1.22,2.76) 1 | 1.71 (1.38,2.21) 1 | 13.65 (7.88,46.53) 55 | 18.72 3.06 237.25 |

4 | 1.84 (1.44,2.48) 0 | 1.36 (1.16,1.61) 0 | 39.8 (21.74,219.32) 39 | 13.06 6.13 153.46 |

5 | 5.62 (4.54,7.31) 30 | 6.18 (5.14,7.72) 30 | 16.56 (10.63,36.42) 38 | 6.61 2.58 54.82 |

6 | 1.67 (1.33,2.19) 0 | 1.55 (1.32,1.84) 0 | 8.98 (5.37,24.85) 50 | 10.64 4.67 49.25 |

7 | 5.5 (4.76,6.48) 28 | 1.71 (1.50,1.98) 0 | N/A | 7.42 2.65 187.9 |

8 | 1.47 (1.21,1.84) 0 | 1.55 (1.34,1.82) 0 | 6.46 (4.19,13.20) 45 | 6.45 5.68 92.53 |

9 | 1.41 (1.16,1.75) 0 | 1.34 (1.16,1.56) 0 | 6.49 (4.3,12.53) 36 | 13.26 4.59 94.41 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 8.43 (6.78,11.08) 0 | 14.86 (11.16,22.06) 16 | 6.26 (5.12,8) 0 | 26.15 9.94 64.51 |

2 | 7.93 (6.12,11.16) 0 | 13.28 (9.03,24.61) 17 | 14.38 (9.9,25.82) 18 | 29.67 1.19 57.56 |

3 | 7.38 (6.08,9.32) 0 | 15.51 (11.87,22.25) 11 | 7.84 (6.58,9.65) 0 | 29.51 6.36 55.36 |

4 | 10.13 (8.31,12.92) 3 | 6.44 (5.24,8.31) 0 | 13.05 (10.51,17.12) 4 | 26.31 10.65 47.97 |

5 | 18.57 (15.27,23.62) 14 | 16.33 (12.16,24.66) 16 | 22.54 (17.85,30.51) 24 | 30.11 7.08 82.57 |

6 | 10.31 (8.40,13.27) 0 | 13.41 (9.80,20.97) 5 | 13.39 (10.63,18.01) 1 | 28 7.14 79.1 |

7 | 13.7 (11.77,16.35) 1 | 29.43 (21.47,46.53) 50 | 41.87 (30.66,65.77) 53 | 29.67 11 63.35 |

8 | 28.15 (21.79,39.64) 48 | 25.51 (17.63,45.67) 50 | 32.9 (23.75,53.22) 53 | 28.22 1.21 64.22 |

9 | 12.05 (9.69,15.87) 2 | 12.26 (9.41,17.43) 6 | 22.84 (17.85,31.58) 36 | 25.75 8.99 729.5 |

10 | 12.13 (10.29,14.74) 2 | 10.73 (8.71,13.9) 2 | 20.78 (16.28,28.6) 35 | 26.01 7.98 67.47 |

Seizure Channels | Non-seizure Channels | |||
---|---|---|---|---|

RAST3 | RAST4 | RMST4 | Half Life | |

Segment | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | HL 95% CI Overlapping Interval Count | Median Min Max |

1 | 8.48 (6.39,12.43) 12 | 10.00 (7.73,14.02) 14 | 37.68 (22.51,112.78) 47 | 20.12 2.20 181.18 |

2 | 17.07 (13.71,22.52) 24 | 29.08 (18.32,69.11) 44 | 23.14 (14.19,60.8) 46 | 18.25 3.08 85.24 |

3 | 6.32 (5.29,7.81) 16 | 8.27 (6.01,13.05) 23 | 6.23 (4.50,9.91) 19 | 15.7 0.49 77.54 |

4 | 12.14 (9.26,17.48) 34 | 181.38 (116.58,407.3) 15 | 36.23 (22.22,96.03) 26 | 5.92 2.71 201.51 |

5 | 8.41 (6.90,10.72) 22 | 8.68 (5.82,16.44) 34 | 6.33 (4.57,10.04) 22 | 11.69 3.09 109.53 |

6 | 18.87 (14.55,26.72) 37 | 22.62 (12.90,85.89) 48 | 10.51 (7.69,16.39) 33 | 13.49 2.67 78.44 |

7 | 15.83 (12.10,22.75) 28 | 72.72 (43.60,216.39) 26 | 6.43 (3.91,16.19) 33 | 8.3 2.47 99.04 |

8 | 20.3 (15.58,28.97) 27 | 380.8 (231.25,1075.45) 6 | 7.92 (4.29,37.49) 47 | 17.17 2.49 54.15 |

9 | 10.76 (8.1,15.83) 25 | 5.94 (4.70,7.99) 15 | 6.42 (3.93,15.72) 25 | 16.65 3.03 59.89 |

10 | 25.37 (16.88,50.32) 35 | 44.89 (29.18,96.25) 25 | 6.39 (3.60,22.43) 41 | 10.08 2.7 98.01 |

Signal | Channel | Mean (s.d.) | Min | Q_{1} | Med | Q_{3} | Max |
---|---|---|---|---|---|---|---|

Seizure | RAST3 | 2.11 (1.31) | 1.18 | 1.45 | 1.70 | 1.89 | 5.56 |

RAST4 | 1.86 (0.68) | 1.44 | 1.59 | 1.75 | 1.91 | 5.64 | |

RMST4 | 22.49 (52.72) | 6.28 | 6.39 | 6.45 | 8.11 | 276.91 | |

Others | 38.33 (73.79) | 0.16 | 6.44 | 17.15 | 42.05 | 2310.14 | |

Awake | RAST3 | 2.27 (1.49) | 1.15 | 1.46 | 1.71 | 1.91 | 5.70 |

RAST4 | 2.61 (1.89) | 1.29 | 1.43 | 1.56 | 2.17 | 6.62 | |

RMST4 | 20.99 (20.92 | 6.17 | 6.51 | 8.51 | 36.66 | 97.28 | |

Other | 26.53 (83.71) | 0.21 | 6.4 | 10.55 | 28.06 | 3465.39 | |

Sleep | RAST3 | 12.92 (6.06) | 5.20 | 9.19 | 11.62 | 15.61 | 50.25 |

RAST4 | 15.76 (6.96) | 5.95 | 10.06 | 14.43 | 19.04 | 33.30 | |

RMST4 | 18.39 (8.15) | 5.84 | 11.72 | 17.8 | 23.03 | 36.52 | |

Other | 56.57 (188.78) | 0.16 | 28.29 | 37.32 | 49.16 | 6931.13 | |

Sleep/Awake | RAST3 | 22.20 (49.86) | 5.35 | 9.05 | 13.93 | 20.34 | 494.76 |

RAST4 | 52.46 (114.55) | 5.10 | 9.27 | 14.36 | 45.83 | 692.80 | |

RMST4 | 14.99 (16.58) | 5.87 | 6.38 | 6.44 | 12.24 | 74.99 | |

Other | 41.48 (152.17) | 0.16 | 6.48 | 20.28 | 42.97 | 6931.13 |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Follis, J.L.; Lai, D.
Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models. *Signals* **2020**, *1*, 26-46.
https://doi.org/10.3390/signals1010003

**AMA Style**

Follis JL, Lai D.
Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models. *Signals*. 2020; 1(1):26-46.
https://doi.org/10.3390/signals1010003

**Chicago/Turabian Style**

Follis, Jack L., and Dejian Lai.
2020. "Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models" *Signals* 1, no. 1: 26-46.
https://doi.org/10.3390/signals1010003