# Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models
## 8 Discussion  
In examining the volatility of EEG signals, two of the three channels where seizures originated (RAST3 and RAST4) had shorter volatility half-life relative to all other channels in the time leading up to the seizure (6 min) and during an awake state. During a sleep state and a sleep-to-awake state, the volatility half-life for the seizure channels were lower for most segments examined, but the respective confidence intervals (constructed from both methods) overlapped a larger proportion of non-seizure confidence intervals. The volatility also tended to persist longer in the sleep state than in the pre-seizure and awake states, but the half-life varied more than in the pre-seizure and awake data. The volatility half-life was similar to the non-seizure channels, especially in the awake data; however, for the sleep data, the half-lives were lower for half the segments. For all of the models, the GARCH coefficient was larger than the ARCH coefficient, indicating larger volatility persistence. However, for some of models, the magnitudes of the coefficients could \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Signal** & **Channel** & **Mean (s.d.)** & **Min** & **Q1** & **Med** & **Q3** & **Max** \\ \hline \multirow{4}{*}{Seizure} & RAST3 & \(\begin{array}{c}2.11\\ (1.31)\end{arrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarrayarray {1}{1}}\) & \(1.18\) & \(1.45\) & \(1.70\) & \(1.89\) & \(5.56\) \\  & RAST4 & \(\begin{array}{c}1.86\\ (0.68)\end{arrayarrayarrayarrayarrayarray}\) & \(1.44\) & \(1.59\) & \(1.75\) & \(1.91\) & \(5.64\) \\  & RMST4 & \(\begin{array}{c}22.49\\ (52.72)\end{arrayarrayarrayarray}\) & \(6.28\) & \(6.39\) & \(6.45\) & \(8.11\) & \(276.91\) \\  & Others & \(\begin{array}{c}38.33\\ (73.79)\end{array}\) & \(0.16\) & \(6.44\) & \(17.15\) & \(42.05\) & \(2310.14\) \\ \hline \multirow{4}{*}{Awake} & RAST3 & \(\begin{array}{c}2.27\\ (1.49)\end{arrayarrayarray}\) & \(1.15\) & \(1.46\) & \(1.71\) & \(1.91\) & \(5.70\) \\  & RAST4 & \(\begin{array}{c}2.61\\ (1.89)\end{array}\) & \(1.29\) & \(1.43\) & \(1.56\) & \(2.17\) & \(6.62\) \\  & RMST4 & \(\begin{array}{c}20.99\\ (20.92\end{array}\) & \(6.17\) & \(6.51\) & \(8.51\) & \(36.66\) & \(97.28\) \\  & Other & \(\begin{array}{c}26.53\\ (83.71)\end{array}\) & \(0.21\) & \(6.4\) & \(10.55\) & \(28.06\) & \(3465.39\) \\ \hline \multirow{4}{*}{Sleep} & RAST3 & \(\begin{array}{c}12.92\\ (6.06)\end{array}\) & \(5.20\) & \(9.19\) & \(11.62\) & \(15.61\) & \(50.25\) \\  & RAST4 & \(\begin{array}{c}15.76\\ (6.96)\end{array}\) & \(5.95\) & \(10.06\) & \(14.43\) & \(19.04\) & \(33.30\) \\  & RMST4 & \(\begin{array}{c}18.39\\ (8.15)\end{array}\) & \(5.84\) & \(11.72\) & \(17.8\) & \(23.03\) & \(36.52\) \\  & Other & \(\begin{array}{c}56.57\\ (188.78)\end{array}\) & \(0.16\) & \(28.29\) & \(37.32\) & \(49.16\) & \(6931.13\) \\ \hline \multirow{4}{*}{Sleep/Awake} & RAST3 & \(\begin{array}{c}22.20\\ (49.86)\end{arrayarrayarray}\) & \(5.35\) & \(9.05\) & \(13.93\) & \(20.34\) & \(494.76\) \\  & RAST4 & \(\begin{array}{c}52.46\\ (114.55)\end{arrayarray}\) & \(5.10\) & \(9.27\) & \(14.36\) & \(45.83\) & \(692.80\) \\  & RMST4 & \(\begin{array}{c}14.99\\ (16.58)\end{array}\) & \(5.87\) & \(6.38\) & \(6.44\) & \(12.24\) & \(74.99\) \\ \cline{1-1}  & Other & \(\begin{array}{c}41.48\\ (152.17)\end{array}\) & \(0.16\) & \(6.48\) & \(20.28\) & \(42.97\) & \(6931.13\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of half-life estimates based on random starting points. be an issue. While the coefficients were significant at the 5% level, some of the models had GARCH coefficients greater than 0.95 and ARCH coefficients smaller 0.01. The small ARCH coefficients make the effect of the past innovations negligible, which is an issue since GARCH models must have at least one non-zero ARCH parameter for the GARCH parameters to be identified. Additionally, the sum of the coefficients was greater than 0.99 (but less than 1.00), which is very close to being an integrated model.