# A Study of EEG Feature Complexity in Epileptic Seizure Prediction
## 3 Results and Discussion
### Analysis of the Complexity of the Pre-Ictal and Inter-Ictal Features
\begin{table}
\begin{tabular}{l l} \hline \hline
**Complexity Metrics** & **Lower and Upper Thresholds** \\ \hline Fisher discriminant ratio F1 & \{\textless{}0.001,**0.017**\} \\ Volume of overlap region F2 & \{\textless{}0.020, 0.987\} \\ Feature efficiency F3 & \{\textless{}0.001,**0.0048**\} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thresholds for the complexity metrics. Figure 2: Empirical and fitted distribution for the complexity measures: (**a**) Fisher discriminant ratio F1, (**b**) the volume of the overlap interval F2, and (**c**) the individual feature efficiency F3. Figure 3: Evaluation of the complexity metrics for each extracted value: (**a**) Fisher discriminant ratio, F1: a high value of F1 shows that the feature is discriminant, (**b**) the volume of the overlap interval, F2: a low value of F2 indicates small amount of overlap among the classes, and (**c**) the individual feature efficiency, F3: a high value of F3 implies a good separability between the classes. The horizontal dotted blue represents the threshold values. \begin{table}
\begin{tabular}{l l c} \hline \hline
**Feature** & **Type** & \(\mathbf{p}\)**-Value** \\ \hline Hjorth parameter, HM & linear, univariate & \textless{}0.001 \\ Hjorth parameter, HC & linear, univariate & \textless{}0.001 \\ Relative gamma band power spectral, \(\gamma_{r}\) & linear, univariate & \textless{}0.001 \\ Bivariate spectral power characteristics, \(b_{18}\) & linear, bivariate & \textless{}0.001 \\ Bivariate spectral power characteristics, \(b_{19}\) & linear, bivariate & \textless{}0.001 \\ Bivariate spectral power characteristics, \(b_{44}\) & linear, bivariate & \textless{}0.001 \\ Mutual information, \(MI\) & linear, bivariate & \textless{}0.001 \\ Phase synchronization index based on conditional & & \\ probability using the wavelet transform, \(\lambda_{cp}^{W}\) & non-linear, bivariate & \textless{}0.001 \\ Phase synchronization index based on conditional & & \\ probability using the Hilbert transform, \(\lambda_{cp}^{H}\) & non-linear, bivariate & \textless{}0.001 \\ Mean phase coherence, \(R\) & non-linear, bivariate & \textless{}0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Retained attributes from the feature complexity analysis, their types, and the corresponding \(p\)-value of the \(t\)-test. Despite of extracting many linear and non-linear both univariate and bivariate features from EEG records, we found that a restricted number of attributes shown to be promising to distinguish the inter-inctal and pre-ictal classes. **Comparison with reference databases**: To have some insight into the complexity of the classification problem of inter-ictal and pre-ictal states, we compared our results against known relatively simple classification problems from the UC-Irvine Machine Learning Depository [50] and other randomly labeled synthetic data. We estimate the complexity measures of three binary classification problems from the _Iris_ dataset and a linearly non-separable problem using the _Letter_ database of 20,000 samples with 16 attributes. We also used a more complex dataset for the human activity recognition (HAR) using sensor signals (accelerometer and gyroscope) recorded from a waist-mounted smartphone. The dataset contains 10,299 samples with 561 features. Moreover, we evaluate the complexity metrics on two artificial classification problems, obtained from randomly labeling uniformly distributed data points, containing, respectively, 10,000 samples with a single feature, and 600,000 with 88 dimensions. The results summarized in Table 3 show that the maximum Fisher discriminant ratio of the epilepsy dataset is lower than simpler problems, for instance, classification tasks from the _iris_ and _letter_ sets and higher than results from the random noise sets.