# 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  
To illustrate the feature complexity analysis, three metrics are evaluated on the extensive list of the extracted features: F1, F2, and F3. The lower and higher threshold values have been identified, for each complexity metric, using a probability density which best fit each complexity data. Figure 2 illustrates the empirical and fitted distribution for each complexity measure. The Fisher discriminant ratio F1 can be modeled by a Weilbull minimum extreme value distribution (Figure 2a), the volume of overlap region F2, follows a Johnson SB distribution (Figure 2b) and the feature efficiency values F3, can be approximated by an exponentiated Weilbull distribution (Figure 2c). Following this modeling, the lower and upper threshold values are determined using the 5 th and the 95 th percentiles of each complexity measure empirical density, as shown in Table 1.  
The results of evaluating the Fisher discriminant ratio, F1, on various features, are shown in Figure 3a. The threshold was set to 0.017 (Table 1 line 1). As shown in Figure 3a, three bivariate spectral power attributes, \(b_{18}\), \(b_{19}\), and \(b_{44}\) and the mutual information, \(MI\), surpass the threshold value. Because higher values indicate better class separation, only these features are retained. Likewise, Figure 3b presents the results obtained by the assessment of the volume of overlap region, F2. Given the threshold 0.02 (Table 1, line 2), only the two Hjorth parameters, HM and HC, are retained since lower values indicate a small amount of overlap among the classes. In Figure 3c, the individual feature efficiency, F3, values for each characteristic are shown. For this complexity metric, the threshold is set to 0.0048 (Table 1, line 3). The features having a higher value of F3 than the threshold value are the relative gamma band spectral power, \(\gamma_{r}\), the Hjorth parameter, HM, the mean phase coherence using the Hilbert transform, \(R^{H}\), and the indexes measures for phase synchronization based on conditional probability, \(\lambda_{cp}^{H}\) and \(\lambda_{cp}^{W}\). Thus, the features are retained because high values of F3 claim a good feature separability between the categories. In summary, the analysis of the extracted features complexity reveals that only ten out of 88 attributes has been picked as not complex. We observed also an overlap between the different feature obtained by each complexity metrics, such as the Hjorth parameter, HM, which has a low value of volume of overlap, F2, and high value of the feature efficiency, F3. According to Table 2 summarizing the list of the retained features, all ten features have a p-value corresponding to the statistical t-test lower than the critical value for 5%significance level, which approves that the features are statistically significantly different, thus confirming their ability to discriminate between the two classes. We observe that only three non-linear features, the mean phase coherence using the Hilbert transform and the two measures for phase synchronization, the indexes based on conditional probability, \(\lambda_{cp}^{W}\) and \(\lambda_{cp}^{H}\), all bivariate, are retained as discriminant, declaring they are the best discriminant non-linear bivariate features and that most of the non-linear other features especially univariate characteristics are incapable to distinguish between the inter-ictal and pre-seizure times, as also observed by [6,47,48]. The Hjorth parameters, HM, and HC, are also shown to be the most uni-variate linear discriminant features of the two states as substantiated in Reference [6]. The other linear univariate recalled feature as discriminant is the relative gamma-band spectral power, \(\gamma_{r}\), which supports the studies [23,49] saying that the characteristics from the gamma band are more relevant than other bands for the epilepsy prediction.