## 4 Results and Discussion
### EEG Data Results
#### 4.2.1 Motor Imagery Results  
Figure 6 depicts the average classification accuracy for all subjects in MI database as a function of the number of selected features during the training-validation stage, for \(\text{TE}_{\text{xx}}\) and \(\text{TE}^{\theta}_{\text{rx}}\). These results show there is a small improvement in the ability to discriminate between the MI tasks when using features extracted through phase TE, as compared with real-valued TE. In addition, they reveal that the CKA-based feature selection strategy successfully identified the most relevant connections for MI task classification. That is to say, the classification system has a stable performance even for a very reduced number of connectivity features. This is fundamental for any practical BCI application that intends to use phase TE as a characterization strategy, since estimating single-trial phase TE is computationally expensive [8]. Therefore, it is important to reduce as much as possible the number of channel pair connectivity features required to achieve peak classification performance. Additionally, it is important to highlight that while classification accuracies in Figure 6, and in Table 1, are in the same range of those obtained through other connectivity-based characterization approaches [10; 23], they are far below those obtained from methods such as common spatial patterns [59; 60; 61]. A possible explanation is that bivariate TE might be more robust at describing long-range interactions rather than local ones [41], like those arising from MI-related activity, centered on the sensorimotor area. In addition, the differences with the results in [10], where we used \(\text{TE}_{\text{xx}}\) to characterize the same database, lay mostly in the fact that in this study we select and analyze one 2 s long time window covering the period right after the end of the visual cue, while in [10] we report results from multiple overlapping windows covering the entirety of the task. Lastly, the large standard deviations from the average accuracies in Figure 6 point to disparate performances for different subjects. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & \multicolumn{2}{c}{**Motor Imagery (acc \%)**} & \multicolumn{2}{c}{**Working Memory (acc \%)**} \\ \cline{2-4}  & **Cross-Validation** & **Testing** & **Cross-Validation** \\ \hline
**GC**[11] & 64.3 \(\pm\) 11.7 & 57.1 \(\pm\) 11.0 & 53.0 \(\pm\) 7.4 \\
**TE\({}_{\text{xx}}\)[10]** & 65.5 \(\pm\) 11.4 & 62.8 \(\pm\) 11.7 & 67.5 \(\pm\) 4.2 \\ \hline
**PSI**[12; 51] & 62.4 \(\pm\) 7.8 & 58.8 \(\pm\) 8.3 & 75.2 \(\pm\) 5.2 \\
**GC\({}^{\theta}\)** & 67.0 \(\pm\) 11.9 & 63.5 \(\pm\) 14.4 & 74.5 \(\pm\) 4.4 \\
**TE\({}^{\theta}_{\text{rx}}\)** & **70.4 \(\pm\) 12.5** & **69.0 \(\pm\) 14.8** & **93.0 \(\pm\) 5.9** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MI and WM classification results in terms of the classification accuracy for all the effective connectivity measures considered. Figure 6: Average classification accuracies, and their standard deviations, for all subjects in the MI database as a function of the number features selected to train the classifiers. Figure 7A shows the highest average classification accuracy per subject for \(\text{TE}^{\theta}_{\text{ex}}\), \(\text{GC}^{\theta}\) and PSI, during the training-validation stage. The subjects are ordered from highest to lowest performance. The analogous information for the testing stage is presented in Figure 7. In both stages, the \(\text{TE}^{\theta}_{\text{ex}}\)-based classifier performs slightly better than those based on alternative connectivity estimation strategies in most subjects. In addition, as inferred from Figure 6, there are large variations in performance for the different subjects in the database, consistent across the two classification stages. This behavior has been reported elsewhere [10; 59; 60; 61; 62].