## 4 Results and Discussion
### Limitations  
In this study, we employed Morlet wavelets as filters for instantaneous phase extraction prior to phase TE estimation, as proposed in [8]. However, as discussed by the authors in [8], the choice of filter can influence the behavior of phase TE. This is an aspect we have yet to explore for our proposal. In the same line, in [42] the authors showed, using the Kraskov-Stogbauer-Grassberger TE estimator on real-valued filtered signals, that filtering  
Figure 12: (**A**) Two-dimensional representation of the relevance vectors for each subject in the WM database obtained after applying t-SNE on \(\theta_{n}\). (**B**) Groups identified by k-means. For the TE\({}_{\text{ex}}^{\theta}\)-based classifier the subjects grouped in G. I, have an average accuracy of \(0.94\pm 0.04\), while those in G. II and G.III have average accuracies of \(0.92\pm 0.08\) and \(0.93\pm 0.08\), respectively. Figure 13: Topoplots of the average node (channel) relevance for each group of clustered subjects and frequency band of interest in the WM database (see Figure 12). The arrows represent the most relevant connectivities for each group. For visualization purposes, only the 1% of the connections, those with the highest average relevance values per group, are depicted. and downsampling are deleterious for TE, since they can lead to altered time delays and hide certain causal interactions. Furthermore, from a conceptual perspective, while filtering dampens spectral power, it does not always remove the information contained in specific frequencies [25]. This would hinder the isolation of frequency specific interactions in TE estimates from real-valued filtered data, the most common approach to obtain spectrally resolved TE values. Whether those effects are also present in the case of phase TE is yet to be analyzed; however, as pointed out in [25], phase TE is conceptually different from spectrally resolved TE. Additionally, the results obtained with our phase TE estimator for the NMM data closely follow those obtained with the PSI, a measure that does not rely on data filtering, which points to a certain degree of robustness to the negative effects that might be associated with phase extraction through complex filtering. A related issue is the possible effects on our proposal of the preprocessing pipelines employed on the EEG data, which involve spectral and spatial filtering. Regarding the former, we have not studied its effects in this work; while for the latter, surface Laplacian positively impacted the discrimination capability of the connectivity features obtained from the different measures considered. In addition, we are yet to examine the effects of the parameter \(\alpha\) in [PERSON]'s entropy on the proposed phase TE estimator. In [10], we showed that the choice of \(\alpha\) indeed modified the performance of the \(\text{TE}_{\text{ex}}\). The same must hold true for \(\text{TE}_{\text{ex}}^{\theta}\). Additionally, we selected the autocorrelation time and Cao's criterion to obtain the embedding parameters for all the TE estimation methods. More complex approaches such as time-delayed mutual information and Ragwitz criterion may yield better results [34]. However, since our motivation was to propose a single-trial phase TE estimator suited as characterization method for BCI applications, the choice of simple parameter estimation methods is justified. As a matter of fact, a practical implementation of a phase TE-based BCI system would likely require further simplifications regarding parameter estimation, in order to facilitate the computation of phase TE in real time. Furthermore, our proposed phase TE estimator inherits the limitations of \(\text{TE}_{\text{ex}}\)[10]. Namely, it is ill suited to analyze long time series (several thousands of data points) because of the increase in computational cost, especially for non-integer values of the parameter \(\alpha\). In addition, it assumes stationary or weakly non-stationary data.