## 5 Conclusions  
In this work, we proposed a single-trial phase TE estimator. Our method combines a kernel-based TE estimation approach, which defines effectivity connectivity as a linear combination of Renyi's entropy measures of order \(\alpha\), with instantaneous phase time series extracted from the data under analysis. We tested the performance of our proposal on synthetic data generated through NMMs and on two EEG databases obtained under MI and WM paradigms. We compared it with commonly used single-trial TE estimators, applied to phase time series, and the PSI and GC. Our results show that the proposed phase TE estimator successfully detects the direction of interaction between individual pairs of signals, capturing the differences in coupling strength and displaying statistically significant results around the frequencies corresponding to the main oscillatory components present in the data. It also succeeds in detecting bidirectional interactions of localized frequency content and is robust to realistic noise and signal mixing levels. Moreover, our method, coupled with a CKA-based relevance analysis, revealed discriminant spatial and frequency-dependent patterns for both the MI and WM databases, leading to improved classification performance compared with approaches based on real-valued TE estimation. In all our experiments, the proposed single-trial kernel-based phase TE estimator is competitive with the comparison methods previously listed in terms of the performance assessment metrics employed. As future work, we will look into developing a cross-spectral representation for our phase TE estimator to study directed interactions between oscillations of different frequencies [65]. We will also explore the effects of the choice of filter on the proposed estimator as well as those of the parameters involved in time embedding and in our kernel-based TE estimation approach. Conceptualization, I.D.L.P.P. and A.A.-M.; methodology, I.D.L.P.P., A.A.-M. and P.M.H.G. ; software, I.D.L.P.P., A.A.-M. and D.C.-P.; validation, I.D.L.P.P., A.A.-M. and D.C.-P.; formal analysis, I.D.L.P.P. and A.O.-G.; investigation, I.D.L.P.P., A.A.-M. and J.I.R.P. ; resources, I.D.L.P., A.A.-M. and A.O.-G.; data curation, I.D.L.P.P. and P.M.H.G. ; writing--original draft preparation, I.D.L.P.P. and A.A.-M.; writing--review and editing, A.A.-M., P.M.H.G. and D.C.-P.; visualization, I.D.L.P.P. ; supervision, A.O.-G. and A.A.-M.; project administration, A.O.-G.; funding acquisition, I.D.L.P.P., A.A.-M. and J.I.R.P. All authors have read and agreed to the published version of the manuscript. Under grants provided by: The Minicencias project (111080763051)-Herramienta de apoyo al diagnostic del TDAH en ninos a partir de multiples caracteristicals de actividad cerebral desde registros EEG; Maestria en ingenieria electrica and Maestria en Ingenieria de Siemens y Computacion--Universidad Tecnologica de Pereira. Author [PERSON] was supported by the program \"Doctorado Nacional en Empresa-Convoctoria 758 de 2016\", funded by Minicencias. In this study, we use public-access EEG databases introduced in previously published works and made freely available by the respective authors [44; 48]. We did not collect any data from human participants ourselves. This study uses anonymized public databases introduced in previously published works by other groups [44; 48]. The databases used in this study are public and can be found at the following links: BCI Competition IV database 2a [[http://www.bbci.de/competition/iv/index.html](http://www.bbci.de/competition/iv/index.html)]([http://www.bbci.de/competition/iv/index.html](http://www.bbci.de/competition/iv/index.html)) (accessed on 2 June 2021), database from brain activity during visual working memory [[https://data.mendeley.com/datasets/j2v7](https://data.mendeley.com/datasets/j2v7) btchdy/2]([https://data.mendeley.com/datasets/j2v7](https://data.mendeley.com/datasets/j2v7) btchdy/2) (accessed on 2 June 2021).