## 5 Conclusions  
The paper describes theoretical foundations and experimental demonstration of a novel workflow for developing a classifier for cognitive state identification using multivariate bio-physiological signals with robust embedding of spectral connectivity properties. The contribution lies in harnessing the low-dimensional spectral dynamic embedding potential of the Koopman transfer operators, borrowing from the complex system dynamics modelling studies. The potential of the model lies in its demonstrated capabilities to classify the cognitive states with acceptable average accuracy higher than 80% across multiple subjects as well as in facilitating explainability of the metastable substructures of the connectivity. The proposed graph kernel Koopman embedding (GKKE) framework for EEG classification advances the state of the art by addressing critical gaps in latent connectivity analysis, class imbalance, and feature explainability. The demonstrated efficacy of the mPLV-WL-DT workflow, coupled with the robustness of the WL kernel and Koopman spectral operators, establishes this framework as a valuable tool for EEG-based cognitive state detection and seizure prediction. Knowledge of the dynamics of these connectivity patterns of the brain-regions related to localised cognitive states and switching thereof equips more accurate clinical prognosis. However, the understanding that the kernel-classifier combination works well may require fine tuning to improve the score for this dataset, and particularly, investigation is needed to fine-tune and generalise performance for different datasets. Hence, this proposed workflowforms a foundation for further exploration with more datasets and combinations of different classifiers' spatio-temporal-spectral kernels for graph evolution embedding. _Implications for Future Research_  
The findings from this study lay the foundation for several future research directions:  
* **Dataset generalisation:** While the framework was tested on epilepsy and cognitive workload datasets, its applicability to other EEG datasets remains an area for exploration. Future studies can extend this work to include datasets related to mental health, neurodegenerative diseases, or motor imagery tasks. * **Incorporation of deep learning models:** Although ensemble classifiers such as Decision Trees and Random Forests performed well, integrating Koopman embeddings with deep learning architectures, such as Convolutional Neural Networks (CNNs) or Long Short-Term Memory (LSTM) networks, may further improve classification performance. * **Refinement of kernels for EEG dynamics:** The development of specialised graph kernels tailored to EEG dynamics, potentially incorporating additional spatial, temporal, and spectral features, could enhance the framework's robustness and generalisability. * **Real-time applications:** The low computational complexity of the WL kernel makes this framework suitable for real-time applications, such as seizure prediction or cognitive load monitoring in adaptive learning environments. _Limitations and Challenges_  
While the proposed framework demonstrates significant advancements, certain limitations warrant further investigation:  
* **Dependency on connectivity measures:** The choice of connectivity measure (e.g., mPLV, correlation) influences the construction of graphs, and alternative measures could yield different results. Further studies are required to determine the optimal connectivity metric for specific EEG tasks. * **Scalability to large datasets:** Although the WL kernel is computationally efficient, the scalability of the framework to larger EEG datasets with higher temporal resolution remains to be evaluated. * **Subject-specific variability:** EEG signals are highly variable across subjects, which may affect the generalisability of the findings. Incorporating subject-specific normalisation or transfer learning techniques could address this issue.