## 2 Materials and Methods
### Evaluation Metrics
#### Novel Contributions
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**Algorithm 2**Algorithm Outline for Graph Koopman Kernel Embedding  
_Practical computation of the Koopman operator approximation--In practice, the reduced order system representation using the Koopman operator is often approximated using data-driven methods like Dynamic Mode Decomposition (DMD), e.g., as described in [19; 25; 26; 27], which provides a finite-dimensional approximation of the Koopman operator. The steps typically involved are presented in Algorithm 3._  
```
Step 1. Data collection--Collect time-series data of the state \(x(t)\) and observables \(g(x(t))\)
2. DMD algorithm--Apply the DMD algorithm to the data to approximate the Koopman operator and compute its eigenvalues and eigenfunctions: i. collect data matrices \(X_{0}\) and \(X_{1}\); ii. compute the Singular Value Decomposition (SVD) of \(X_{0}\); iii. project \(X_{1}\) onto the subspace spanned by the left singular vectors to compute \(A_{\textit{tilde}}\); iv. perform eigendecomposition on \(A_{\textit{tilde}}\) to obtain eigenvalues and eigenvectors; v. map the eigenvectors back to the original space to obtain DMD modes. Step 3. Analysis--Use the obtained eigenvalues and eigenfunctions to analyse the system dynamics: i. interpret eigenvalues; ii. analyse DMD modes in terms of spatial structures; iii. reconstruct the system dynamics; iv. visual analysis. ```  
**Algorithm 3** DMD Algorithm for Koopman operator ApproximationMetastable connectivity substructuresThe close relation of the eigenvalue problem of determining the connectivity substructures to kernel canonical correlation analysis (CCA) was described in [28; 29; 30]. Kernel CCA computes eigenfunctions of the forward-backwards dynamics to identify so-called coherent sets. Coherent sets are a generalisation of metastable sets and are regions of the state space that are not distorted over a specific time interval. All the information about the long-term behaviour of the time-evolving graph G is contained within the eigenfunctions associated with s dominant eigenvalues close to 1. Evaluation of efficacy of Koopman embedding for classification tasksThe Koopman embedding of each EEG segment is first passed to a nonlinear function, e.g., WL kernel; the output of this function is a vector, possibly of a different dimension from the embedding. The vectors for all segments are then combined into a training set, which is then passed into a standard classification scheme based on Convolutional Neural Networks. The result then predicts the clinical phenomenon of interest (e.g., brain cognitive states from EEG). We performed separate cross-validation runs on the training. The accuracies achieved with the setting demonstrated the overall robustness of the model. To ensure that the [PERSON] model generalises reasonably well to unseen test data, we allocate the first two-thirds of the data to build the model and the last third for testing. DatasetTwo datasets were considered to understand the model behaviour across the cognitive load and seizure datasets. First, EEG data related to cognitive states and mental load were used to investigate the method. The dataset is available at [[https://github.com/BCI-HCI-IITKGP/Cognitive-Mental-world-EEG-Data](https://github.com/BCI-HCI-IITKGP/Cognitive-Mental-world-EEG-Data)]([https://github.com/BCI-HCI-IITKGP/Cognitive-Mental-world-EEG-Data](https://github.com/BCI-HCI-IITKGP/Cognitive-Mental-world-EEG-Data)) accessed on 6 February 2025. The EEG data were collected in the BCI-HCI lab [31]. Five experiments were conducted, and data for each subject were considered. The dataset includes five tasks for each subject: Idle, 1-Back, 2-Back, Dual-1-Back, and Dual-2-Back. For this paper, the Idle and Dual-1-Back workload tasks were selected from the dataset. The experiments involved various cognitive tasks to assess an individual's cognitive workload or mental effort. The Idle condition serves as a baseline for comparison, where the participant is not actively engaged in any task.