## 3 Methodology
### Embedded-Based Feature Selection Using LASSO  
LASSO, categorized as an embedded-based method, incorporates feature selection as an integral part of the model training process. This approach is particularly effective in handling high-dimensional datasets [31]. Its primary objective is to add a penalty term to the loss function to encourage sparsity in the model. This is achieved by minimizing the following objective function:  
\[L_{Lasso}(\omega)=\frac{1}{2n}{\sum_{i=1}^{n}}\Big{(}\mathcal{Y}_{i}-\chi_{i}^ {T}\omega)^{2}+\lambda\Sigma_{j=1}^{p}|\omega j| \tag{5}\]  
Figure 1: EEG brainwave dataset training.  
where \(\mathcal{Y}_{i}\) represents the observed output for the \(i\)-th instance, \(x_{i}\) denotes the input feature vector, and \(\omega\) is the vector of coefficients to be estimated. The first term \(\frac{1}{2\textit{m}}\sum_{i=1}^{n}(\mathcal{Y}_{i}-\chi_{i}^{T}\omega)^{2}\) represents the ordinary least squares loss, which aims to minimize the squared differences between the predicted and observed values. The second term \(\lambda\Sigma_{j=1}^{p}|\omega j|\) introduces the LASSO penalty, where \(\lambda\) is a hyperparameter controlling the strength of regularization. The key innovation of LASSO regularization lies in the regularization term \(\lambda\Sigma_{j=1}^{p}|\omega j|\), which enforces sparsity by penalizing the absolute values of the coefficients [32].  
This motivates some coefficients to become exactly zero for effective performance. This is beneficial because it eliminates less important predictors, thereby simplifying the model and enhancing interpretability. The zero-out coefficients correspond to features that do not significantly contribute to the model's predictive power. By excluding these features, LASSO reduces model complexity and helps prevent overfitting, especially in scenarios with high-dimensional data. Thus, the presence of zero coefficients is crucial to achieving an effective and robust predictive model. Consequently, LASSO not only aids in fitting the model to the data but also serves as a valuable tool for identifying and emphasizing the most relevant features in the process. The objective of LASSO is to minimize the mean squared error between the predicted and actual values while imposing a penalty on the absolute values of the model coefficients [33]. LASSO can help in selecting the most relevant EEG features by pushing some feature coefficients to zero, effectively performing feature selection. This is crucial for optimizing the performance of emotion classification models based on EEG data, as it reduces overfitting and enhances the interpretability of the model [34]. It is particularly useful when dealing with high-dimensional data, as it helps create more capable models, making them easier to interpret and potentially more efficient. Owing to its absolute value, LASSO provides a nondifferentiable term, but despite that, there are methods to minimize it. As we show below, LASSO is also robust to outliers. LASSO can effectively handle noisy data by eliminating less important features and preventing the inclusion of irrelevant features [35]. Additionally, it addresses multicollinearity, a strong correlation between features that affect the label concurrently, by tending to select only one feature from highly correlated features [36].