## 3 Methodology
### Classification  
For the classification of the extracted features from the EEG signals into the ASD or TD category, this study utilized support vector machine (SVM) supervised machine learning for both linear and non-linear problems. SVMs are widely used in classification tasks, as they define a linear decision boundary in a high-dimensional space where input vectors are transformed non-linearly. Based on the nature of the optimal hyperplane [39], SVMs can be categorized into three types: linearly separable, linearly non-separable, and non-separable. For non-linear mappings, a kernel function is used to implicitly map the inputs into a higher-dimensional space.  
* Linear kernel (L): The SVM linear classifier aims to find the optimal hyperplane that separates the data into classes. The SVM linear classifier seeks to find the optimal hyperplane that separates the data into distinct classes. The parameters used in MATLAB 2022b were as follows: Kernel Function = 'linear' and Standardize = true. The decision boundary is determined by Equation (8): \[0=w\cdot x+b\] (8) where \(w\) is the weight vector perpendicular to the hyperplane, \(x\) is the input feature vector, and \(b\) is the bias term. The classifier minimizes the following objective function, as shown in Equation (9): \[0.5\ |w|^{2}+\lambda\left(\sum_{i=1}^{n}max(0,1-y_{i}(w\cdot x_{i})+b)\right)\] (9) Here, \(\lambda\) controls the trade-off between maximizing the margin and minimizing the classification errors, and \(y\) represents the label of the data.
* Gaussian radial basis function (RBF) kernel: A kernel is commonly used for non-linear mappings. The kernel function, \(k(x_{i},y_{j})\), is expressed as follows in Equation (10): \[k(x_{i},y_{j})=\phi(x_{i})\cdot\phi(y_{j})\] (10) where \(k\) is the kernel function, and the weight vector to the hyperplane is defined by Equation (11): \[w=\sum_{i}y_{i}\ \ a_{i}\ \phi(x_{i})\] (11) The SVM classifier minimizes the following objective function, shown in Equation (12): \[\lambda|\mathbf{w}|^{2}+\left(\frac{1}{m}{\sum_{i=1}^{m}max\left(0,1-\left(w^{T} \cdot x_{i}\ \right)-b\right)}\right)\] (12) where \(\lambda>0\) represents the trade-off between the size of the margin and its flexibility in classification. The MATLAB parameters were as follows: Kernel Function = 'rbf' and Standardize = true.  
* Quadratic or polynomial kernel (quadratic): A quadratic or polynomial kernel of degree (\(d=2\)) is used in the SVM to map the original features into a higher-dimensional space. This enables SVM to classify data that are not linearly separable in the original space [39]. The kernel function is defined in Equation (13): \[k(x_{i},y_{j})=(x_{i}\cdot y_{j})^{2}+2c(x_{i}\cdot y_{j})^{2}+c^{2}\] (13) where \(c\) is a constant used to adjust the influence of the higher-order terms in the polynomial. Typically, \(c\geq 0\) and \(\bigwedge^{2}\) indicate that this is a quadratic kernel (degree 2). The MATLAB hyperparameters were as follows: Kernel Function = 'polynomial', Polynomial Order = 2, Kernel Scale = 'auto', Box Constraint = 1, and Standardize = true. This expansion ensures that the kernel incorporates not only the dot product of the original features but also squared terms and cross-products, enabling the SVM to capture more complex patterns in the data. For all of the SVM models, the features were standardized to have zero mean and unit variance. Within the EEG signal classification, the SVM utilized the maximum, mean, and minimum energy computations from each band and brain hemisphere as features. These features were fed into the SVM, which calculated the optimal decision boundary to distinguish ASD from TD cases. Figure 7 illustrates the workflow of the proposed method, including feature extraction, standardization, and classification using an SVM with different kernels.