## 3 Methods
### Feature Extraction and Selection
Wavelets are characterized by small waves with limited duration and zero average values [41]. WT is good for analyzing data at a specific time and frequency or revealing information with different scales [42]. It extracts local information with different frequencies to reveal trends, discontinuities, and repeated patterns underlying the data. Through the analysis of low- and high-frequency information, data trends and deviations (representing low and high fluctuations) can be identified accordingly [43]. DWT decomposes the EEG data at a predefined level by splitting each level into two sub-bands containing different frequency ranges (i.e., low and high frequencies) using the formula  
\[\phi_{\alpha,\tau}(\mathcal{X}^{t,c})=\frac{1}{\sqrt{a}}(\mathcal{X}^{t,c}_{ \frac{t-\tau}{a}})\]  
where \(\mathcal{X}^{t,c}\) represents the EEG data at the segment \(t,1\leq t\leq T\) of the channel \(c\), and \(\phi_{\alpha,\tau}(\mathcal{X}^{t,c})\) is a scaling function, called wavelet basis functions, generated by the mother wavelet \(\phi(\mathcal{X}^{t,c})\). \(a\) and \(\tau\) indicate the scale and translation factor, respectively, controlling the scaling and translation of the wavelet. In our study, DWT was employed to extract features by decomposing EEG data into high- and low-frequency information at each level. High-frequency information produces detail coefficients, presenting any rapid or sudden changes in the data, while low-frequency information produces approximate coefficients, showing slow changes in the data. The detail coefficients are particularly useful for detecting rapid changes, such as discontinuities or sudden shifts. Using the coefficients, the following wavelet features are computed at each level:  
\[\omega_{1}^{j}=\sum_{1}^{n}\frac{d_{i,j}}{n},\quad\omega_{2}^{j}=\sqrt{\frac{ \sum_{i=1}^{n}(d_{i,j}-\overline{d_{i,j}})^{2}}{n}},\quad\omega_{3}^{j}=\sum( n_{\mu}\times\sqrt{(2*log(n))})\]  
\[\omega_{4}^{j}=\frac{\sum_{1}^{n}(\delta_{i,j})^{2}}{n},\quad\omega_{5}=\frac{ \mu_{d_{i,5}}}{\mu_{a}},\quad\omega_{6}=\frac{\mu_{d_{i,4}}}{\mu_{a}}\]  
where \(d_{i,j}=\{d_{1,j},d_{2,j},\cdots,d_{i,j}\}\) represent the wavelet coefficients at the \(j\)th level (\(j=1,2,\cdots,(l+1)\)), \(l\) denotes a pre-defined decomposition level, \(i\) indicates the elements of the coefficients (\(i=1,2,\cdots,n\)), \(\overline{d_{i,j}}\) indicates the mean of \(d_{i,j}\), \(\mu_{a}\) represents the sum of the coefficients in level \(l\), and \(n\) denotes the length of the coefficients. \(n_{\mu}\), representing the noise variance, can be computed as \(n_{\mu}=\frac{e_{d_{i,j}}}{\overline{d_{i}}}\), where \(e_{d_{i,j}}\) indicates the median of the coefficients, and \(T_{d}=0.6745\) presents the wideband neuronal signal for Gaussian noise [44]. We evaluated various mother wavelets and decomposition levels to examine their similarities and differences for analyzing neurological disabilities. Once all features were extracted, we performed feature selection to remove any redundancies. In detail, we employed one-way analysis of variance (ANOVA) [45] to determine statistically significant features (\(p<0.05\)). Only the statistically significant features were utilized for classification and visual analysis.