## 2 Proposed Methodology
### New Signal Combination Model with Four Common Methods
#### 2.2.4 PCA Using New Combinations of Signals  
The fourth proposed model increases the good distinctive features extracted separately from single channels. In this model, using PCA, we sort the data without dimensional reduction based on the best features before the classification. A single electrode is used for sorting the features and transforming data to a new space, followed by classification, as illustrated in Figure 8 and explained in the following steps:  
Filtering data with Butterworth filter: The first phase is the same as the first phase of the previous model. Using filter bank (sub-band) signals or new combinations of signals: The second phase is the same as the second phase of the previous model. Sorting features by PCA: The purpose of PCA, as an orthogonal linear transformation method, is to reduce the dimensionality by transferring data to a new space and sorting them. The greatest variance lies on the first coordinate as the first principal component, the second-greatest variance is on the second coordinate, and so on. PCA is introduced as one method for reducing primary features using the artificial transformation of data to a new space [89; 90; 91]. Figure 8: Checking new hybrid signals by PCA. The basic description of the PCA approach is as follows. Let us define a data matrix \(\mathrm{X}(\mathrm{M}\times\mathrm{N})\), where \(\mathrm{M}\) is the number of observations, and \(\mathrm{N}\) is the number of features. Then, \(\mathrm{C}(\mathrm{N}\times\mathrm{N})\) is defined as the covariance matrix of matrix \(\mathrm{X}\):  
\[\mathrm{C}= \begin{bmatrix}\mathrm{Var}(\mathrm{X}_{1})&\mathrm{Cov}(\mathrm{ X}_{1},\mathrm{X}_{2})&\cdots&\mathrm{Cov}(\mathrm{X}_{1},\mathrm{X}_{\mathrm{N}})\\ \mathrm{Cov}(\mathrm{X}_{2},\mathrm{X}_{1})&\mathrm{Var}(\mathrm{X}_{2})& \cdots&\mathrm{Cov}(\mathrm{X}_{2},\mathrm{X}_{\mathrm{N}})\\ \ldots&\ddots&\cdots&\cdots\\ \mathrm{Cov}(\mathrm{X}_{\mathrm{N}},\mathrm{X}_{1})&\mathrm{Cov}(\mathrm{X}_{ \mathrm{N}},\mathrm{X}_{2})&\cdots&\mathrm{Var}(\mathrm{X}_{\mathrm{N}})\\ \end{bmatrix}  \tag{23}\]  
where \(\mathrm{Cov}\big{(}\mathrm{X}_{i},\mathrm{X}_{j}\big{)}=\frac{1}{\mathrm{M}} \Sigma_{1}^{-}\big{(}\mathrm{X}_{i}(\mathrm{k})-\overline{\mathrm{X}}_{i} \big{)}\big{(}\mathrm{X}_{j}(\mathrm{k})-\overline{\mathrm{X}}_{j}\big{)}\), and \(\mathrm{var}(\mathrm{X}_{i})=\sigma_{\mathrm{X}_{i}}^{2}\). Eigenvectors and the eigenvalue matrix are produced by matrix \(\mathrm{C}\):  
\[\mathrm{C}\Gamma=\Lambda\Gamma\Rightarrow\mathrm{C}=\Gamma\Lambda\Gamma^{{}^{ \prime}} \tag{24}\]  
where \(\Lambda(\mathrm{N}\times\mathrm{N})\) is a matrix of sorted eigenvalues (\(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{\mathrm{N}}\)) :  
\[\Lambda= \begin{bmatrix}\lambda_{1}&0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \ldots&\ddots&\ldots&\cdots\\ 0&0&\cdots&\lambda_{\mathrm{N}}\\ \end{bmatrix}  \tag{25}\]  
And \(\Gamma(\mathrm{N}\times\mathrm{N})\) is a matrix of eigenvectors (each column corresponds to one eigenvalue from matrix \(\Lambda\)):  
\[\Gamma= \begin{bmatrix}\gamma_{11}&\gamma_{12}&\cdots&\gamma_{1\mathrm{N}}\\ \gamma_{21}&\gamma_{22}&\cdots&\gamma_{2\mathrm{N}}\\ \cdots&\ddots&\cdots&\cdots\\ \gamma_{\mathrm{N}1}&\gamma_{\mathrm{N}2}&\cdots&\gamma_{\mathrm{NN}}\\ \end{bmatrix}  \tag{26}\]  
Finally, the matrix of principal components is defined as:  
\[\mathrm{Y}=\mathrm{X}\Gamma= \begin{bmatrix}\gamma_{11}&\gamma_{12}&\cdots& \gamma_{1\mathrm{N}}\\ \gamma_{21}&\gamma_{22}&\cdots&\gamma_{2\mathrm{N}}\\ \cdots&\ddots&\cdots\\ \gamma_{\mathrm{N}1}&\gamma_{\mathrm{N}2}&\cdots&\gamma_{\mathrm{NN}}\\ \end{bmatrix}  \tag{27}\]  
Each of the \(\mathrm{N}\) features \(\mathrm{x}_{1},\mathrm{x}_{2},\cdots,\mathrm{x}_{n}\) can be explained with parameters from matrix \(\mathrm{Y}\) as a linear combination of the first \(\mathrm{K}\) principal components: \(\mathrm{x}_{mn}=\sum_{k=1}^{\mathrm{K}}\gamma_{nk}\mathrm{y}_{nk}\). Performing classification using classifiers: The fourth phase is the same as the fourth phase of the previous model, with differences in the scale of input data.