## 2 Proposed Methodology
### New Signal Combination Model with Four Common Methods
#### 2.2.3 Lagrangian Polynomial Equation Using New Combinations of Signals  
The third proposed model increases the number of good distinctive features extracted from every single channel separately. In this model, we use a Lagrangian polynomial model to transform the data into formulas, which are then used as features for classification. A single channel/electrode is involved in the extraction of features and classification, as illustrated in Figure 7 and explained in the following steps.  
Figure 7: Converting data to polynomial formulas and selecting equation coefficients as features with and without selection for classification.  
(1) Filtering data with Butterworth filter: In the first phase, using frequency filtering, the noise and artifacts are removed with a filter bank. This process is the same as the first step in Section 2.2.1, with the difference being that it involves single channels individually.  
(2) Using filter bank (sub-band) signals or new combinations of signals: Two models are considered input data, including sub-bands in the names of filter banks and new combinations of signals. New combinations of signals are created based on the following formula for a single electrode. This formula is used for the calculation of a single electrode or channel.  
\[\text{S\_FB(i,j)}=\text{S\_FB(i)}+\text{S\_FB(j)}.\ \ \ \text{i,j}=1\ldots 9. \tag{17}\]  
where S\(\_\)FB(i,j) represents the filter bank signal.  
(3) Converting data to a formula of coefficients with different orders by using the Lagrangian polynomial equation: The input data are about 1 s from 3.5 s of imagination time. The Lagrangian polynomial equation converts these input data of the two single channels to coefficients for classification.  
The Lagrangian polynomial is described below.  
There are \(\text{k}+1\) data assets \(\left(x_{0},y_{0}\right),\cdots,\left(x_{\text{j}},y_{\text{j}}\right),\cdots, \left(x_{\text{k}},y_{\text{k}}\right)\), while each \(\text{x}_{\text{j}}\) is unique. The interpolation polynomial, as a linear combination, is in  
(5) Performing classification using classifiers: In this phase, we perform classification based on features to determine accuracy using classifiers. This phase is similar to the last phase of the two above models.  
This model was tested on one public dataset (including left and right hands) for examination. We describe this public dataset in more detail in the next section. Single channels are applied for processing. Then, the combination of two filter banks based on Formula (17) for single electrodes/channels is calculated. Each combination is considered separately in the next steps. In the next step, two procedures are considered for processing. The first procedure uses fixed filter banks of a single channel, and the second procedure uses the new combinations of signals for a single channel. Then, each procedure uses the Lagrangian polynomial equation to convert data to formula structures. The Lagrangian coefficients are considered features for the classification. Feature selection is not used in the first procedure, but it is used in the second procedure. Mutual information feature selection selects the best features from 1 to 30 features. And finally, the coefficients selected as the most effective features are sent to the LDA, KNN, and ELM classifiers for classification.