## 2 Proposed Methodology
### New Signal Combination Model with Four Common Methods
#### 2.2.2 FBCSP [82; 83; 84] Using New Combinations of Signals  
The second proposed model increases the good distinctive features extracted from the whole channel, which has the additional step of feature selection compared to the first model. In this model, new combinations of signals are created, and the best signals from both primary and new signals are selected using feature selection and used for classification. All of the electrodes cooperate for noise and artifact filtering and the extraction of features. This model consists of the following phases, presented in Figure 6 and explained below:  
(1) Filtering data with Butterworth filter: This step is the same as the first step in the previous section, i.e., Section 2.2.1. (2) Creating new combinations of signals: This step is the same as the second step in the previous section, i.e., Section 2.2.1. (3) Applying CSP as the spatial filter and performing feature extraction: In this phase, the CSP algorithm extracts m pairs of CSP features from new combinations of signals. This is performed through spatial filtering by linearly transforming the EEG data. Then, the features of all new signals are collected in a feature vector for the ith trial, i.e.,  
\[\mathrm{X_{i}}=\left[\mathrm{cf_{1}},\ \ \mathrm{cf_{2}},\ \ \cdots,\ \ \ \mathrm{cf_{9}}\right] \tag{13}\]  
where \(\mathrm{cf_{b}}\in\mathbb{R}^{2m}\) denotes m pairs of CSP features for the bth band-pass-filtered EEG measurement, \(\mathrm{X_{i}}\in\mathbb{R}^{1\times(9*2m)}\). (4) Collecting and selecting features: In this phase, an algorithm called Mutual Information Best Individual Features (MIBIF) is used for feature selection from the extracted features, and it selects the best features, which are sorted (in descending order) using class labels. In general, the purpose of the mutual information index in MIBIF [85; 86] is to obtain maximum accuracy with k features selected from a subset of S \(\subset\) F, in which the primary set F includes d features. This approach maximizes the mutual information I(S; \(\Omega\)). In the classification of features, input features, i.e., X, are usually continuous variables, and the Omega class (\(\Omega\)) has discrete values. So, the mutual information between the input features X and class \(\Omega\) is calculated as:  
\[\mathrm{I(X;\Omega)=H(\Omega)-H(Y|\Omega)} \tag{14}\]  
Figure 6: Feature selection from the entire set of features extracted from new signals like FBCSP. where \(\omega\in\Omega=\{1,..,\mathbf{N}_{\omega}\}\)  
And the conditional entropy is calculated as:  
\[\mathrm{H}(\Omega|\mathrm{X})=-\int_{\mathrm{X}}\sum\
olimits_{\omega=1}^{ \mathrm{N}_{\omega}}\mathrm{P}(\omega|\mathrm{X})\mathrm{log}_{2}\mathrm{P}( \omega|\mathrm{X})\mathrm{d}\mathrm{x}, \tag{15}\]  
where the number of classes is \(\mathrm{N}\omega\). In general, there are two types of feature extraction approaches in the mutual information technique, namely, the wrapper and filter approaches. With the wrapper feature selection approach, the conditional entropy in (14) is simply:  
\[\mathrm{H}(\Omega|\mathrm{X})=-\sum\
olimits_{\omega=1}^{\mathrm{N}_{\omega}} \mathrm{P}(\omega|\mathrm{X})\mathrm{log}_{2}\mathrm{P}(\omega|\mathrm{X}), \tag{16}\]  
\(\mathrm{P}(\omega|\mathrm{X})\) can be easily estimated from data samples that are classified as class \(\omega\) using the classifier over the total sample set. Mutual information based on the filter approach is described briefly in three steps:  
Step 1: Initialize set \(\mathrm{d}\) feature \(\mathrm{F}=\{\mathrm{f}_{1},\mathrm{f}_{2},\ldots,\mathrm{f}_{\mathrm{d}}\}\) and select feature set \(\mathrm{S}=\mathrm{Null}\). Step 2: Compute MI features based on \(\mathrm{I}(\mathrm{f}_{1}\,\mathrm{I}\,\mathrm{I})\) for each \(\mathrm{i}=1\ldots\mathrm{d}\), where \(\mathrm{f}_{\mathrm{i}}\) belongs to F.  
Step 3: Select the \(\mathrm{k}\) best features that maximize \(\mathrm{I}(\mathrm{f}_{\mathrm{i}}\,\mathrm{I}\,\mathrm{I})\)[84; 85]. Step 4: Repeat the previous steps until \(|\mathrm{S}|=\mathrm{k}\).