## 3 Description of Feature Extraction  
Based on the idea of independent optimization of parameters, the normal MFE is improved, and the improved MFE (IMFE) method is applied to the feature extraction of MI-EEG. The specific steps can be summarized as follows:  
1. The optimal selection of time interval for MI-EEG Suppose that the original MI-EEG signal of the \(L\)th channel in a trial is \(X_{L}^{0}=\left[x_{L,1},\cdot\cdot\cdot,x_{L,d},\cdot\cdot\cdot,x_{L,e}\cdot \cdot\cdot x_{L,k}\right]^{T}\in R^{K\times 1}\), \(L=1,2,\cdot\cdot,C\), where \(C\) and \(K\) are the number of channels and sampling points per trial, respectively. The FE time series of every channel of MI-EEG is calculated for each training sample of different imaginary tasks. To obtain the mean FE time series, they are superimposed and then averaged for every imaginary task. The optimal sampling interval may be determined to ensure there is a significant difference between the mean fuzzy entropies of MI-EEG on two channels for every imaginary task. A new EEG signal \(X_{L}^{1}\) can be constituted by selecting the optimal sampling interval of the data points from the original EEG signal \(X_{L}^{0}\) and it can be expressed as \(X_{L}^{1}=\left[x_{L,d},\cdot\cdot\cdot,x_{L,e}\right]^{T}\in R^{N\times 1}\), where \(d\) is the first selected point from \(X_{L}^{0}\), \(e\) is the last selected point from \(X_{L}^{0}\), \(N\) is the number of selected data points from \(X_{L}^{0}\), and \(N=d-e+1\). 2. The Coarse-Grained Procedure of MI-EEG The coarse-grained MI-EEG signals of \(X_{L}^{1}\) on multiple scale factors \(\tau=1,2,\cdot\cdot,\tau_{\max}\) can be obtained according to Section 2.2 and denoted as \(X_{L,1}^{1}\), \(X_{L,2}^{1}\), \(\cdot\cdot\cdot X_{L,j^{\prime}}^{1}\), \(\cdot\cdot\cdot,X_{L,\tau_{\max}}^{1}\) in turn, where \(L=1,2,\cdot\cdot\cdot,C\), \(\tau_{\max}\) is the maximum of scale factor \(\tau\) and \(X_{L,j}^{1}\) represents the coarse-grained MI-EEG of the \(L\)th channel for the \(j\)th scale. Figure 1: The coarse-grained process of time series for scale factor \(\tau\). The Calculation of MFE  
The FE of each coarse-grained MI-EEG signal can be calculated according to Section 2.1. The FE of coarse-grained sequences \(X^{1}_{L,1},X^{1}_{L,2},\cdots X^{1}_{L,j},\cdots,X^{1}_{L,\tau_{\text{max}}}\) can be denoted as \(FE^{1}_{L,1}\), \(FE^{1}_{L,2},\cdots\), \(FE^{1}_{L,\tau_{\text{max}}}\), \(L=1,2,\cdots,C\) in turn. Thus, the MFE of the \(L\)th channel MI-EEG is given by Equation (12):  
\[MFE^{1}_{L}=[FE^{1}_{L,1},FE^{1}_{L,2},\cdots,FE^{1}_{L,\tau_{\text{max}}}] \in R^{1\times\tau_{\text{max}}}. \tag{12}\]  
4. The parameters' independent optimization of MFE for different scale factors \(\tau\)  
On different scale factors, the parameters, including embedding dimension \(m\), boundary gradient \(n\), and boundary width \(r\), will directly influence the MFE value. To obtain the feature vectors that are beneficial to classification, the relevant parameters will be optimized independently. For multiple scale factors \(\tau\), when any two parameters of \(m\), \(n\), and \(r\) remain relatively fixed, the variation curves of the average and standard deviation of MI-EEG's MFE with a parameter are calculated for different imaginary tasks, respectively. The optimal values of the parameters may be determined by considering the fluctuation of the error line for each imaginary task, the overlapping degree of error lines, and the difference of means between different tasks. After the independent optimization of the parameters, the MFE of the \(L\)th channel MI-EEG is expressed as  
\[IMFE^{1}_{L}=[IFE^{1}_{L,1},IFE^{1}_{L,2},\cdots,IFE^{1}_{L,\tau_{\text{max}}}] \in R^{1\times\tau_{\text{max}}}, \tag{13}\]  
where \(IFE^{1}_{L,\tau}(\tau=1,2,\cdots,\tau_{\text{max}})\) represents the improved FE of the \(L\)th channel MI-EEG for scale factor \(\tau\). 5.