## 4 Experimental Research
### The Parameters' Independent Optimization of MFE  
In the course of calculating MFE, the four parameters, i.e., scale factor \(\tau\), embedding dimension \(m\), boundary gradient \(n\), and boundary width \(r\), should be determined in advance. For scale factor \(\tau\), when it was too large, the calculation of MFE would raise the problem of insufficient data points, while a too small scale factor would not be good for accessing the deeper information of MI-EEG. From Section 4.2 we can see that the change of FE is significant when the time range is [451,900]. Meanwhile, to ensure that the calculation of FE is not affected by the data length, the length of time series is at least 100 points. As a result, \(\tau_{\text{max}}\) was set to 4. The remaining three parameters would be determined by experiments. Firstly, when \(\tau\), \(m\), \(n\), and \(r\) were given fixed values, we calculated the fuzzy entropy of 140 trials of MI-EEG on channel C3 and C4 for imaginary left hand movement. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature Vector** & **Classifier** & **Top Classification Rate (\%)** & **Average Classification Rate with 10-fold CV (\%)** \\ \hline \(F^{1}\) & SVM & 96.43 & 88.93 \\ \(F^{2}\) & SVM & 100 & 90.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of feature vector construction. Then, we calculated 140 differences between the FE on channel C3 and the FE on channel C4, and we defined \(FED=FE_{\text{C3}}-FE_{\text{C4}}\), where \(FE_{\text{C3}}\) and \(FE_{\text{C4}}\) stand for the FE of MI-EEG on channels C3 and C4, respectively. Finally, we calculated the mean and standard deviation of 140 \(FED\), and they were noted as \(M_{FED}\) and \(SD_{FED}\), respectively. For a given \(\tau\), we could obtain the variation curve of \(M_{FED}\) and \(SD_{FED}\) with any one parameter while the others were kept constant. When scale factor \(\tau\) was \(1\), we obtained three curves, which are presented in Figure 6. The solid pink line represents the situation of imaginary left hand movement. Similarly, the results associated with imaginary right hand movement are displayed with a green dotted line. Figure 6a gives the variation curves of \(M_{FED}\) and \(SD_{FED}\) with parameter \(m\) when \(\tau=1\), \(n=2,r=0.1 SD\), and \(SD\) is the standard deviation of original MI-EEG. When \(m\) equals \(1\), although \(SD_{FED}\) is small for any one of the imaginary tasks, which means the MI-EEG signals are more intensive for any one of two tasks, the \(M_{FED}\) values corresponding to two imaginary tasks are quite close, which means the two tasks show a poor distinction. With the increase of \(m\), for any one of two imaginary tasks the \(M_{FED}\) first increases and then remains stable. The bigger \(m\) is, the more accurate the calculation of FE is and the more detailed information is implied. Meanwhile, the more complex the computation, the more data points are needed. Taking into account the constraints of the experimental dataset, the parameter \(m\) is set to 2. Figure 6b displays the variations of \(M_{FED}\) and \(SD_{FED}\) with parameter \(n\) when \(\tau=1,m=2,r=0.1 SD\), and \(SD\) is the standard deviation of the original MI-EEG. When parameter  
Figure 6: For \(\tau=1\), the mean and standard deviation of FED curves for input variables (**a**) embedding dimension \(m\); (**b**) boundary gradient \(n\); and (**c**) boundary width \(r\). \(n\) is 1, the \(M_{FED}\) values corresponding to the two imaginary tasks are quite different, which means they show a better distinction, but the \(SD_{FED}\) values are also very big for the two tasks, which means the MI-EEG signal is too scattered for any one task. When parameter \(n\) equals 3 or 4, \(SD_{FED}\) is small for any one imaginary task. On the other hand, the \(M_{FED}\) values corresponding to the two imaginary tasks are very close. When parameter \(n\) equals 2, \(SD_{FED}\) is moderate for any one imaginary task; meanwhile the \(M_{FED}\) values corresponding to the two imaginary tasks are quite different.