## 4 Experimental Research
### Statistical Analysis  
The IMFE is developed in this paper on the basis of MFE. It is necessary to analyze the differences between IMFE and MFE statistically. In the following, a paired \(t\)-test is applied to identify whether there is a significant difference when they are used for feature extraction of MI-EEG.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **Feature Extraction** & **Classifier** & \
\begin{tabular}{c} **Top Classification** \\ **Rate (\%)** \\ \end{tabular} &
\begin{tabular}{c} **Average Classification Rate** \\ **with 10-Fold CV (\%)** \\ \end{tabular} \\ \hline
4 & WT & BP & 92.4 & - \\
36 & WPT & LDA & 88.57 & - \\
37 & CSP & SVM & 82.86 & - \\
38 & CSP & BPR & 90 & - \\
39 & CSSP & SVM & 87.14 & - \\
40 & WT-AR & LDA & 92.86 & - \\
41 & WT-ICA & GA-SVM & 90.71 & - \\
42 & WT-PSD & LDA & 89.29 & - \\
43 & WT-WE & Kmeans & 90.1 & - \\
44 & MOWT & SVM & 91.8 & - \\
45 & SSE & KNN & 85.16 & - \\
46 & WPE & BP & 88.57 & - \\
47 & EMD & FCM & 83 & - \\
48 & EMD & PSO-SVM & 87.6 & - \\
49 & PCA & CHSOM & 96 & - \\ this paper & IMFE & SVM & 100 & 92.14 \\ \hline \hline \end{tabular} Note: “\(-\)” represents the combination or optimization of methods for feature extraction or classification; “\(-\)” means that average classification rate with 10-fold CV is not given in the reference. BP: Back propagation, WPT: Wavelet Packet Transform, BPR: Biomimetic Pattern Recognition, ICA: Independent Component Analysis, PSD: Power Spectral Density, WE: Wavelet Entropy, MOWT: Maximum Overlap Wavelet Transform, SSE: Singular Spectrum Entropy, KNN:-X-Nearest Neighbor, EMD: Empirical Mode Decomposition, FCM: Fuzzy C-means, PSO: Particle Swarm Optimization, PCA: Principal Component Analysis, GHSOM: Growing Hierarchical Self-organizing Map.  
\end{table}
Table 4: Comparison with other recognition methods.  
Figure 12: A comparison of time consumption using the proposed IMFE and conventional feature extraction methods.  
Suppose that \(IFE_{L,\tau}^{lh}\) and \(IFE_{L,\tau}^{rh}\) stand for the improved fuzzy entropy of the \(L\)th channel MI-EEG for scale factor \(\tau\) corresponding with imaginary left hand and right hand movements, respectively. Similarly, \(FE_{L,\tau}^{lh}\) and \(FE_{L,\tau}^{rh}\) denote the normal fuzzy entropy of the \(L\)th channel MI-EEG for scale factor \(\tau\) corresponding with imaginary left hand and right hand movements, respectively. Define \(D_{IMFE}=IFE_{L,\tau}^{lh}-IFE_{L,\tau}^{rh}\), \(D_{MFE}=FE_{L,\tau}^{lh}-FE_{L,\tau}^{rh}\), and \(D=D_{IMFE}-D_{MFE}\), and calculate D for each channel of C3, C4, and Cz and each one of scale factor \(\tau=2,3\) and \(4\). Then, we tested that D is a sample from a normal population \(N(\mu_{D},\sigma_{D}^{2})\). The null hypothesis is \(\mathrm{H}_{0}:\mu_{D}\leq 0\); the alternative hypothesis is \(\mathrm{H}_{1}:\mu_{D}>0\). The one-tailed paired \(t\)-test was chosen (\(\alpha=0.05\)). The decision rule is to reject \(\mathrm{H}_{0}\) if:  
\[\frac{\overline{d}}{\frac{s_{D}}{\sqrt{n}}}>t_{\alpha}(n-1) \tag{17}\]  
or  
\[p=P\{t>t_{\alpha}(n-1)\}\leq 0.05, \tag{18}\]  
where \(\overline{d}\) and \(s_{D}\) denote the mean and standard deviation of sample \(D\), respectively; \(n\) is the number of elements in sample \(D\). The \(t\)-test results are shown in Table 5.  
From Table 5, we see that all the \(p\) values are less than 0.05. Therefore, the null hypothesis \(\mathrm{H}_{0}\) is rejected at the 0.05 significance level. Therefore, the fuzzy entropy values obtained by IMFE and MFE are significantly different and IMFE outperforms MFE in discriminating between two imaginary tasks.