## 3 Results  
We consider the value of \(\max_{\rho\in[0,1]}|S^{\alpha}(\rho)-\rho|\) as a function of the order \(\alpha\) and find the optimized index. As shown in Figure 1, the circle indicates the minimum when \(\alpha\) changes from 0 to 100 with 0.01 as the increment. When taking the optimized order \(\alpha\) as 1.79 (\(\alpha=1.79\)), the optimized index can achieve good performance.  
Figure 1: Value of \(\max_{\rho\in[0,1]}|S^{\alpha}(\rho)-\rho|\) as a function of order \(\alpha\) for the value of \(N=19\) (correspond to the number of EEG channels).  
Figure 2A shows an example of 19-channel selected EEG data of a child with ASD with a 4-s period after preprocessing. Figure 2D shows an example of 19-channel selected EEG data of a healthy child with a four-second period after preprocessing. To measure and compare the global synchronization of multichannel EEG, we perform the correlation matrix analysis on 19-channel EEG data from subjects with ASD and healthy control subjects, respectively. The correlation matrix results for ASD subjects and healthy control subjects are shown in Figure 2B,E, respectively. Then, their corresponding eigenvalues are computed to show different correlation structures of EEG signals between ASD and healthy controls, and the results are given in Figure 2C,F respectively.  
Figure 2: Examples of multichannel electroencephalograph (EEG) analysis. (**A**) The multichannel EEG epochs; (**B**) its corresponding correlation matrix based on S estimator and (**C**) its eigenvalues distribution for autism spectrum disorder (ASD) subject; (**D**) The multichannel EEG epochs; (**E**) its correlation matrix based on S estimator and (**F**) its eigenvalues distribution for controls subject.  
From Figure 2B,E, the correlation matrix reveals increasing synchronization activities in ASD than healthy control. The results shown in Figure 2C,F reveal that the maximal eigenvalues in ASD are higher than those in healthy controls. The maximal eigenvalue of correlation matrix of EEG of ASD group is 9.0376, while the maximal eigenvalue of healthy controls is 6.0938.  
The Renyi entropy and Shannon entropy results for all the 160 EEG epochs between ASD and control groups are shown in Figure 3A,B, respectively. In both figures, each box shows the interquartile range containing 50% of values with a line denoting the median. The whiskers presents the overall data range. The averaged Renyi entropy values for the EEG epochs were averaged by 0.5460 \(\pm\) 0.0571 and 0.4994 \(\pm\) 0.0580 (mean \(\pm\) SD) in ASD and Control subjects, respectively. The averaged Shannon entropy values for the EEG epochs were averaged by 0.4135 \(\pm\) 0.0554 and 0.3710 \(\pm\) 0.0542 (mean \(\pm\) SD) in ASD and Control subjects, respectively. According to the statistical analysis, the \(t\)-test of Renyi entropy shows that synchronization activities of ASD significantly increases than in healthy control (\(p\) = 3.2894 \(\times\) 10\({}^{-12}\)). The \(t\)-test of Shannon entropy shows that synchronization activities of ASD significantly increases than in healthy control (\(p\) = 2.2836 \(\times\) 10\({}^{-11}\)). Both of the Renyi entropy and Shannon results reveal increasing synchronization activities in ASD than in healthy control. Moreover, the Renyi entropy possesses high distinguishing degree.