# Variation Trends of Fractal Dimension in Epileptic EEG Signals
## 3 Results and Discussion
\(D_{\text{c}}\) values were below 1 in the seizure status of RSE-f1 sc, RSE-f2 and RSE-f2 sc algorithms, as shown in Figure 5b. For the RSE algorithm, the \(D_{\text{c}}\) values became lower after using the scaling region interception method. Meanwhile, Figure 5b showed higher absolute differences between seizure and normal statuses but also larger error bars. However, the corresponding standard deviation was small, which meant that the error bars come from few data, the distribution of the overall data tended to the mean value. Additionally, the values of \(D_{\text{c}}\) difference between seizure and normal statuses became larger under the same algorithm after being denoised. Therefore, the statistical analysis on the \(D_{\text{c}}\) difference needed to be conducted to further illustrate the effect of noise reduction, the values of \(D_{\text{c}}\) difference were divided into ten groups under various algorithms of raw and denoised signals, respectively. Figure 4: \(D_{\text{c}}\) mean values by using various algorithms: (**a**,**c**,**e**,**g**,**i**) raw EEG signals; (**b**,**d**,**f**,**h**,**j**) denoised EEG signals. Red and black symbols represented seizure and normal statuses, respectively. First, the distribution of each group was tested, where the Shapiro-Wilk test in R language was used (Shapiro-Wilk normality test, shapiro.test), the results were summarized in Table 2, which was mainly used to test the normal distribution of \(D_{\text{c}}\) differences value between seizure and normal statuses [39]. \(W\) and \(p\)-value are the test statistic, and the null hypothesis of this test is that the sample is originated from a normally distributed parent. When \(p\)-value is less than the selected significance level (usually 0.05), the null hypothesis is rejected. As shown in Table 2, not all of the ten groups were normally distributed with a significance level below 0.05, therefore the \(D_{\text{c}}\) difference of the same algorithm between raw and denoised signals were tested with Wilcoxon Two-sample Rank sum test. Second, since each group was independent, the different groups of the same algorithm were paired data, the commonly used Wilcoxon Two-sample Rank sum test in R language was used to test the difference between groups (Wilcoxon rank sum test, Wilcox.test) [40]. \(W\) and \(p\)-value are the test statistic, when p-value is less than the selected significance level (usually 0.05), the \(D_{\text{c}}\) difference value of the two groups are considered to be significantly different. As shown in Table 3, the \(p\)-value of each algorithm was less than \(3\times 10^{-3}\), revealing the highly distinguished differences between seizure and normal statuses under the same algorithm after denoising, and the results were consistent with Figures 4 and 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data** & **Algorithm** & \(W\) & \(p\)**-Value** \\ \hline \multirow{4}{*}{EEG-raw} & Higuchi & 0.98944 & \(5.58\times 10^{-1}\) \\  & RSE-f1 & 0.98895 & \(5.18\times 10^{-1}\) \\  & RSE-f1 sc & 0.98831 & \(4.68\times 10^{-1}\) \\  & RSE-f2 & 0.98811 & \(4.52\times 10^{-1}\) \\  & RSE-f2 sc & 0.98258 & \(1.66\times 10^{-1}\) \\ \hline \multirow{4}{*}{EEG-denoised} & Higuchi & 0.98921 & \(5.39\times 10^{-1}\) \\  & RSE-f1 & 0.96064 & \(2.65\times 10^{-3}\) \\ \cline{1-1}  & RSE-f1 sc & 0.96723 & \(8.68\times 10^{-3}\) \\ \cline{1-1}  & RSE-f2 & 0.95819 & \(1.74\times 10^{-3}\) \\ \cline{1-1}  & RSE-f2 sc & 0.97522 & \(3.95\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Shapiro–Wilk test results of the values of \(D_{\text{c}}\) difference between seizure and normal statuses. Figure 5: \(D_{\text{c}}\) mean values based on Figure 3 for seizure and normal statuses calculated by (**a**) raw signals and (**b**) denoised signals using various algorithms. The red lines were represented the error bar, and red solid boxes on each column were represented the corresponding standard deviation.