## 5 Results and Discussions
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Parameters** & **IMF1** & **IMF2** & **IMF3** & **IMF4** & **IMF5** & **IMF6** & **IMF7** & **IMF8** & **IMF9** & **IMF10** & **IMF11** \\ \hline  & 3 & 3 & 5 & 12 & 16 & 18 & 28 & 35 & 45 & 40 & 23 \\ m & 8 & 9 & 6 & 5 & 3 & 2 & 2 & 2 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The appropriate time delay and embedding dimension of IMFs. Figure 12: The time delay, embedding dimension, and phase space of IMF1: (**a**) Time delay. (**b**) Embedding dimension. (**c**) Phase space. The bispectrum-based method [46] is used to test for linearity of IMF1 and IMF7, which can further prove the effectiveness of the above nonlinear indicators. The bi-spectrum-based test results are shown in Figure 16. The figure has shown that the estimated (full line) and theoretical (dashed line) values of R, the interquartile range of the estimated bi-coherence values, are very close, which indicates that the test of IMF1 show strong evidence of linearity. Contrary to the former, the estimated and theoretical values of R of IMF7 are clearly different and the estimated value is large, which indicates that the test of IMF7 show strong evidence of nonlinearity. The results are consistent with that of the proposed nonlinear test indicators in this paper. Figure 14: The statistical RQA index of IMF7. Figure 13: The statistical RQA index of IMF1. The analysis process has repeated for other IMFs, and the statistical nonstationary and nonlinear RQA measures pairs of all IMFs have been listed in Table 2. In the table, \(Z_{i}(i=1,\cdots,6)\) refers to the median value of the statistical RQA measures. \(D_{i}(i=1,\cdots,6)\) refers to the binary statistical RQA measures index. From Table 2, it can be clearly seen that the binary statistical RQA measures of IMF1-IMF2 are zero and that of IMF3-IMF11 are one, which indicates that IMF1-IMF2 are linear and stationary and \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**IMFs**} & \multicolumn{2}{c}{**DET**} & \multicolumn{2}{c}{**ENTR**} & \multicolumn{2}{c}{**Lmax**} & \multicolumn{2}{c}{**LAM**} & \multicolumn{2}{c}{**VENTR**} & \multicolumn{2}{c}{**Vmax**} \\ \cline{2-13}  & **Z1** & **D1** & **Z2** & **D2** & **Z3** & **D3** & **Z4** & **D4** & **Z5** & **D5** & **Z6** & **D6** \\ \hline IMF1 & 0.61 & 0 & 0.69 & 0 & 86 & 0 & 0.34 & 0 & 0.44 & 0 & 93 & 0 \\ IMF2 & 0.32 & 0 & 0.37 & 0 & 43 & 0 & 0.46 & 0 & 0.57 & 0 & 45 & 0 \\ IMF3 & 0.64 & 1 & 0.79 & 1 & 260 & 1 & 0.82 & 1 & 1.14 & 1 & 80 & 1 \\ IMF4 & 0.86 & 1 & 1.84 & 1 & 1501 & 1 & 0.91 & 1 & 2.51 & 1 & 256 & 1 \\ IMF5 & 0.92 & 1 & 3.79 & 1 & 3667 & 1 & 0.93 & 1 & 3.93 & 1 & 1201 & 1 \\ IMF6 & 0.95 & 1 & 4.21 & 1 & 4123 & 1 & 0.94 & 1 & 4.23 & 1 & 764 & 1 \\ IMF7 & 0.98 & 1 & 4.46 & 1 & 5012 & 1 & 0.98 & 1 & 4.58 & 1 & 664 & 1 \\ IMF8 & 0.98 & 1 & 4.75 & 1 & 5056 & 1 & 0.98 & 1 & 5.14 & 1 & 458 & 1 \\ IMF9 & 0.98 & 1 & 4.99 & 1 & 5072 & 1 & 0.99 & 1 & 5.25 & 1 & 489 & 1 \\ IMF10 & 0.99 & 1 & 5.43 & 1 & 5030 & 1 & 0.99 & 1 & 5.74 & 1 & 547 & 1 \\ IMF11 & 0.99 & 1 & 5.64 & 1 & 5206 & 1 & 0.99 & 1 & 6.51 & 1 & 552 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The statistical RQA measures of IMFs. Figure 16: The interquartile range of the estimated bi-coherence values of IMF1 and IMF7: (**a**) IMF1. (**b**) IMF7. Figure 15: Time-frequency diagram of IMF1 and IMF7: (**a**) IMF1. (**b**) IMF7. IMF3-IMF11 are nonlinear and non-stationary. To prove the correctness of the conclusion based on binary statistical RQA measures, the energy [47] of each IMF is calculated, which is shown in Figure 17.  
According to References [24; 47], the first group of IMFs whose energies are monotonically decreasing can be viewed as white noises. From Figure 17, it can be clearly seen that the first groups of IMFs, which can be regarded as noises are IMF1-IMF2.