## 5 Empirical Results  
### Descriptive Statistics  
It is a well-known fact that the statistical properties of time series vary with time and depend on time windows. Table 2 presents the descriptive statistics for the original ship flow series at different time scales. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time Scale** & **Port** & **Average** & **Standard Deviation** & **Skewness** & **Kurtosis** & **ADF(\(p\))** & **KPSS(\(p\))** \\ \hline \multirow{3}{*}{Daily} & Shanghai & 31 & 8.437 & \(-\)0.420 & 0.464 & \(-\)8.152(0.105) & 14.418(0.01) \\  & Singapore & 34 & 7.567 & \(-\)0.393 & 2.156 & \(-\)7.639(0.108) & 0.598(0.01) \\  & Rotterdam & 15 & 4.557 & 0.263 & 0.161 & \(-\)8.989(0.290) & 15.130(0.01) \\ \hline \multirow{3}{*}{Weekly} & Shanghai & 218 & 42.618 & \(-\)0.177 & \(-\)0.926 & \(-\)4.962(0.301) & 5.777(0.01) \\  & Singapore & 235 & 34.244 & \(-\)0.367 & 2.004 & \(-\)3.675(0.226) & 0.307(0.01) \\  & Rotterdam & 104 & 19.488 & \(-\)0.008 & \(-\)0.820 & \(-\)4.085(0.201) & 6.009(0.01) \\ \hline \multirow{3}{*}{Monthly} & Shanghai & 950 & 176.878 & \(-\)0.110 & \(-\)1.053 & \(-\)2.385(0.419) & 1.443(0.01) \\  & Singapore & 1019 & 134.381 & 0.164 & \(-\)0.581 & \(-\)2.684(0.299) & 0.102(0.01) \\ \cline{1-1}  & Rotterdam & 453 & 81.193 & \(-\)0.003 & \(-\)1.115 & \(-\)2.494(0.375) & 1.640(0.01) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Descriptive statistics of the original ship flow series. Figure 1: Original data of ship flow of different time scales in three ports. According to Table 2, the average number of ships arriving in the port of Singapore is greater than that of the Shanghai port at any time scale. Therefore, although Shanghai's container port ranks first in the world list, the port of Singapore is still the busiest container port in the world. This is mainly due to its unique geographical location and its role as the largest transit port in the Asia-Pacific region. Besides, the standard deviation of ship flow sequences in port of Shanghai is larger than that of Singapore and Rotterdam ports, regardless of the time scale. This shows that the numbers of arriving ships in the ports of Rotterdam and Singapore are relatively stable compared to Shanghai. The results from skewness and kurtosis analysis demonstrated that the different time scales significantly affect the temporal structure of the ship flow sequence. Indeed, skewness reflects the degree of symmetry in the distribution. Table 2 shows that the skewness in almost all the ship flow sequences are negative, except for the daily ship flow sequence of Rotterdam (0.263) and monthly ship flow sequence of Singapore (0.164). This indicates that all of the ship flow sequences are leftward distributions, except for the daily ship flow sequence of Rotterdam and monthly ship flow sequence of Singapore. On the other hand, the kurtosis reflects the sharpness of the image. The higher the kurtosis, the sharper the center point on the image; in this sense, the kurtosis measures the degree of data aggregation in the center. In this study, the traditional kurtosis is replaced by the \"super kurtosis\" calculation method, which is to subtract the kurtosis 3 of the normal distribution from the original kurtosis so that the comparison benchmark is zero. Table 2 shows that the super kurtosis values of daily ship flow sequences are greater than 0, indicating that the daily ship flow sequence distributions are more concentrated and have a longer tail than the normal distribution. In the weekly ship flow sequences, only the super kurtosis of Singapore is greater than 0. The super kurtosis of all the monthly ship flow sequences are less than 0, indicating that these sequences are scattered and have a shorter tail than the normal distribution. Therefore, only the daily ship flow sequence of three ports and the weekly ship flow sequence of Singapore exhibited the characteristic of \"sharp peak or fat tail\". Further, the ship flow sequences of three ports with different time scales deviated from the normal distribution.