## 5 Empirical Results
### Long-Range Dependence and Multifractality  
Figure 5 depicts the fluctuation function \(F\) versus scale for the ship flow sequences of three ports with different time scales in log-coordinates and the OLS linear regression for these curves when \(q\) is equal to two. We calculated the generalized Hurst exponents from the slopes of these straight lines. The Hurst exponent of all the ship flow sequences was greater than 0.5, which indicated a positive long-range dependence and persistence in these ship flow sequences. Figure 4: Frequency and probability density distribution of monthly ship flow sequences. Table 3 presents the results for long-range dependence in ship flow sequences of three ports when \(q\) is equal to two. Empirical results suggested high Hurst exponents for ship flow sequences. An important feature of these results implied that, in general, Hurst exponents becomes larger as the time scale increases. The port of Singapore has the highest Hurst exponent in weekly ship flow sequence, which is close to 1. However, in the monthly ship flow sequences, Singapore obtains the lowest Hurst exponent, which is just 0.751. Therefore, ship flow sequences in the port of Singapore have a higher Hurst exponent when the time scale is small. On the contrary, the ship flow sequences in the ports of Shanghai and Rotterdam have a higher Hurst exponent when the time scale is larger. Moreover, the differences in Hurst exponents under different time scales seem to be higher for the port of Singapore. Figure 6 presents the results for the MF-DFA methodology for \(q=[-10:10]\). Qualitative results showed that the generalized Hurst exponent for all ship flow sequences decreased with the increase in \(q\); the generalized Hurst exponent of these ship flow sequences is dependent on the selection of \(q\). However, for the monthly ship flow sequence  
Figure 5: Generalized Hurst exponent \((q=2)\) for three ports with different time scales. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Ship Flow Sequences**} & \multicolumn{3}{c}{**Time Scale**} \\ \cline{2-4}  & **Daily** & **Weekly** & **Monthly** \\ \hline Shanghai \((m=2)\) & 0.677 & 0.836 & 0.866 \\ Singapore \((m=2)\) & 0.761 & 0.924 & 0.751 \\ Rotterdam \((m=2)\) & 0.573 & 0.759 & 0.958 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Generalized Hurst exponents \((q=2)\) for ship flow sequences with different time scales. in the ports of Shanghai and Rotterdam, this dependence on \(q\) was not significant. The differences in generalized Hurst exponents of daily and weekly series for three ports seem to be smaller as \(q\) increases. The generalized Hurst exponents for port of Singapore decreased faster than those for Shanghai and Rotterdam with the increase in \(q\) in all time scales. In particular, there was no substantial change in generalized Hurst exponent of the monthly ship flow series for Shanghai and Rotterdam; \(H(q)\) remained between 0.8 and 1.2 with the increase in \(q\). For the monthly sequence, when \(q\) is less than zero, the generalized Hurst exponents of Singapore and Rotterdam are larger than that of Shanghai; meanwhile, when \(q\) is larger than zero, the situation may change to the opposite, and the generalized Hurst exponent of ship flow for Shanghai becomes the largest. These results suggest that these differences are not spurious or due to error measures. An important additional comment is that the degrees of multifractality of ship flow sequences in Shanghai and Singapore container ports are much higher than the ones found for ship flow in Rotterdam port. The leveling of \(q\)-order Hurst exponent reflects that the \(q\)-order root-mean-square (RMS) is insensitive to the magnitude of local fluctuations. The multifractal spectrum will have a long left tail when the time series have a multifractal structure that is insensitive to the local fluctuations with small magnitudes.