## 6 Conclusions  
In this study, we investigated the statistical properties of container ship flow time series and made a detailed investigation on long-range behaviors and fractal characteristics of ship flow sequences for three representative container ports in the world--Shanghai, Singapore, and Rotterdam. This study concludes three main findings. Firstly, the empirical evidence given in this study emphasize the significance of long-range dependence behaviors and multifractal property in all ship flow sequences at different time scales for three container ports. Secondly, the empirical evidence from comparisons among these ship flow sequences at different time scales implies that the long-range dependence becomes larger for each port as the time scale increases, except for the port of Singapore. Shanghai and Rotterdam were identified as the ports with the highest degree of long-range dependence in monthly ship flow sequences, while Singapore was identified as having the highest degree of long-range dependence in weekly ship flow sequence. Finally, the empirical evidence confirmed the multifractal property as an impact factor on the ship flow sequences of container ports. The analysis on the shuffled data indicated that the presence of multifractality in the ship flow sequences of container ports is due to the correlation properties as well as to the probability density function of the ship flow series. The findings of this study provide some interesting implications. First, the existence of long-range dependence and multifractality in container ship flow could be exploitable and helpful for shipping companies and policy makers. In other words, the presence of chaotic structure such as long-range dependence and multifractality in container ship flow sequences implies that the volume and direction of container ship flow may demand certain rules. Therefore, the shipping company can carry out short-term capacity allocation and adjustment according to container ship flow prediction. Second, the presence of long-range dependence and multifractality in the data suggests that container ship flow forecasting models should account for existing nonlinearities in the data, otherwise, their results may be biased and highly misleading. Port groups can use these findings in forecasting the expected volatility in the number of arriving container ships, and thereby, in developing and carrying out the layout planning of the port infrastructure, shipping date planning, and even port expansion investment. Moreover, some advanced modeling approaches can be employed for ship flow sequence forecasting, such as statistical modeling [40; 41] and machine learning methods [42; 43]. In addition, we regarded the trends in the ship traffic flow time series in this study as caused by external conditions. We identified and filtered out these trends in MF-DFA analysis. However, these trends may not be completely caused by external conditions, and some trends may carry endogenous power from data. At this time, when we explore the long-range power-law dependence of the time series, whether to filter out the trend  
Figure 8: Hurst exponents of the shuffled ship flow sequences for \(q=[-10:10]\). needs further discussion, as pointed out by [PERSON] et al. [38]. The effect of trends on detrended fluctuation analysis for ship traffic flow time series of ports can be further studied in the future. C.-J.L. : conceptualization, methodology, software, validation, formal analysis, investigation, visualization; J.W. : writing--review and editing; H.L.J. : writing--review and editing; Z.-H.H. ; resources, data curation, writing--original draft preparation, supervision. All authors have read and agreed to the published version of the manuscript. This work was supported in part by the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), under grant number CE140100049, the National Natural Science Foundation of China (No.