## 5 Discussion
### Underreporting of Vessel Traffic Accidents  
The level of quality in accident data has an important impact on vessel traffic risk analysis. However, it has been argued that a certain number of vessel traffic accidents may be unreported [30]. Although the underreporting of vessel traffic accidents is an important factor for the quantification of the actual vessel traffic risk level, there are few studies on this issue [31]. The underreporting of vessel traffic accidents is not considered in the process of uncertainty analysis in this study because the assumption of the underreporting rate will introduce epistemic uncertainties. The rate of underreporting can be defined as the ratio of the unreported accidents to the true number of accidents [31]. The possible effects of the underreporting of vessel traffic accidents on the time window selection are discussed below.  
According to the research [32], the percentage of accidents, which are missing from the records, varies from database to database and the average number of unreported accidents makes up roughly 50% of all accidents. In addition, the underreporting rate of the United Kingdom's maritime accident databases is estimated at 11% to 61% in the period from 2005 to 2009. Regardless of the degree of underreporting, the available data in this study will be complemented by additional data. When the rate of underreporting is assumed to be 50%, the number of vessel traffic accidents in Table 1 will be doubled. Similarly, the accident data will be complemented when the underreporting rate ranges from 11% to 61%. In such cases, the number of vessel traffic accidents is no longer an exact value but an interval of possible values.  
Considering the underreporting of vessel traffic accidents, two uncertain time series of the vessel traffic accident frequency can be formed. The first uncertain time series is established when the rate of underreporting is 50%, which is shown in Figure 6a, whereas the second uncertain time series is formed when the rate of underreporting ranges from 11% to 61%, which is shown in Figure 6b. In Figure 6a, the accident frequencies for each year 2000-2016 are represented by diamond icons and are connected by a line. As the accident frequency is represented by the interval of possible values in Figure 6b, the upper and lower limits of the accident frequency are represented by square icons and circle icons, and are connected by lines, respectively. The limits of the confidence interval are shown and connected by dotted lines in Figure 6a,b.  
Figure 5: (**a**) Uncertain time series calculated by every two years of accident data; (**b**) Uncertain time series calculated by every three years of accident data.  
By comparison, the uncertain time series in Figure 6a is similar to the uncertain time series in Figure 2. According to Section 3, the uncertain time series in Figure 6a is split into four segmentations, which are the years 2000-2001, 2002-2004, 2005, and 2006-2016. The latest segmentation, 2006-2016, is considered as the statistical time window, which is the same as the result of time window selection in Figure 2. Thus, no effects of the underreporting of vessel traffic accidents on the time window selection have been found when the underreporting rate is assumed to be an exact value. Using the sliding window approach described in Section 3 to split the uncertain time series in Figure 6b, we find that there is always an intersection between the interval of the vessel traffic accident frequency in the next year and the confidence interval of the former segmentation. In other words, the algorithm for time window selection proposed in this study may not work well in the case that the rate of underreporting is assumed to be an interval of possible values.