## 4 Results  
Using the vessel traffic accident data from the United Kingdom's marine accident investigation branch [28], the case study is carried out on the algorithms for uncertainty quantification and time window selection that were illustrated in Sections 2 and 3. The vessel traffic accident data include the number of United Kingdom's vessels, vessel traffic accidents and fatalities caused by the accidents for each year 2000-2016, which are shown in Table 1.  
The vessel traffic accident frequency can be measured in the number of accidents per ship year. A ship year is defined as one ship sailing for one year, which can be understood as the corresponding exposure of the vessel traffic being analyzed. Then the vessel traffic accident frequency for each year between 2000 and2016 can be calculated by dividing the number of vessel traffic accidents by the number of vessels listed in Table 1. According to Equation (1), the confidence intervals of the vessel traffic accident frequency for each year between 2000 and2016 can be estimated using a given confidence level. Different confidence values can be set for different research purposes. In this study, the confidence level is set as 0.95 based on the a priori knowledge of the European Maritime Safety Agency [18]. When the confidence intervals are arranged in chronological order, an uncertain time series is formed. The uncertain time series is represented in Figure 2, in which the accident frequency for each year 2000-2016 are represented by diamond icons and are connected by a line, the limits of the confidence interval are shown and connected by dotted lines. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Year** & **2000** & **2001** & **2002** & **2003** & **2004** & **2005** & **2006** & **2007** & **2008** \\ \hline Number of vessels & 1050 & 1047 & 1210 & 1343 & 1406 & 1443 & 1480 & 1518 & 1578 \\ Number of vessel traffic accidents & 139 & 139 & 129 & 146 & 144 & 197 & 130 & 116 & 131 \\ Number of fatalities & 48 & 47 & 48 & 50 & 49 & 38 & 38 & 46 & 46 \\ \hline
**Year** & **2009** & **2010** & **2011** & **2012** & **2013** & **2014** & **2015** & **2016** & \\ \hline Number of vessels & 1564 & 1520 & 1521 & 1450 & 1392 & 1361 & 1385 & 1365 & \\ Number of vessel traffic accidents & 128 & 141 & 115 & 133 & 122 & 120 & 118 & 107 & \\ Number of fatalities & 38 & 36 & 35 & 26 & 19 & 21 & 23 & 21 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vessel traffic accident data. Figure 2: Uncertain time series of the vessel traffic accident frequency. According to Section 3, the uncertain time series in Figure 2 is split by the sliding window approach. The first year (2000) in the uncertain time series is regarded as the first segmentation. As can be seen in Figure 2, the vessel traffic accident frequency and the corresponding confidence interval in the year 2000 are 0.132 and [0.111, 0.156]. Since the vessel traffic accident frequency in the year 2001 is 0.133 and it does not exceed the confidence interval [0.111, 0.156], the year 2001 should be merged into the first segmentation. Then the confidence interval is updated to [0.117, 0.149] based on Equation (1). The vessel traffic accident frequency in the year 2002 is 0.107 and it exceeds the confidence interval [0.117, 0.149]. As a result, the second segmentation starts with the year 2002. The above-mentioned splitting procedure is repeated until the year 2016. Then four segmentations are obtained, which are the years 2000-2001, 2002-2004, 2005, and 2006-2016. The year 2005 fits the required situation that a single segmentation only contains one year. The mean value of the vessel traffic accident frequency during the years 2006-2016 can be calculated as 0.084 and it exceeds the confidence interval of the segmentation from the year 2002 to 2004, which is [0.096, 0.116]. This indicates that there is remarkable change between the segmentation from the year 2002 to 2004 and the segmentation from the year 2006 to 2016.