# Modeling of Vessel Traffic Flow for Waterway Design-Port of Swinoujscie Case Study
## 3 Results
### Regression Analysis of Traffic Flow Parameters  
#### 3.3.1 The Model Including Drought T  
To construct regression models of marine traffic flow, all passages for consecutive gates were considered. It can be observed that there was a difference between the mean and standard deviation of vessel position from the centre of the route for selected groups of vessels at different gates. It was decided to make these groups permeable and see if there would be any statistically significant difference in the results. To compare three independent groups with respect to the quotient variables, a nonparametric Kruskal-Wallis test was applied. The Kruskal-Wallis test could cover the null hypothesis for three or more samples. The null hypothesis states that all samples come from the same distribution. In this case, continuous distributions were assumed. On the other hand, if it can be assumed that all population distributions have the same shape (normal or not), the hypothesis states that the medians of the populations are identical [29]. The H-statistic for the Kruskal-Wallis Test is calculated using the defined formula:  
\[H=\frac{1}{C}\left(\frac{12}{N(N+1)}\sum_{j=1}^{k}\left(\frac{\left(\sum_{i=1 }^{n_{j}}R_{ij}\right)^{2}}{n_{j}}\right)-3(N+1)\right) \tag{6}\]  
where:  
* \(H\) is the the value of Kruskal-Wallis test;
* \(N=\sum_{j=1}^{k}n_{j}\);
* \(N\) is the number of all observations;
* \(k\) is the number of compared groups;
* \(n_{j}\) is the sample size for (\(j=1,2,\ldots,k\));
* \(R_{ij}\) is the the rank assigned to the value of the variable, for (\(i=1,2,\ldots,n\)),(\(j=1,2,\ldots,k\));
* \(C=1-\frac{\sum(t^{3}-t)}{N^{3}-N}\) is the adjustment factor for ties, t-number of cases included in tied rank. The H statistic has an asymptotic (for a large sample size) distribution of the chi-square with the number of degrees of freedom determined by the formula: \(df=(k-1)\). The next step is to compare the H-value with the critical chi-square value \(\chi^{2}_{0.05}\) for \(df\). If the critical chi-squared value is less than the H statistic, we can reject the null hypothesis that the medians are equal. If the chi-squared value is not less than the H-statistic, there is insufficient evidence that the medians are unequal. The Kruskal-Wallis test indicates significant differences between categories but does not specifically say which categories differ, so a post-hoc test was performed with the Bonferroni correction. The Bonferroni correction is an adjustment to \(p\)-values made when multiple statistical tests are run simultaneously on the same data. Figure 7: The scatterplot of vessel speed against draught. The test showed that the samples for legs N1 to N6 were from the same distribution, so there was no significant difference between the samples. For samples N7, N8, and N9, the null hypothesis was rejected. The results presented in the remainder of this paper are for gates N1-N6. Figure 8a shows the relationship between the mean distance of ships from the centre and the draught \(T\) of the ship. Simple linear regression models were constructed for three groups of ships. It can be seen that there was a linear relationship between these parameters, with a coefficient of determination of 0.8 for GC, 0.7 for OPT, and 0.27 for PAS. For the GC and OPT vessels, the regression coefficient was satisfactory according to the accepted interpretation. It can be concluded that the higher the value of the draught, the closer to the middle of the track the navigator is sailing. The correlation between standard deviation and draught of the vessel was lower than for the mean value. The best coefficient of determination was for PAS (\(R^{2}\) = 0.60), as shown in Table 1. Although the coefficient of determination for the GC group was low, the independent parameter \(T\) was statistically significant for the model (\(p\) = 0.004). For OPT vessels, the parameter \(T\) was not statistically significant (\(p\) > 0.05). An interesting result is a correlation between standard deviation and draught for passenger vessels.