## 3 Case Study
### Data Collection and Pre-Processing  
In this study, an assessment is conducted on diesel-powered Z-Drive harbor tugboats including azimuth stern drive (ASD), Rotortug, and Z-Tractor tugboats, equipped with azimuthing propulsors, encompassing diverse main dimensions, speeds, and bollard pull values. A dataset containing the main characteristics of over 200 tugboats which are designed by well-known tugboat designers and built in leading shipyards in the tug building industry is collected and compiled for analysis. To refine the dataset, pre-processing steps are implemented, removing sister vessels that share identical designs to avoid adversely impacting analysis distribution. Table 1 illustrates the range of parameter variability within the dataset. As depicted in Table 1, a total of 476 data samples were obtained from 68 distinct diesel-powered Z-drive harbor tugboat series equipped with azimuthing propulsors. These tugboats exhibit an average length of 29.1 m, ranging from 18.7 to 42 m, and an average bollard pull (BP) capacity of 71.25 metric tons, ranging from 31 to 120 metric tons. This investigation aims to explore the relationship between vessel length and the distribution of bollard pull and speed parameters. Figure 6 presents the distribution of bollard pull and speed parameters, as well as the conditional probability distribution relative to vessel length for the collected dataset. When examining these conditional probability distributions, the peak probability is observed around 70 metric tons BP at approximately 30 m in length and a speed of approximately 13 knots. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & **Maximum** & **Minimum** & **Mean** & **Median** & **Mode** & **Std** & **Var** \\ \hline Length [m] & 42.00 & 18.70 & 29.13 & 29.05 & 24.40 & 4.40 & 19.32 \\ \hline Beam [m] & 16.00 & 9.20 & 12.21 & 12.02 & 12.00 & 1.24 & 1.55 \\ \hline Draft [m] & 6.50 & 3.40 & 5.12 & 5.10 & 5.30 & 0.67 & 0.45 \\ \hline Draft, Max. [m] & 7.20 & 3.95 & 5.58 & 5.60 & 5.70 & 0.67 & 0.45 \\ \hline Power [kW] & 7600 & 1939 & 4269.22 & 4190 & 5050 & 951.64 & 905,609.22 \\ \hline Speed, Max. [knot] & 15.10 & 11.50 & 13.05 & 13.00 & 13.00 & 0.75 & 0.57 \\ \hline Bollard Pull [mT] & 120.00 & 31.71 & 71.25 & 70.00 & 70.00 & 16.04 & 257.41 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics of the dataset for listed tugboats. Figure 5: Bayesian network for tugboats. Regression analysis scrutinizes the mathematical relationship among two or more variables, while correlation analysis assesses the direction and magnitude of this relationship. Correlation analysis, as a statistical method, furnishes insights into the relationship, direction, and strength of the correlation between variables. The correlation coefficient serves as a metric indicating the strength of the relationship between the dependent and independent variables. It gauges the linear relationship between two variables and is unit-independent, ranging between \(-1\) and \(1\). A coefficient nearing \(0\) signifies a weak correlation, while a proximity to \(1\) indicates a strong correlation. In this study, the Pearson correlation coefficient is computed from the collected data and visually represented in Figure 7. The intricate relationship between these variables is delineated in Figure 8. The figures illustrate a robust correlation between the data of length (LOA), beam (B), draft (T), draft-maximum (T\({}_{\text{max}}\)), and power (P) and bollard pull (BP), whereas the correlation with speed (V) is relatively weaker. Upon detailed examination of the correlation analysis, it becomes evident that the most robust correlations are evident between the P and BP variables, showcasing a substantial correlation coefficient of \(0.97\). Additionally, a notable correlation emerges between the T and Tmax variables, boasting a strong coefficient of \(0.88\). Subsequently, the correlations between the P-Tmax and BP-Tmax variables follow closely behind with coefficients of \(0.86\).