# Near-Optimal Weather Routing by Using Improved A* Algorithm
## 4 Case Study
### Prediction of Ship's Estimated Time of Arrival
#### 4.1.3 Learning-Based Regression Model
The dependent variables with high VIF values are dropped before the regression model is generated. Generally, a VIF value of 5 or higher for a descriptive variable is likely to cause an over-fitting problem; therefore, in this study, a variable of 5 or higher was eliminated, and a learning model was created. Table 7 summarizes the feature selection process through VIF analysis. After the number of candidate features was reduced through VIF analysis, the features associated with the wind wave and the swell wave were deleted in first phase. Even though the low VIF values for the direction and period of total wave, wind wave, and swell wave are observed, the VIF analysis, which quantifies the extent of linear correlation based on the least-square method, does not accurately estimates their nonlinear correlation [28]. As mentioned in Section 4.1.2, the total waves are highly correlated with wind waves and swell waves because total waves correspond to the combination of the wind waves and swell waves. As a result, three parameters (direction, height, and period) related to the total wave remain to maintain independence between input variables, and others related to wind and swell are excluded from the candidate group for the simplicity of the learning model. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Phase 1**} & \multicolumn{2}{c}{**Phase 2**} & \multicolumn{2}{c}{**Selected**} \\ \cline{3-6}  & & **VIF** & **Decision** & **VIF** & **Decision** & **VIF** \\ \hline
1 & COG & 1.2962 & - & 1.2944 & - & 1.2931 \\
2 & Draught & 2.3784 & - & 2.3739 & - & 2.3738 \\
3 & Total wave height & 24.4058 & - & 4.0210 & - & 3.9379 \\
4 & Total wave direction & 3.0208 & - & 1.4624 & - & 1.4596 \\
5 & Total wave period & 6.4452 & - & 3.3574 & - & 3.3115 \\
6 & Wind wave height & 9.2041 & Dropped & - & - & - \\
7 & Wind wave direction & 3.2017 & Dropped & - & - & - \\
8 & Wind wave period & 3.1675 & Dropped & - & - & - \\
9 & Swell wave height & 14.0707 & Dropped & - & - & - \\
10 & Swell wave direction & 1.5663 & Dropped & - & - & - \\
11 & Swell wave period & 3.1215 & Dropped & - & - & - \\
12 & Pressure MSL & 243.8376 & - & 243.4247 & Dropped & - \\
13 & Pressure surface & 244.5687 & - & 244.1839 & - & 1.5887 \\
14 & Ambient temperature & 17.6496 & - & 17.45677 & Dropped & - \\
15 & Sea surface salinity & 1.5039 & - & 1.4976 & - & 1.4838 \\
16 & Sea surface temperature & 15.7028 & - & 15.5211 & - & 1.7525 \\
17 & Wind speed & 4.2820 & - & 2.4308 & - & 2.4033 \\
18 & Wind angle & 1.8113 & - & 1.3921 & - & 1.3850 \\
19 & Current speed & 1.1089 & - & 1.0993 & - & 1.0959 \\
20 & Current angle & 1.1038 & - & 1.0995 & - & 1.0991 \\
21 & Gross tonnage & 2.1512 & - & 2.1498 & - & 2.1474 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: VIF analysis results. Because the pressure MSL, pressure surface, the ambient temperature and sea surface temperature are highly correlated, one of them should be removed from the candidate features to avoid multicollinearity of those features. The sea surface temperature and pressure surface were directly measured at sea level and therefore had a direct effect on SOG; however, the pressure MSL and ambient temperature were not. Thus, the pressure MSL and ambient temperature were excluded from the candidate features. Consequently, 13 out of 21 features were selected as final input variables, and the results are presented in Table 8.