# Analyzing and Forecasting Vessel Traffic Through the Panama Canal: A Comparative Study
## 3 Results
### Diebold-Mariano (DM) Test  
The results obtained from the Diebold-Mariano test are presented below (Table 3), in order to assess significant differences in the forecasting accuracy of the various models. Based on the results presented in Table 3, observations can be made as the following: the positive DM statistic and the low \(p\)-value (0.012 \(<\) 0.05) indicate a statistically significant difference in performance between the SARIMA and Holt-Winters models. The positive sign suggests that the SARIMA model outperforms the Holt-Winters model in terms of forecasting accuracy. When comparing Holt-Winters and NNAR, the difference was even more pronounced (\(p<0.001\)), confirming a significant disparity in accuracy between the two models at the 5% significance level, with NNAR clearly performing better. In contrast, the comparison between NNAR and SARIMA yielded a DM value close to zero (-0.07) and a high \(p\)-value (0.9442), which exceeds the 5% significance level, indicating no statistically significant difference; therefore, both models exhibited comparable predictive performance. The forecasted number of vessels and the 95% confidence intervals for each model over the next twelve months (year 2023) are presented in Table 4. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**Effectiveness**} & \multicolumn{3}{c}{**Box–Ljung Test**} \\ \cline{2-6}  & **RMSE** & **MAE** & **MAPE** & \(\chi^{2}\) & \(p\) \\ \hline SARIMA (1,0,1)(0,1,1) [12] & 87.58 & 67.49 & 6.31 & 15.90 & 0.72 \\ Holt–Winters & 106.17 & 82.36 & 7.71 & 79.77 & 0.00 \\ NNAR (13,1,7) [12] & 83.96 & 68.21 & 6.33 & 17.46 & 0.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model Effectiveness, Diagnostic Tests, and Fitting Results. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**DM Test**} & \multirow{2}{*}{\(p\)**-Value DM**} & \multicolumn{2}{c}{**Is There a Significant Difference?**} \\ \hline SARIMA and & & & \\ Holt–Winters & & & \\ Holt–Winters and NNAR & 5.3064 & 0.000125 & Yes \\ NNAR and SARIMA & \(-0.0716\) & 0.9442 & No \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The Diebold–Mariano Test. Maritime traffic through the Panama Canal remains steady during the forecast period of 2023, with January and March representing the months with the highest vessel transit compared to the rest of the year. The time series plots of the forecasts generated by the SARIMA, Holt-Winters, and NNAR models (Table 4), together with the training and testing data of vessel transits through the Panama Canal, are shown in Figures 12-14. As shown in Figure 12, during the training period, represented in yellow, a consistent upward trend in maritime traffic was observed, accompanied by cyclical variations that suggest the presence of seasonality or recurring patterns in vessel flow. This evolution provides a robust foundation for modeling historical behavior. In the testing phase, marked in blue, the dynamic pattern is maintained, although with slight stabilization toward the later years. This behavior allowed for assessing the model's generalization capability, indicating that it accurately replicates real conditions without signs of overfitting. Finally, the forecast segment, shown in red, projects a moderate continuation of the observed  
Figure 14: Forecast Using the NNAR (13,1,7)\({}_{12}\) Model. Figure 13: Forecast Using the Holt–Winters Model. pattern. The model anticipates traffic levels similar to those previously recorded, reflecting an expectation of persistent behavior in the maritime system under similar conditions. As shown in Figure 13, during the model training phase, highlighted in yellow, a sustained upward trend in the number of vessels is evident, accompanied by cyclical fluctuations that suggest a strong seasonal component in maritime traffic. In the testing phase, marked in blue, a gradual stabilization in vessel flow was observed.