## 3 Results and Discussion
### Model Assessments
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator** & **PSR** & **PMR** & **HWSS** & **HWDS** & **HWTS** \\ \hline MAPE & 3.18\% & 2.77\% & 3.29\% & 1.83\% & 1.76\% \\ MAE & 120.10 & 78.23 & 127.82 & 54.01 & 46.82 \\ RMSE & 153.81 & 104.38 & 165.08 & 75.09 & 67.05 \\ CVRMSE & 22.92 & 16.26 & 24.50 & 11.22 & 10.45 \\ R\({}^{2}\) & 0.9709 & 0.9743 & 0.9641 & 0.9893 & 0.9899 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calculated values of different performance statistical metrics for the used models. The RMSE value of the Holt-Winters model's triple seasonality was also the lowest with a value of 67.05 among the other models, followed by the double seasonality with a value of 75.09, whereas the single seasonality of the Holt-Winters model was the maximum with a value of 165.06. Similarly, the CVRMSE percentage of the Holt-Winters model's triple seasonality was the least compared to other models, followed by the double seasonality. Therefore, based on the statistical performance metrics, it is vividly clear that the triple seasonality of the Holt-Winters model has a superior fitting and better accuracy compared to other used models followed by the double seasonality of the Holt-Winters model. A model's generalization is measured by a model's capability to estimate samples beyond the training zone. The five models' relative error variations are shown in Figure 4 with a maximum relative error value of less than 40% for all models. The triple seasonality of the Holt-Winters model values was less than 21%, and the maximum was for the single seasonality of the Holt-Winters model. The worst relative error was for the single seasonality of the Holt-Winters model. Figure 4: Relative errors percentage for the five models. Figure 3: The coefficient of determination R\({}^{2}\) and the fitting characteristics of real and simulated data for all the models. The models' robustness is assessed by incorporating different Gaussian white noise intensities into the real tested data similar to relevant previous studies [44; 45; 46]. The data were divided into a training dataset and several testing sets with each set corresponding to a different noisy intensity. Four different intensities of 20%, 40%, 60%, and 80% were utilized to investigate the robustness of the five models. The models' robustness was evaluated through the accuracy of the coefficient of determination R\({}^{2}\) by studying the random distributed Gaussian white noise on the training dataset. Table 4 shows the variance of R\({}^{2}\) affected by different noise intensities. The effects of different noise intensities are also plotted in Figure 5 to show the effects of noise on the five models' overall trends. The general trends showed an association between the increase in noise intensity and a reduction in the determination coefficient's value as expected. High robustness is noticed by the low variance of R\({}^{2}\) of the triple seasonality of the Holt-Winters model with the least value of 0.9816 at 80% noise intensity. The Holt-Winters model's single seasonality showed the least robustness among the other models with a minimum value of 0.9552. The other models showed excellent robustness across the different noise intensities. These results drive to the same previous conclusion that the triple seasonality then double seasonality of the Holt-Winters model outperformed the other models in terms of accuracy, generalization, and robustness. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Noise Intensity** & **PSR** & **PMR** & **HWSS** & **HWDS** & **HWTS** \\ \hline
0\% & 0.9709 & 0.9743 & 0.9641 & 0.9893 & 0.9899 \\
20\% & 0.9697 & 0.9710 & 0.9619 & 0.9853 & 0.9872 \\
40\% & 0.9671 & 0.9705 & 0.9592 & 0.9847 & 0.9867 \\
60\% & 0.9670 & 0.9684 & 0.9587 & 0.9820 & 0.9858 \\
80\% & 0.9667 & 0.9682 & 0.9552 & 0.9798 & 0.9816 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The accuracy reduction in coefficient of determination for different models under various noise intensities.