## 5 Discussion  
In the literature, various time series forecasting models with different capabilities exist. The forecasting performance of these methods, similar to other artificial intelligence techniques, is contingent upon the dataset. Testing various methods is crucial to identify the model that best suits the dataset. This study comparatively analyzes the Ecological Foot-print (EF) forecasting capabilities of time series models, including LSTM from deep learning methods, ARIMA (1,1,0) and Holt-Winters from classical methods, and the ARIMA-SVR  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Year**} & \multicolumn{2}{c}{**ARIMA (1,1,0)**} & \multicolumn{2}{c}{**Holt-Winters**} & \multicolumn{2}{c}{**LSTM**} & \multicolumn{2}{c}{**ARIMA-SVR**} \\ \cline{2-9}  & **RMSE (10\({}^{4}\) gha)** & **MAPE (\%)** & **RMSE (10\({}^{4}\) gha)** & **MAPE (\%)** & **RMSE (10\({}^{4}\) gha)** & **MAPE (\%)** & **RMSE (10\({}^{4}\) gha)** & **MAPE (\%)** \\ \  
from hybrid methods. Figure 7 illustrates forecasted EF values (10\({}^{10}\) gha) by different time series models and actual EF values (10\({}^{10}\) gha). The zoomed-in section in Figure 7 displays the actual EF values (test set) for 2013-2022, along with the forecasted EF values by LSTM, ARIMA (1,1,0), Holt-Winters, and ARIMA-SVR for these years. Despite the growing inclination towards employing deep learning and hybrid methods for achieving more accurate forecasts, our findings demonstrate that the classical time series method ARIMA, known for its simplicity in implementation, yields more accurate EF forecasts compared to other methods. The forecasting performances of the models were statistically tested using the Wilcoxon Signed-Rank and Friedman Tests. The significance level for both tests was set at 0.05. Thus, a robust statistical analysis of the models' forecasting capabilities based on mean MAPE values was ensured. The results obtained from these tests are given in Table 7.  
The Friedman test was employed to determine if there was a statistically significant difference among four different time series models. On the other hand, the Wilcoxon Signed-Rank Test was used to identify differences between pairs of models. The Friedman test statistic yielded a statistic of 14.879, with a corresponding \(p\)-value of 0.0019. Since the \(p\)-value is \(<\)0.05, it is concluded that there is a statistically significant difference in the overall performance of the models. In the pairwise comparative analysis of forecasting models, according to the results of the Wilcoxon Signed-Rank Test, statistical differences emerged between several model pairs. Specifically, the ARIMA model exhibited statistically significant differences when compared to the ARIMA-SVR model (\(p\) = 0.00195), the Holt-Winters model (\(p\) = 0.00390), and between the ARIMA-SVR model and the Holt-Winters model (\(p\) = 0.00976). These significant findings emphasize not only the overall importance of model performance but also highlight specific cases where one model performs better than another. The remarkable statistical discrepancy between the ARIMA model and both the ARIMA-SVR and Holt-Winters models (\(p\) = 0.00195 and \(p\) = 0.00390, respectively) signifies the ARIMA model's superior predictive capabilities in comparison. Moreover, the notable difference observed between the ARIMA-SVR model and the Holt-Winters model (\(p\) = 0.00976) reveals specific aspects of performance, offering valuable insights into the strengths and weaknesses of each model. These results indicate that the ARIMA model stands out, demonstrating higher forecast accuracy and reliability compared to other models. Advances in time series forecasting are characterized by innovative approaches, including the adoption of deep learning techniques, the use of big data, the widespread use of ensemble methods, and the integration of hybrid models in addition to classical time series methods.