## 4 Results
### Comparison of Time Series Forecasting Models
The results of the Ljung-Box test for the ARIMA (1,1,0) model are provided in Table 4. If the \(p\)-value is greater than 0.05, the hypothesis is not rejected, indicating that autocorrelations are not statistically significant in this case. Since the \(p\)-value in the test result is greater than 0.05, it has been demonstrated that the ARIMA (1,1,0) model is suitable for Ecological Footprint time series data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & \(\mathbf{Q}\) & \(\mathbf{p}\)**-Value** \\ \hline
**ARIMA (1,1,0)** & 0.25 & 0.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of the Ljung–Box test for the ARIMA (1,1,0) model. Figure 6: Residuals, ACF, and PACF diagrams for the ARIMA (1,1,0) model. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Model** & **AIC** & **BIC** & **AICc** & \
\begin{tabular}{c} **RMSE** \\ **(10\({}^{4}\) gha)** \\ \end{tabular} &
\begin{tabular}{c} **MAPE** \\ **(\%)** \\ \end{tabular} \\ \hline ARIMA(2,1,2) & 1336 & 1347 & 1337 & 59,696 & 2.27 \\
**ARIMA(0,1,0)** & **1345** & **1346** & **1345** & 59,696 & 2.27 \\
**ARIMA (1,1,0)** & 1349 & 1350 & 1346 & **59,536** & **2.12** \\ \hline ARIMA(0,1,1) & 1345 & 1350 & 1346 & 59,609 & 2.16 \\ ARIMA(1,1,1) & 1347 & 1350 & 1345 & 60,398 & 2.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of different ARIMA models on the test set. After identifying the ARIMA (1,1,0) model that best fits the dataset, the objective is to determine the LSTM model that is most suitable for the dataset. LSTM models possess the capability to capture more complex patterns and nonlinear relationships in time series data [32]. However, due to their numerous parameters, LSTM models can exhibit variable performance. Therefore, a series of trial and validation processes are necessary to identify an appropriate LSTM model. This process involves evaluating different configurations of LSTM models and selecting the one that achieves the best performance. Different combinations of parameters were used to create a series of LSTM models, and these models were then trained with the dataset from 1961 to 2012, predicting values for the years 2013\(-\)2022. During the training of the LSTM model, the Mean Squared Error (MSE) performance metric was utilized for loss evaluation. The Rectified Linear Unit (ReLU), the most widely utilized activation function in deep neural networks, was utilized [33]. The Adam optimizer with a learning rate of 0.001 was chosen LSTM models were trained with randomly selected batch sizes, epochs, neurons, and dropout rate values. The trained models were used to forecast the Ecological Footprint for the last 10 years, and their performance was evaluated. The parameters and test set results of the LSTM models, including RMSE and MAPE, are presented in Table 5. According to the test results, the LSTM\({}_{12}\) model, with parameters batch size = 32, epoch = 250, neuron = 128, dropout rate = 0.0, demonstrated the most successful outcome, yielding a 3.19% error and 64,030 \(\times\) 10\({}^{4}\) gha. Finally, the hybrid ARIMA-SVR model was developed to forecast the Ecological Footprint. In developing the ARIMA-SVR model, the Grid Search method was employed to determine hyperparameters for support vector regression (SVR). Through various trials with Grid Search, the values C = 1000, epsilon = 0.5, and the 'linear' kernel function were determined for SVR. The performances of the four different time series prediction models used in the study were compared based on the RMSE and MAPE metrics. In Table 6, annual MAPE and RMSE values calculated for the predicted values by the models for the test set (2013\(-\)2022 EF values) are presented. Additionally, the average MAPE and RMSE results for the test set are provided below the table.