## 4 Results
### Comparison of Time Series Forecasting Models  
Classical time series-based ARIMA and Holt-Winters models, the deep learning-based LSTM model, and our developed hybrid ARIMA-SVR time series prediction models were used to assess the predictability of the Ecological Footprint worldwide. The ADF test was performed to assess the stationarity of the time series, and the results are presented in Table 2. The ADF test statistic for the original series was calculated as \(-0.787\), and the critical values for the significance levels of 1%, 5%, and 10% were \(-3.542\), \(-2.910\), and \(-2.593\), respectively. Additionally, the calculated \(p\)-value (0.823), being higher than the significance alpha level (\(\alpha\) = 0.05), indicates that the series is non-stationary. Therefore, to make the time series stationary, the first difference was taken (d = 1), and the ADF test was performed again. The calculated \(p\)-value for the first-differenced series is 0.000, indicating that it is stationary since this value is less than the 0.05 significance level, implying the absence of a unit root. With the series made stationary, the parameters of the ARIMA model can be determined. To determine ARIMA model parameters, two approaches can be employed. Parameters are determined by examining Autoregressive Function (ACF) and Partial Autoregressive Function (PACF) plots [31]. Nevertheless, it may not always be feasible to observe the model's structure through these plots. Therefore, model parameters can also be determined using automatic model selection. In Python programming, the 'auto_arima' function can be used to identify the most successful model. This function defaults to the AIC metric, but model selection can also be carries out based on BIC or AICc metrics. However, criteria like AIC, while generally helpful in model selection, may not always yield accurate results when strictly adhered to. Therefore, different ARIMA models were tested on the test set for Ecological Footprint predictions, and their performances were compared (Table 3). Using the same method, the 10-year data of the test set was predicted individually, and the metrics in the table were calculated. Subsequently, the averages of AIC, BIC, AICc, RMSE, and MAPE values for this 10-year test set were calculated and compared in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Data**} & \multirow{2}{*}{**ADF Test**} & \multicolumn{4}{c}{**Critical Value**} & \multirow{2}{*}{**\(p\)-Value**} & \multirow{2}{*}{**Stationarity**} \\ \cline{3-3} \cline{5-6}  & & **1\%** & & & & \\ \hline Original & \(-0.787\) & \(-3.542\) & \(-2.910\) & \(-2.593\) & 0.823 & Non-Stationary \\ \hline First differenced & \(-6.703\) & \(-3.546\) & \(-2.912\) & \(-2.594\) & 0.000 & Stationary \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ADF unit root test and difference results. The auto_arima function determines the best model based on AIC, BIC, and AICc metrics. The model selected by auto_arima, which has the lowest values for these metrics highlighted in bold in Table 3, is ARIMA(0,1,0). However, when different ARIMA models are tested on the EF dataset, it is revealed that the ARIMA (1,1,0) model has lower RMSE and MAPE values (Table 3, underlined bold). Therefore, the parameters \(p\), \(d\), \(q\) for the ARIMA model have been chosen as 1, 1, 0, respectively. The suitability of the ARIMA (1,1,0) model was assessed utilizing the Ljung-Box test and ACF with PACF graphs. The Ljung-Box test, which relies on the sum of the squares of autocorrelation coefficients, is a statistical test that examines the significance of autocorrelation in the residuals of a time series. This test attempts to determine whether autocorrelation at a specific lag level exceeds a level of randomness. The ACF and PACF residual graphs shown in Figure 6 indicate the absence of significant autocorrelation, as the error correlations fall within the estimated threshold boundaries at various lags.