## 4 Results and Analysis
### Analysis of ARIMA Model Construction and Prediction Results
#### 4.3.2 Model Order Determination  
The order of the ARIMA model was determined by observing the autocorrelation and partial autocorrelation plots, selecting an appropriate model and lag order for estimation. Figure 9 shows that the autocorrelation coefficients are significantly non-zero at lag 4. Considering subjectivity, the range for q was set between one and five. The partial autocorrelation coefficients also appeared significantly non-zero after lag 3, setting the range for p between one and four. Due to the uncertainty in the number of forest fires, modeling was complex. To identify the best model, multiple ARIMA (p, d, q) models were established, and their test results are listed in Table 7. In selecting the model, criteria such as R\({}^{2}\), AIC, SC, HQC, and the standard error of regression were considered [57]. Lower AIC, SC, and HQC values, along with a larger residual, typically indicate a more optimal ARIMA model.  
Although the Akaike Information Criterion (AIC), Schwarz Criterion (SC), and Hannan-Quinn Criterion (HQC) are commonly used for model order selection, their minimum values are not the sole criteria for determining the optimal model [58]. In this study, we initially selected the ARIMA model with the lowest AIC, SC, and HQC values and conducted a randomness test on the estimated residuals of this model. If the model passed this test, it was considered optimal. If not, the model with the next lowest AIC, SC, and HQC values was considered, and this iterative process continued until the best model was identified. The residuals of the fitted ARIMA (3,1,5) model were subjected to a white noise test. The autocorrelation and partial autocorrelation plots of the residual series are shown in Figure 10, indicating that the residuals are white noise, which validates the effectiveness of this model. Table 8 presents the estimation results of the ARIMA (3,1,5) model.  
Figure 9: Autocorrelation and partial autocorrelation plots.  
The ARIMA (3,1,5) model was selected as the final model for fitting the forest fire occurrence data. The results are presented in Figure 11, where the actual data are depicted with a solid red line, and the fitted data are shown with a solid blue line. The upper and lower dashed lines correspond to the fitted values and residuals of the model, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**(p,d,q)** & **AIC** & **SC** & **HQC** & \
\begin{tabular}{c} **Adjusted** \\ **R-Squared** \\ \end{tabular} &
\begin{tabular}{c} **S.E. of** \\ **Regression** \\ \end{tabular} \\ \hline (1,1,1) & 13.53940 & 13.63369 & 13.56893 & 0.453248 & 203.8786 \\ (1,1,2) & 13.31108 & 13.45252 & 13.35537 & 0.578236 & 179.0653 \\ (1,1,3) & 12.96592 & 13.15451 & 13.02499 & 0.710097 & 148.4578 \\ (1,1,4) & 13.33599 & 13.57173 & 13.40982 & 0.591917 & 176.1372 \\ (1,1,5) & 13.40452 & 13.68741 & 13.49312 & 0.574360 & 179.8863 \\ (2,1,1) & 13.62994 & 13.77268 & 13.67358 & 0.440748 & 209.6740 \\ (2,1,2) & 13.42280 & 13.61312 & 13.48099 & 0.559083 & 186.1740 \\ (2,1,3) & 13.31421 & 13.55210 & 13.38694 & 0.615712 & 173.8079 \\ (2,1,4) & 13.08547 & 13.37095 & 13.17275 & 0.702421 & 152.9473 \\ (2,1,5) & 13.34588 & 13.67894 & 13.44770 & 0.623401 & 172.0602 \\ (3,1,1) & 13.58022 & 13.77220 & 13.63731 & 0.505039 & 209.9782 \\ (3,1,2) & 13.29464 & 13.53461 & 13.36600 & 0.638857 & 171.6736 \\ (3,1,3) & 13.36663 & 13.65460 & 13.45226 & 0.622447 & 175.5305 \\ (3,1,4) & 13.17323 & 13.50919 & 13.27313 & 0.696608 & 157.3498 \\ (3,1,5) & 12.79705 & 13.18101 & 12.91122 & 0.796418 & 128.8941 \\ (4,1,1) & 13.67274 & 13.91469 & 13.74241 & 0.486292 & 206.8332 \\ (4,1,2) & 13.31952 & 13.60985 & 13.40312 & 0.649172 & 170.9264 \\ (4,1,3) & 13.67903 & 14.01774 & 13.77656 & 0.510113 & 201.9808 \\ (4,1,4) & 12.93370 & 13.32081 & 13.04517 & 0.772763 & 137.5630 \\ (4,1,5) & 12.98354 & 13.41903 & 13.10894 & 0.765825 & 139.6470 \\ \hline \hline \end{tabular}
\end{table}
Table 7: ARIMA model test results.  
Figure 10: Plot of residual series autocorrelation and partial autocorrelation functions.