## 3 Results and Discussion
### Forecasting Performance for Each Provincial-Level Administrative Region in 2023  
Among all the abovementioned models, the TNLI-power exponential, LPQI-power exponential, ELM, and PSO-ELM four models have better overall forecasting performance \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **R\({}^{2}\)** & **MRE** \\ \hline MLR & 0.6253 & 67.48\% \\ ELM & 0.8968 & 33.60\% \\ PSO-ELM & 0.8974 & 23.42\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The modeling and forecasting results based on multiple NTL indices. on the TOVAFAF than other models, in which TNLI-power exponential and LPQI-power exponential represent the power exponential model established based on the single TNLI and LPQI index, respectively. To further demonstrate the effectiveness of the aforementioned four models, the RE was used to evaluate the forecasting effect for 29 provincial-level administrative regions in 2023. It should be noted that five provincial-level administrative regions of China, i.e., Beijing, Shanghai, Hong Kong, Macao, and Taiwan, are not included in the scope of this study, and the specific reasons have been explained in Section 2.1. Figure 5 shows the actual value and model forecasting value of TOVAFAF for each provincial-level administrative region in 2023. It can be qualitatively seen the forecasting effect of each model on the TOVAFAF of each provincial-level administrative region. For the PSO-ELM model, except for Shandong and Zhejiang provinces, the forecasting values of TOVAFAF in most provincial-level administrative regions are close to the actual value or fluctuate near the actual value. The forecasting effect of the ELM model on the TOVAFAF in many provinces is similar to that of PSO-ELM. For the TNLI-power exponential and LPQI-power exponential models, the forecasting values of TOVAFAF are close to the actual value in some provinces, while the forecasting values are much different from the actual value in some provinces. To quantitatively evaluate the forecasting effect of various models on the TOVAFAF, Figure 6 shows the RE of the four models for each provincial-level administrative region in 2023, and Figure 7 further shows the number of provincial-level administrative regions at various levels of RE in 2023. For the PSO-ELM model, the RE values of 22 provincial-level administrative regions are below 50%, and the RE values of 18 provincial-level administrative regions are below 30%, which makes the fluctuation amplitude of the RE curve smaller than that of other models. For the ELM model, there are also 22 provincial-level administrative regions with RE below 50%. However, the number of provincial-level administrative regions with RE below 30% for the ELM model is 15, which is less than that of the PSO-ELM model. In addition, the RE values of Guangdong, Hainan, Qinghai, Shanxi, and Zhejiang provinces even exceed 90% for the ELM model. That makes the fluctuation amplitude of the RE curve of the ELM model larger than that of the PSO-ELM model. There are 11 and 14 provincial-level administrative regions with RE above 50% for the TNLI-power exponential model and the LPQI-power exponential model, respectively. And the RE values of some provincial-level administrative regions even significantly exceed 100% for the TNLI-power exponential and LPQI-power exponential models. Therefore, the fluctuation amplitude of the RE curves of the TNLI-power exp  
Figure 5: The actual value and model forecasting value of TOVAFAF for each provincial-level administrative region in 2023.  
exponential models is much larger than that of the ELM and PSO-ELM models. The above results indicate that the forecasting performance of the PSO-ELM model is superior to that of other models. Furthermore, MRE was used to evaluate the overall forecasting performance of the four models in 2023. The linear relationship between the actual values and the forecasting values of each model for 29 provincial-level administrative regions in 2023 was analyzed.