## 3 Results and Discussion
### Forecasting Model Based on Single NTL Index
From the perspective of the established model, Table 3 and Figure 3 show that the R\({}^{2}\) value of the power exponential model established based on each NTL index (TNLI, LPQI, LPRI, and NLSDSI) is higher than those of the other models (linear, exponential, logarithmic, and polynomial models) established by the corresponding NTL index. Table 3 and Figure 4 show that the MRE value of the power exponential model established based on each NTL index is lower than those of the other models established by the corresponding NTL index. The above results indicate that the forecasting performance of the power exponential model is superior to that of the other models established based on each NTL index. Furthermore, among all the models established based on each NTL index, the power exponential model established by the LPQI index achieves the best forecasting performance with the R\({}^{2}\) value for 0.7113 and the MRE value for 48.34%. However, the MRE of the optimal model for forecasting the TOVAFAF is close to 50%, indicating that the forecasting models established based on single NTL index cannot commendably reflect the relationship between NTL data and the TOVAFAF. forecasting models based on multiple NTL indices. As with the modeling process based on single NTL index, the TOVAFAF for each provincial-level administrative region from 2013 to 2020 and the corresponding multiple NTL indices were used as the training set, and the relevant data from 2021 to 2022 were used as the validation set. Due to the extremely significant correlation between the four NTL indices (TNLI, LPQI, LPRI, and NLSDSI) and the TOVAFAF, all the four NTL indices were taken as the independent variables together, and the TOVAFAF was taken as the dependent variable. Before modeling, each NTL index was normalized to the range of 0-1 to avoid the impact of different orders of magnitude on modeling results. The specific mathematical expression of the MLR model in this paper is as follows:  
\[t=11577.99*x_{1}+7915.59*x_{2}-10210.69*x_{3}-2189.15*x_{4}+1494.33 \tag{9}\]  
where \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) represent the value of TNLI, LPQI, NLSDSI, and LPRI, respectively. For the ELM model, its mathematical expression is shown in Equations (1) to (5). The activation function \(g(\cdot)\) and the number of hidden layer neurons \(L\) produce important effects on model performance. This paper took the sigmoid function as the activation function of the ELM model, as shown in Equation (10). By optimization, the number of hidden layer neurons was set to 35. Since the input weights and hidden layer bias were generated randomly, the ELM model was run 20 times, and the average value was taken as the final result to avoid the influence of random errors on the forecasting results. \[g(\phi)=\frac{1}{1+e^{-\phi}} \tag{10}\]  
To further improve the forecasting performance of the ELM model, the particle swarm optimization (PSO) algorithm was used to optimize the input weights and hidden layer biases of the ELM model, thereby establishing the PSO-ELM model. In this model, the number of iterations of the PSO algorithm was set to 200, and MRE of the validation set was taken as the fitness function to evaluate the optimization effect. The modeling and forecasting results based on multiple NTL indices are shown in Table 4.  
As can be seen from Table 4, among the forecasting models established based on multiple NTL indices, the MLR model obtains the worst forecasting performance with the R\({}^{2}\) value for 0.6253 and the MRE value for 67.48%. The MRE value of the MLR model is relatively large, and its forecasting performance is even inferior to that of the power exponential models established based on the single TNLI and LPQI index. The R\({}^{2}\) and MRE values of the ELM model reach 0.8968 and 33.60%, respectively, so its forecasting performance is significantly better than that of the MLR model. This indicates that the relationship between the NTL data and the TOVAFAF is not a simple linear relationship, but a more complex nonlinear relationship.