## 3 Results and Discussion
### Forecasting Model Based on Single NTL Index  
The TOVAFAF for each provincial-level administrative region from 2013 to 2020 and the corresponding NTL indices were used as the training set to establish the forecasting models. Taking the four NTL indices (TNLI, LPQI, LPRI, and NLSDSI) as the single independent variable, respectively, and the TOVAFAF as the dependent variable, linear model, exponential model, logarithmic model, power exponential model, and polynomial model were established. Furthermore, the established models were used to forecast the TOVAFAF for each provincial-level administrative region from 2021 to 2022, and the effectiveness of the established models was evaluated using R\({}^{2}\) and MRE. The modeling and forecasting results are shown in Table 3, and Figures 3 and 4. From the perspective of NTL index, Table 3 and Figure 3 show that the R\({}^{2}\) values of the five models (linear, exponential, logarithmic, power exponential, and polynomial models) established based on LPQI index are higher than those of the corresponding models established by other NTL indices (TNLI, LPRI, and NLSDSI). Table 3 and Figure 4 show that the MRE values of the five models established based on LPQI index are lower than those of the corresponding models established by other NTL indices. The above results indicate that the forecasting performance of the five models established based on LPQI index is superior to that of corresponding models established by other NTL indices. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Index** & **TNLI** & **ANLI** & **LPQI** & **LPRI** & **ALPLI** & **NLSDI** & **NLVI** & **NLSDSI** \\ \hline Correlation & 0.6587 & 0.0570 & 0.7676 & 0.2754 & 0.0752 & 0.1092 & \(-\)0.0553 & 0.5409 \\ Coefficient & \(p\)-value & \(1.829\times 10^{-37}\) & \(3.331\times 10^{-1}\) & \(1.430\times 10^{-57}\) & \(1.917\times 10^{-6}\) & \(2.014\times 10^{-1}\) & \(6.339\times 10^{-2}\) & \(3.481\times 10^{-1}\) & \(1.955\times 10^{-23}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of correlation analysis and significance test. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Index Name** & **Model** & **Expression** & \(\mathbf{R}^{2}\) & **MRE** \\ \hline \multirow{4}{*}{TNLI} & Linear & \(t=~{}0.0046x+~{}1730.5\) & 0.4257 & 88.11\% \\  & Exponential & \(t=~{}1284.5\mathrm{e}^{(2\times 10^{-6})x}\) & 0.3415 & 141.06\% \\  & Logarithmic & \(t=2218.8\mathrm{ln}(x)~{}-~{}24467\) & 0.5375 & 62.29\% \\  & Power exponential & \(t=~{}(0.0083\mathrm{x}^{0.9971})\) & 0.6518 & 58.12\% \\  & Polynomial & \(t=~{}(-5\times 10^{-9})x^{2}+~{}0.012x+~{}20.84\) & 0.5438 & 66.68\% \\ \hline \multirow{4}{*}{LPQI} & Linear & \(t=~{}0.0238x+~{}784.37\) & 0.5782 & 67.51\% \\  & Exponential & \(t=~{}886.57\mathrm{e}^{(9\times 10^{-6})x}\) & 0.4851 & 71.34\% \\  & Logarithmic & \(t=2589.4\mathrm{ln}(x)~{}-~{}26065\) & 0.5820 & 60.51\% \\  & Power exponential & \(t=~{}0.0039x1.1682\) & 0.7113 & 48.34\% \\  & Polynomial & \(t=~{}(-3\times 10^{-8})x^{2}+~{}0.0321x+~{}312.52\) & 0.5846 & 62.47\% \\ \hline \multirow{4}{*}{LPRI} & Linear & \(t=~{}3477x+~{}3061\) & 0.0705 & 138.57\% \\  & Exponential & \(t=~{}2378.2\mathrm{e}^{0.6514x}\) & 0.0149 & 112.83\% \\ \cline{1-1}  & Logarithmic & \(t=~{}817.64\mathrm{ln}(x)~{}+~{}5465.1\) & 0.1585 & 109.69\% \\ \cline{1-1}  & Power exponential & \(t=~{}5951.9x^{0.381}\) & 0.2066 & 98.55\% \\ \cline{1-1}  & Polynomial & \(t=-18965x^{2}+~{}16187x+~{}1871.3\) & 0.1965 & 94.51\% \\ \hline \multirow{4}{*}{NLSDSI} & Linear & \(t=~{}0.0002x+~{}2099.3\) & 0.2758 & 104.49\% \\ \cline{1-1}  & Exponential & \(t=~{}1397\mathrm{e}^{(8\times 10^{-5})x}\) & 0.2591 & 109.15\% \\ \cline{1-1}  & Logarithmic & \(t=~{}1857.3\mathrm{ln}(x)~{}-25254\) & 0.3736 & 91.44\% \\ \cline{1-1}  & Power exponential & \(t=~{}0.0038x^{0.8622}\) & 0.4835 & 78.86\% \\ \cline{1-1}  & Polynomial & \(t=~{}(-1\times 10^{-11})x^{2}+~{}0.0005x+~{}882.81\) & 0.3428 & 98.79\% \\ \hline \end{tabular}
\end{table} 
Table 3: The modeling and forecasting results based on single NTL index.