## 4 Numerical Examples
### Analysis and Discussion of Model Performance Comparison Using LSE Estimation Parameter Values
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & \(MSE_{prdict}\) & \(RMSE_{prdict}\) & \(TS_{prdict}\) & \(Bus_{prdict}\) \\ \hline G-O model & \(a\) = 15,384; \(b\) = 0.000093 & 384.31 & 19.6 & 16.89 & 2.88 \\ \hline Delayed S-shaped model & \(a\) = 69,278; \(b\) = 0.000963 & 860.68 & 29.34 & 25.27 & 3.98 \\ \hline Inflection S-shaped model & \(a\) = 169,49; \(b\) = 0.09902; _β_ = 156.02 & 721.22 & 26.86 & 23.13 & 3.91 \\ \hline Yamada Imperfect-2 model & \(a\) = 550.62; \(b\) = 0.00016; \(a\) = 0.6667 & 809.63 & 28.45 & 24.51 & 3.84 \\ \hline P-N-Z model & \(a\) = 116,44; \(b\) = 0.095367; _β_ = 103.89; \(a\) = 0.007597 & 832.66 & 28.86 & 24.86 & 4.16 \\ \hline Weibull distribution model & \(a\) = 774.76; \(b\) = 0.002447; \(c\) = 0.93203 & 644.44 & 25.39 & 21.87 & 3.79 \\ \hline Wang model & \(a\) = 220; \(b\) = 0.000497; _β_ = 14.432; \(d\) = 2.0004 & 473.97 & 21.77 & 18.75 & 3.22 \\ \hline Li model & \(a\) = 126.26; \(b\) = 0.019206; \(b\) = 13.13; \(N\) = 0.0543 & 795.44 & 28.2 & 24.29 & 4.22 \\ \hline PM & \(a\) = 188; \(b\) = 0.006066; _β_ = 0.000013; \(d\) = 0.96689; & **211.38** & **14.54** & **12.52** & **2.05** \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Model performance comparison using 85% of the data (DS1-2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & \(MSE_{prdict}\) & \(RMSE_{prdict}\) & \(TS_{prdict}\) & \(Bus_{prdict}\) \\ \hline G-O model & \(a\) = 4340; \(b\) = 0.000502 & 302.96 & 17.41 & 20.67 & 2.6 \\ \hline Delayed S-shaped model & \(a\) = 159,89; \(b\) = 0.050676 & 108.09 & 10.4 & 12.35 & 1.48 \\ \hline Inflection S-shaped model & \(a\) = 529.13; \(b\) = 0.043241; _β_ = 14.957 & 49.52 & 7.04 & 8.36 & 0.86 \\ \hline Yamada Imperfect-2 model & \(a\) = 1821.4; \(b\) = 0.0008092; _β_ = 0.045222 & 74.46 & 8.63 & 10.25 & 1.1 \\ \hline P-N-Z model & \(a\) = 137.86; \(b\) = 0.033775; _β_ = 2.3587; \(a\) = 0.033423 & 54.95 & 7.41 & 8.8 & 0.91 \\ \hline Weibull distribution model & \(a\) = 559.16; \(b\) = 0.006495; \(c\) = 0.83948 & 530.95 & 23.04 & 27.37 & 3.48 \\ \hline Wang model & \(a\) = 133.01; \(b\) = 0.000027; _β_ = 14.432; \(d\) = 2.865 & 343.62 & 18.54 & 22.02 & 2.78 \\ \hline Li model & \(a\) = 132.98; \(b\) = 0.026009; _β_ = 0.13769; \(N\) = 8.4145 & 868.29 & 29.47 & 35 & 4.46 \\ \hline PM & \(a\) = 180; \(b\) = 0.024217; _β_ = 0.00196; \(d\) = 2.0528; & **45.63** & **6.76** & **8.02** & **0.84** \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Model performance comparison using 85% of the data (DS1-1). model comes in last. The PM has the greatest fitting and predictive performance among them, as shown in Figure 1f. From Tables 15-17, we can see that using 85% of the data (DS2-1, DS2-2, and DS2-3), the \(MSE_{p\text{\_}dict}\), \(RMSE_{p\text{\_}dict}\), \(TS_{p\text{\_}dict}\), and \(Bias_{p\text{\_}dict}\) values of the PM are 1789.1, 42.3, 10.72, and 5.48; 92.41, 9.61, 8.74, and 1.19; and 1484.8, 38.53, 19.6, and 5.1, respectively. Comparing the PM to other models, it performs predictions more accurately. The second best is the P-N-Z model, and the worst is the Li model. According to Table 15, the \(MSE_{p\text{\_}dict}\), \(RMSE_{p\text{\_}dict}\), and \(Bias_{p\text{\_}dict}\) values of the second-ranked Yamada Imperfect-2 model are around 1.16, 0.47, and 0.6 times larger than those of the PM, respectively. Table 16 shows that the \(MSE_{p\text{\_}dict}\), \(RMSE_{p\text{\_}dict}\), and \(Bias_{p\text{\_}dict}\) values of the Yamada Imperfect-2 model, which came in second place, are 1.05, 0.43, and 0.42 times larger than those of the PM, respectively. Table 17 reveals that the \(MSE_{p\text{\_}dict}\), \(RMSE_{p\text{\_}dict}\), and \(Bias_{p\text{\_}dict}\) values of the second-placed ISS model are approximately 3.14, 1.03, and 0.76 times greater than those of the PM, respectively. Figure 2b demonstrates the generality of the PM's fitting performance. However, the PM performs prediction better than the other models.