## 4 Numerical Examples
### Analysis and Discussion of Model Performance Comparison Using LSE Estimation Parameter Values
From Figure 3b, we can see that the fitting and predictive performance of the PM is better than the other models' performances. Using 85% of the data (DS3-2), the second best is the GGO model, and the worst is the Wang model. From Figure 3d, we can see that the fitting and predictive performance of the PM is better than those of the other models. Using 85% of the data (DS3-3), the second best is the ISS model, and the worst is \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & \(MSE_{prict}\) & \(RMSE_{prict}\) & \(T_{prict}\) & \(Bias_{prict}\) \\ \hline G-O model & \(a=12,240\); \(b=0.000039\) & 972.08 & 31.18 & 28.35 & 4.67 \\ \hline Delayed S-shaped model (DSS) & \(a=189,750\); \(b=0.00022\) & 317.25 & 17.81 & 16.2 & 2.18 \\ \hline Inflection S-shaped model (ISS) & \(a=146.14\); \(b=0.063423\); \(\beta=2379.3\) & 720.84 & 26.85 & 24.42 & 4 \\ \hline Yamada Imperfect-2 model & \(a=1191.2\); \(b=0.000045\); \(a=0.14894\) & 189.53 & 13.77 & 12.52 & 1.69 \\ \hline P-N-Z model & \(a=170.5\); \(b=0.008725\); \(\beta=11.167\); \(\alpha=0.02813\) & 1152.1 & 33.94 & 30.87 & 4.57 \\ \hline Weibull distribution model (GGO) & \(a=984.25\); \(b=0.000719\); \(c=0.91933\) & 1247.1 & 35.31 & 32.11 & 5.31 \\ \hline Wang model & \(a=120\); \(b=0.008146\); \(\beta=0.66208\); \(d=1.0133\) & 4943.9 & 70.31 & 63.94 & 10.61 \\ \hline Li model & \(a=120\); \(b=0.012852\); \(\phi=0.20877\); \(N=3.0644\) & 4369.6 & 66.1 & 60.11 & 9.97 \\ \hline PM & \(a=199.99\); \(b=0.001332\); \(\theta=0.000941\); \(d=1.3562\); \(\sigma=0.07682\); \(\beta=1.0428\) & **92.41** & **9.61** & **8.74** & **1.19** \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Model performance comparison using 85% of the data (DS2-2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & \(MSE_{prict}\) & \(RMSE_{prict}\) & \(T_{prict}\) & \(Bias_{prict}\) \\ \hline G-O model & \(a=13,587\); \(b=0.000028\) & 18,870 & 137.37 & 69.88 & 20.41 \\ \hline Delayed S-shaped model (DSS) & \(a=199,080\); \(b=0.000195\) & 11,278 & 106.2 & 54.02 & 15.77 \\ \hline Inflection S-shaped model (ISS) & \(a=1121.3\); \(b=0.050711\); \(\beta=10275\) & 6141.7 & 78.37 & 39.86 & 9.02 \\ \hline Yamada Imperfect-2 model & \(a=1405\); \(b=0.000009\); \(a=0.58784\) & 10,835 & 104.09 & 52.95 & 15.45 \\ \hline P-N-Z model & \(a=64.107\); \(b=0.009432\); \(\beta=3.6832\); \(\alpha=0.022014\) & 12,738 & 112.86 & 57.41 & 16.76 \\ \hline Weibull distribution model (GGO) & \(a=932.41\); \(b=0.000008\); \(c=1.8644\) & 11,953 & 109.33 & 55.61 & 16.24 \\ \hline Wang model & \(a=220\); \(b=0.008994\); \(\beta=0.34156\); \(d=0.81963\) & 25,696 & 160.3 & 81.54 & 23.82 \\ \hline Li model & \(a=219.94\); \(b=0.27305\); \(\phi=0.021584\); \(N=5.5202\) & 29,722 & 172.4 & 87.7 & 25.62 \\ \hline PM & \(a=202.02\); \(b=0.031377\); \(\theta=0.003222\); \(d=1.1128\); \(\sigma=0.11074\); \(\beta=400.97\) & **1484.8** & **38.53** & **19.6** & **5.1** \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Model performance comparison using 85% of the data (DS2-3). the Wang model. Figure 3f shows that the PM outperforms other models in terms of fitting performance. Moreover, the PM has the best predictive performance among them.