## 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** & \multicolumn{2}{c}{**Parameter Estimation Values**} & \(MSE_{p\text{\_}dict}\) & \(RMSE_{p\text{\_}dict}\) & \(TS_{p\text{\_}dict}\) & \(Bias_{p\text{\_}dict}\) \\ \hline G-O model & \(a=57\),779; \(b=0.000119\) & 20,925 & 144.66 & 36.65 & 22.19 \\ \hline Delayed S-shaped model & \(a=1150.4\); \(b=0.025851\) & 13,432 & 115.9 & 29.37 & 17.62 \\ \hline Inflection S-shaped model & \(a=578.36\); \(b=0.27254\); \(\beta=2737.9\) & 13,260 & 115.15 & 29.18 & 17.2 \\ \hline Yamada Imperfect-2 model & \(a=301.53\); \(b=0.008722\); \(a=0.1615\) & 3863.1 & 62.15 & 15.75 & 8.75 \\ \hline P-N-Z model & \(a=460.31\); \(b=0.065729\); \(\beta=17.129\); \(a=0.071921\) & 6923 & 83.2 & 21.08 & 10.43 \\ \hline Weibull distribution model & \(a=2836.8\); \(b=1\times 10^{-6}\); \(c=3.4407\) & 50,403 & 224.51 & 56.88 & 32 \\ \hline Wang model & \(a=420\); \(b=0.003043\); \(\beta=15.064\); \(d=1.7471\) & 28,218 & 167.98 & 42.56 & 25.81 \\ \hline Li model & \(a=420.01\); \(b=0.03163\); \(\phi=0.29135\); \(N=3.6529\) & 50,748 & 225.27 & 57.08 & 34.65 \\ \hline PM & \(a=399.97\); \(b=0.074046\); \(\theta=0.003679\); \(d=1.5573\); & **1789.1** & **42.3** & **10.72** & **5.48** \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Model performance comparison using 85% of the data (DS2-1). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & \multicolumn{2}{c}{**Parameter Estimation Values**} & \(MSE_{p\text{\_}dict}\) & \(RMSE_{p\text{\_}dict}\) & \(TS_{p\text{\_}dict}\) & \(Bias_{p\text{\_}dict}\) \\ \hline G-O model & \(a=4124.8\); \(b=0.000117\) & 657.47 & 25.64 & 41.34 & 3.44 \\ \hline Delayed S-shaped model & \(a=162.24\); \(b=0.013341\) & 332.53 & 18.24 & 29.4 & 2.33 \\ \hline Inflection S-shaped model & \(a=93.789\); \(b=0.031273\); \(\beta=10.262\) & 336.23 & 18.34 & 29.56 & 2.35 \\ \hline Yamada Imperfect-2 model & \(a=89.907\); \(b=0.003484\); \(a=0.024927\) & 406.58 & 20.16 & 32.51 & 2.63 \\ \hline P-N-Z model & \(a=163.03\); \(b=0.008491\); \(\beta=2.697\); \(a=0.006232\) & 454.22 & 21.31 & 34.36 & 2.8 \\ \hline Weibull distribution model & \(a=500.27\); \(b=0.000441\); \(c=1.2008\) & 555.69 & 23.57 & 38 & 3.13 \\ \hline Wang model & \(a=220\); \(b=0.000431\); \(\beta=10.457\); \(d=1.758\) & 694.05 & 26.34 & 42.47 & 3.54 \\ \hline Li model & \(a=117.61\); \(b=0.009977\); \(\phi=0.038583\); \(N=15.754\) & 991.27 & 31.48 & 50.76 & 4.29 \\ \hline PM & \(a=149.9\); \(b=0.019652\); \(\theta=0.00013\); \(d=2.5565\); & **113.91** & **10.67** & **17.21** & **1.22** \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Model performance comparison using 85% of the data (DS1-3). The DSS model comes in second, and the Wang model comes in last, using 85% of the data (DS2-2). We can see from Figure 2d that the suggested model's performance in terms of fitting is better than the G-O model, GGO model, Wang model, and Li model, but worse than the DSS model, ISS model, Yamada Imperfect-2 model, and P-N-Z model. Yet, the PM performs predictions more accurately than the other models. Using 85% of the data (DS2-3), the second best is the ISS model, and the worst is the Li model. Figure 2f shows that the PM's fitting performance is better than the other models, except for the ISS model, which achieves a better performance. The PM also has the highest prediction accuracy of any of them. Tables 18-20 show that using 85% of the data (DS3-1, DS3-2, and DS3-3), the \(MSE_{prict}\), \(RMSE_{prict}\), \(T_{prict}\), and \(Bias_{prict}\) values of the PM are 28.75, 5.36, 1.38, and 0.61; 1469.1, 38.33, 38.71, and 5.28; and 9.13, 3.02, 6.2, and 0.39, respectively. The PM has better predictive performance than the other models. The second best is the ISS model, and the worst is the Wang model. Table 20 shows that the \(MSE_{prict}\), \(RMSE_{prict}\), and \(Bias_{prict}\) values of the ISS model which ranked second are about 60.69, 6.86, and 7.26 times larger than those of the PM, respectively.