## 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 c c} \hline \hline
**Model** & **Parameter Estimation Values** & _MSE_ & \(R^{2}\) & _RMSE_ & _KD_ & _TS_ & _Bias_ \\ \hline G-O model & \(a\) = 13,720; \(b\) = 0.00004 & 910.49 & 0.6332 & 30.17 & 0.5051 & 46.6 & 27.33 \\ \hline Delayed S-shaped model & \(a\) = 45,139; \(b\) = 0.000442 & 476.11 & 0.8082 & 21.82 & 0.5142 & 33.7 & 17.23 \\ Infection S-shaped model & \(a\) = 7594.7; \(b\) = 0.000085; _\(\beta\)_= 0.033645 & 963.68 & 0.6118 & 31.04 & 0.506 & 47.94 & 26.33 \\ \hline Yamada Imperfect-2 model & \(a\) = 2071; \(b\) = 0.000162; _\(\alpha\)_ = 0.010508 & 676.83 & 0.7273 & 26.02 & 0.4051 & 40.18 & 22.1 \\ \hline P-N-Z model & \(a\) = 1498; \(b\) = 0.000545; _\(\beta\)_= 0.30981; _\(\alpha\)_ = 0.00005 & 960.58 & 0.613 & 30.99 & 0.5095 & 47.87 & 26.59 \\ \hline Weibull distribution model & \(a\) = 1347.4; \(b\) = 0.003495; \(c\) = 0.51984 & 1486.4 & 0.4012 & 38.55 & 0.6762 & 59.54 & 36.9 \\ \hline Wang model & \(a\) = 120; \(b\) = 0.005039; _\(\beta\)_ = 2.2287; \(d\) = 1.3032 & 1497.6 & 0.3967 & 38.7 & 0.7125 & 59.77 & 35.45 \\ \hline Li model & \(a\) = 120; \(b\) = 0.012707; _\(\phi\)_ = 0.26143; \(N\) = 3.2208 & 1658.9 & 0.3317 & 40.73 & 0.7479 & 62.9 & 36.45 \\ \hline PM & \(a\) = 349.79; \(b\) = 0.002847; _\(\theta\)_ = 0.000001; \(d\) = 3.0276; & **356.92** & **0.8562** & **18.89** & **0.3842** & **29.18** & **12.93** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Model performance comparison using 100% of the data (DS2-2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & _MSE_ & \(R^{2}\) & _RMSE_ & _KD_ & _TS_ & _Bias_ \\ \hline G-O model & \(a\) = 14,737; \(b\) = 0.00049 & 2219 & 0.5256 & 47.11 & 0.5746 & 56 & 40.5 \\ \hline Delayed S-shaped model & \(a\) = 253.02; \(b\) = 0.00959 & 1813.7 & 0.6123 & 42.59 & 0.5387 & 50.63 & 33 \\ \hline Infection S-shaped model & \(a\) = 191.7; \(b\) = 0.004842; _\(\beta\)_ = 0.1383  
Figure 2: _Cont._  
Figure 2: The cumulative number of detected faults over time. (**a**,**c**,**e**) represent the cumulative number of detected faults using 100% of the data for DS2-1, DS2-2, and DS2-3, respectively. (**b**,**d**,**f**) represent the cumulative number of detected faults using 85% of the data for DS2-1, DS2-2, and DS2-3, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Parameter Estimation Values** & \(MSE\) & \(R^{2}\) & \(RMSE\) & \(KD\) & \(TS\) & \(Bias\) \\ \hline G-O model & \(a=12\),170; \(b=0.000015\) & 604.47 & 0.497 & 24.59 & 0.6019 & 59.28 & 18.57 \\ \hline Delayed S-shaped model & \(a=204\),190; \(b=0.000088\) & 291.13 & 0.7578 & 17.06 & 0.3903 & 41.14 & 12.78 \\ \hline Infection S-shaped model & \(a=247.57\); \(b=0.006256\); \(\beta=14.285\) & 407.72 & 0.6607 & 20.19 & 0.4619 & 48.69 & 16.11 \\ \hline Yamada Imperfect-2 model & \(a=2948.2\); \(b=0.000018\); \(a=0.021904\) & 359.18 & 0.7011 & 18.95 & 0.4337 & 45.7 & 14.84 \\ \hline P-N-Z model & \(a=140.39\); \(b=0.006483\); \(\beta=15.751\); \(a=0.005078\) & 262.13 & 0.7819 & 16.19 & 0.3515 & 39.04 & 12.77 \\ \hline Weibull distribution model & \(a=1114.2\); \(b=0.000129\); \(c=1.033\) & 600.94 & 0.5 & 24.51 & 0.5991 & 59.11 & 18.23 \\ \hline Wang model & \(a=220\); \(b=0.004571\); \(\beta=16.162\); \(d=0.92305\) & 1004.4 & 0.1642 & 31.69 & 0.8387 & 76.42 & 21.3 \\ \hline Li model & \(a=220\); \(b=0.011034\); \(\phi=0.74084\); \(N=0.30068\) & 983.72 & 0.1814 & 31.36 & 0.8284 & 75.63 & 20.59 \\ \hline PM & \(a=148.72\); \(b=0.013007\); \(\theta=0.05343\); \(d=0.19862\); & **113.84** & **0.9053** & **10.67** & **0.2245** & **25.73** & **8.02** \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Model performance comparison using 100% of the data (DS3-3).