## 3 Model Comparisons  
### Comparison Criteria and Parameter Estimation Method  
#### 3.1.1 Criteria for Models' Descriptive Power Comparison  
Here we use three criteria to judge the performance of the models. The first criterion is the mean value of squared error (Mean Square Error, MSE), which can be calculated as follows:  
\[\text{MSE}=\frac{1}{n-N}\sum\limits_{i=1}^{n}{(y_{i}-\hat{m}(t_{i}))^{2}} \tag{11}\]  
where \(n\) represents the number of observations, \(y_{i}\) represents the total number of faults observed by time \(t_{i}\), \(\hat{m}(t_{i})\) denotes the estimated cumulative number of faults up to time \(t_{i}\), \(N\) represents the number of parameters in the model. Therefore, the lower the value of MSE, the better the model performs.  
The second criterion is correlation index of the regression curve equation (\(R^{2}\)), which can be expressed as follows:  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}{(y_{i}-\hat{m}(t_{i}))^{2}}}{\sum\limits_{ i=1}^{n}{(y_{i}-\overline{y})^{2}}} \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Model Name** & **Model Type** & **MVF(\(m\)t) or \(m_{c}\)(\(t\))** \\ \hline
1 & G-O model(also called \(m_{1}(t)\)[14]) & Concave & \(m(t)=a(1-e^{-bt})\) \\
2 & \(m_{2}(t)\)[14] & Concave & \(m_{c}(t)=a(1+\frac{e}{b-c}e^{-bt}-\frac{b}{b-c}e^{-ct})\) \\
3 & \(m_{3}(t)\)[14] & S-shaped & \(m_{c}(t)=a(1-(1+bt)e^{-bt})\) \\
4 & Inflection S-shaped [29] & Concave & \(m(t)=\frac{a(1-e^{-bt})}{1+b\epsilon^{-bt}}\) \\
5 & Yamada exponential [30] & Concave & \(m(t)=a(1-e^{-\gamma a(1-e^{-bt})})\) \\
6 & Yamada Rayleigh [30] & S-shaped & \(m(t)=a(1-e^{-\gamma a(1-e^{-bt}/2)})\) \\
7 & Yamada Weibull [30] & Concave and S-shaped & \(m(t)=a(1-e^{-\gamma a(1-e^{-bt})})\) \\
8 & Delayed S-shaped [30] & S-shaped & \(m(t)=a(1-(1+bt)e^{-bt})\) \\
9 & HD/G-O [31] & Concave & \(m(t)=\log[\left({(e^{t}-c)}\right/\left({e^{\infty-bt}-c}\right)]\) \\
10 & [PERSON] imperfect (1) [32] & Concave & \(m(t)=\frac{ab}{a+b}\left({e^{at}-e^{-bt}}\right)\) \\
11 & [PERSON] imperfect (2) [32] & Concave & \(m(t)=a(1-e^{-bt})(1-\frac{e}{b})+aat\) \\
12 & P-Z(1997) model [33] & S-shaped and Concave & \(m(t)=\frac{1}{(1+\frac{1}{b\epsilon^{-bt}})}\left((c+a)(1-e^{-bt})-\frac{ab}{b -a}(e^{-at}-e^{-bt})\right)\) \\
13 & Fault removal model (2003) [28] & S-shaped & \(m(t)=\frac{a}{p-b}\left\{1-\left(\frac{(1+\epsilon)e^{-bt}}{1+\epsilon^{-bt}} \right)^{\frac{1}{b}(p-\beta)}\right\}\) \\
14 & P-Z coverage model (2003) [24] & S-shaped and Concave & \(m(t)=a(1+at-\frac{bt+1}{\epsilon^{bt}})-\frac{an(1+bt)}{b\epsilon^{bt+1}} \left[\ln(bt+1)+\sum\limits_{i=0}^{\infty}{\frac{(1+bt)^{i+1}-1}{(t+1)(t+1)}}\right]\) \\
15 & SRGM-3 model (2011) [34] & S-shaped & \(m(t)=\frac{A}{1-a}\left[1-\left(\left(1+bt+\frac{b^{2}\epsilon^{2}}{2}\right) ^{\frac{b}{b-bt}}\right)^{P(1-a)}\right]\) \\
16 & proposed model & S-shaped and Concave & \(m_{c}(t)=\frac{a}{1+a}\left(1-\left[1-\frac{A(1-e^{-bt})}{1+\epsilon^{-bt}} \right]^{1-a}\right)\frac{1}{1+\epsilon^{-bt}}\) \\ \hline \end{tabular}
\end{table}
Table 2: Software reliability models and their MVFs.  
where \(\overline{y}=\frac{1}{n}\sum\limits_{i=1}^{n}y_{i}\). Therefore, the larger \(R^{2}\), the better the model explains the variation in the data.  
The last criterion is adjusted \(R^{2}\) (Adjusted \(R^{2}\)), which can be expressed as follows:  
\[\text{Adjusted }R^{2}=1-\frac{(1-R)(n-1)}{n-M-1} \tag{13}\]  
where \(R\) denotes the value of \(R^{2}\) and \(M\) represents the model's predictor number. Therefore, the larger value of Adjusted \(R^{2}\), the better is the model's performance.