## 3 Model Comparisons
### Data Analysis and Model Comparison with Real Application  
#### 3.2.1 A Middle-Size Software System Data  
Here we examine the performance of the proposed model and compare it with several traditional models using data collected from testing a middle-size software system (Data Set 1, DS-1) [11]. The failure data are recorded by week and are shown in Table 3. In contrast to a traditional software reliability data set, this data set includes not only fault-detection data but also fault-correction data. We use all data points to fit the models and get the parameters estimation of all models. The model parameters, MSE values, \(R^{2}\) values and Adjusted \(R^{2}\) values are listed in Table 4.  
All models' fitting results for DS-1 are graphically illustrated in Figure 3. Figure 4 is three dimensional and the coordinates X, Y and Z illustrate the values of MSE, \(R^{2}\) and Adjusted \(R^{2}\) respectively. We can see that the proposed model has the best criteria values of MSE, \(R^{2}\) and Adjusted \(R^{2}\), i.e., the new model shows the best fitting power to the real data set than all other models.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model No.** & **Model Name** & **Model Parameter Estimation Results** & **MSE** & **R\({}^{2}\)** & **Adjusted R\({}^{2}\)** \\ \hline \(1\) & \(m_{1}(t)\) & \(\hat{a}=231.6\), \(\hat{b}=0.05838\) & 148.3333 & 0.9396 & 0.9356 \\ \(2\) & \(m_{2}(t)\) & \(\hat{a}=147.5\), \(\hat{b}=0.2627\), \(\hat{c}=0.263\) & 56.2357 & 0.9786 & 0.9756 \\ \(3\) & \(m_{3}(t)\) & \(\hat{a}=147.5\), \(\hat{b}=0.2627\) & 52.4867 & 0.9786 & 0.9772 \\ \(4\) & Inflection S-shaped & \(\hat{a}=129.9\), \(\hat{b}=0.5074\), \(\hat{c}=17.3\) & 48.1286 & 0.9817 & 0.9791 \\ \(5\) & Yamada exponential & \(\hat{a}=235.8\), \(\hat{b}=0.001389\), \(\hat{\gamma}=6.448\), \(\hat{a}=6.403\) & 171.4615 & 0.9395 & 0.9256 \\ \(6\) & Yamada Rayleigh & \(\hat{a}=142.9\), \(\hat{b}=0.01177\), \(\hat{\gamma}=2.109\), \(\hat{a}=1.638\) & 40.3308 & 0.9858 & 0.9825 \\ \(7\) & Yamada Weibull & \(\hat{a}=141.9\), \(\hat{b}=2.315\), \(\hat{a}=1.197\), \(\hat{b}=0.004362\) & 34.6385 & 0.9878 & 0.9850 \\ \(8\) & Delayed S-shaped & \(\hat{a}=147.5\), \(\hat{b}=0.2627\) & 52.4867 & 0.9786 & 0.9772 \\ \(9\) & HD/G-O & \(\hat{a}=231.6\), \(\hat{b}=0.05837\), \(\hat{c}=4.891\) & 158.9286 & 0.9396 & 0.9310 \\ \(10\) & Yamada imperfect(1) & \(\hat{a}=231.6\), \(\hat{b}=0.05837\), \(\hat{a}=1.436\times 10^{-9}\) & 158.9286 & 0.9396 & 0.9356 \\ \(11\) & Yamada imperfect(2) & \(\hat{a}=231.6\), \(\hat{b}=0.05837\), \(\hat{a}=1.641\times 10^{-9}\) & 158.9286 & 0.9396 & 0.9356 \\ \(12\) & P-Z (1997) model & \(\hat{a}=0.9179\), \(\hat{b}=0.5102\), \(\hat{c}=129.3\), & 55.7917 & 0.9818 & 0.9758 \\ \(13\) & Fault removal model (2003) & \(\hat{a}=99.8\), \(\hat{a}=932.9\), \(\hat{b}=2.483\), & 19.0364 & 0.9943 & 0.9917 \\ \(14\) & P-Z coverage model (2003) & \(\hat{a}=147.5\), \(\hat{b}=0.2627\), \(\hat{a}=4.776\times 10^{-10}\) & 56.2357 & 0.9786 & 0.9772 \\ \(15\) & SRGM-3 model (2011) & \(\hat{A}=126.7\), \(\hat{a}=0.08195\), \(\hat{b}=2.569\), \(\hat{p}=0.1126\) & 34.0154 & 0.9880 & 0.9863 \\ \(16\) & proposed model & \(\hat{a}=230.5\), \(\hat{A}=0.8538\), \(\hat{a}=0.7528\), \(\hat{c}=73.78\) & 5.**9370** & **0.9984** & **0.9974** \\ \(\hat{b}=3.794\), \(\hat{b}=0.05419\), \(\hat{r}=1.093\) & & & & & \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of goodness-of-fit power of SRGMs for \(m_{c}(t)\) of DS-1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Weeks** & \begin{tabular}{c} **Cumulative** \\ **Detected Faults** \\ \end{tabular} & \
\begin{tabular}{c} **Cumulative** \\ **Corrected Faults** \\ \end{tabular} & **Weeks** &
\begin{tabular}{c} **Cumulative** \\ **Detected Faults** \\ \end{tabular} & **Cumovetative** & **Cumovetative** \\ \hline \(1\) & \(12\) & \(3\) & \(10\) & \(114\) & \(109\) \\ \(2\) & \(23\) & \(3\) & \(11\) & \(116\) & \(113\) \\ \(3\) & \(43\) & \(12\) & \(12\) & \(123\) & \(120\) \\ \(4\) & \(64\) & \(32\) & \(13\)