# Default Detection Rate-Dependent Software Reliability Model with Imperfect Debugging
## 5 Numerical Example
### Performance Verification
#### 5.3.2 Performance Analysis of the Type II Imperfect Debugging Model  
It is obvious that the type II imperfect debugging model establishes the mathematical relationship between fault detection, repair and introduction and considers more random factors in the test. To this end, the experiment will be conducted on the performance of the type II imperfect debugging model. The comparisons of the curves of all the models in Table 1 on the failure data are shown graphically in Figure 4.  
As seen directly from Figure 4, only several models have serious deviation compared with raw data, which makes it impossible to describe the trend of the fault detection quantity. To distinguish the performance between the models, Table 4 gives the fitting performance indicators for M-1-5 models in Table 1. The fitting performance indicators for the type I imperfect debugging models and the type II imperfect debugging models can be referred to Tables 2 and 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **DS** & _MSE_ & **R-Square** & _Variation_ & **RMS-PE** & _BMMRE_ \\ \hline M-1 & & 12.99494 & 0.838505 & 3.78606 & 3.81486 & 0.431939 \\ \hline M-2 & & 1.642027 & 0.964688 & 1.32519 & 1.329018 & 0.103872 \\ \hline M-3 & & 1.480686 & 1.042056 & 1.313155 & 1.333504 & 0.365081 \\ \hline M-4 & & 1.417172 & 1.022353 & 1.245078 & 1.252981 & 0.276839 \\ \hline M-5 & & 2.516028 & 0.938045 & 1.679348 & 1.696125 & 0.116178 \\ \hline M-1 & & 124.2525 & 0.743409 & 11.48903 & 11.50564 & 0.22644 \\ \hline M-2 & & 11.61711 & 1.001203 & 3.50065 & 3.501827 & 0.077979 \\ \hline M-3 & DS2 & 25.25638 & 1.211331 & 5.739474 & 5.912213 & 0.319313 \\ \hline M-4 & & 18.92156 & 1.120702 & 4.707464 & 4.782304 & 0.195724 \\ \hline M-5 & & 11.61711 & 1.001135 & 3.500646 & 3.501821 & 0.077979 \\ \hline M-1 & & 29.02455 & 0.720307 & 5.592244 & 5.610177 & 0.894779 \\ \hline M-2 & & 4.474233 & 0.858111 & 2.302146 & 2.341391 & 0.260841 \\ \hline M-3 & DS3 & 0.975441 & 1.001077 & 1.015523 & 1.01578 & 0.065226 \\ \hline M-4 & & 1.305707 & 0.969718 & 1.183524 & 1.18652 & 0.068236 \\ \hline M-5 & & 4.474233 & 0.858111 & 2.302155 & 2.341403 & 0.260842 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of Fitting Performance of M-1–5 Models.  
Figure 4: Fitting Curves of Models. (**a**) Fitting Curves of Models on DS1; (**b**) Fitting Curves of Models on DS2; (**c**) Fitting Curves of Models on DS3.  
Performance Comparison between the Type II Imperfect Debugging Model and the Type I Imperfect Debugging Model  
Here, with the comparison of representatives of the two types, IDII_5 and IDI_5, it can be found that:  
* On DS1, IDII_5 has three indicators (_Variation_, _RMS-PE_, _BMMRE_) that are optimal, one indicator (_MSE_) is suboptimal, and one indicator (_R-square_) is very close to its optimal indicator;
* On DS2, IDII_5 has one indicator (_MSE_) that is optimal, and the other four indicators (_R-square_, _Variation_, _RMS-PE_, _BMMRE_) are suboptimal, and four indicators differ from the optimal value only in the fifth or sixth decimal places.
* On DS3, IDII_5 has three indicators (_MSE_, _variation_, _RMS-PE_) that are optimal. The _R-square_ differs from the optimal value only between 0.01 and 0.02, and the _BMMRE_ differs from the optimal value between 0.03 and 0.04.  
Therefore, it can be seen that IDII_5 is slightly better than IDI_5, which means that type II considering the ID and the introduction of a new fault is superior to the type I model, which only considers ID. This is because the imperfect model that only considers imperfect debugging ignores the phenomena of the introduction of a new fault in debugging and loses important random factors in the real software debugging process, which causes distortion of the function.