# Deep-Learning Software Reliability Model Using SRGM as Activation Function
## 4 Numerical Examples
### Results of Dataset 1  
Table 4 lists the estimated parameter values of the SRGM assuming an NHPP using Dataset 1 and the structure of the software reliability model using deep learning. It also shows the estimated parameter values and structure of the proposed deep-learning software reliability model. The parameter \(b\) of the proposed deep-learning software reliability model is estimated to be 0.52. Based on the estimated parameter \(b\), the structure of the deep-learning software reliability model has four hidden layers. The Adam optimization method is used, and the learning rate is set to 0.000001. Figure 3 shows the estimated value of \(m(t)\) for each model and deep-learning structure. \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Criteria** & **Formula** \\ \hline
1 & MSE & \(\frac{\sum_{i=1}^{n}(\hat{m}(t_{i})-y_{i})^{2}}{n}\) \\
2 & MAE & \(\frac{\sum_{i=1}^{n}(\hat{m}(t_{i})-y_{i})}{n}\) \\
3 & PRR & \(\sum\limits_{i=1}^{n}\left(\frac{\hat{m}(t_{i})-y_{i}}{\hat{m}(t_{i})}\right)^{2}\) \\
4 & PP & \(\sum\limits_{i=1}^{n}\left(\frac{\hat{m}(t_{i})-y_{i}}{y_{i}}\right)^{2}\) \\
5 & \(R^{2}\) & \(1-\frac{\sum_{i=1}^{n}(\hat{m}(t_{i})-y_{i})^{2}}{\sum_{i=1}^{n}(y_{i}-\overline {y_{i}})^{2}}\) \\
6 & RMSPE & \(\sqrt{\text{Variance}^{2}+\text{Bias}^{2}}\). \\
7 & MEOP & \(\frac{\sum_{i=1}^{n}|\hat{m}(t_{i})-y_{i}|}{n+1}\) \\
8 & TS & \(100*\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-\hat{m}(t_{i}))^{2}}{\sum_{i=1}^{n}y^{2}} }\gamma_{o}\) \\
9 & PC & \(\frac{n}{2}*log\left(\frac{\sum_{i=1}^{n}(\hat{m}(t_{i})-y_{i})^{2}}{n}\right) +\left(\frac{n-1}{n}\right)\) \\
10 & preSSE & \(\sum\limits_{i=k+1}^{n}(\hat{m}(t_{i})-y_{i})^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Criteria for model comparisons. \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Model** & **Estimation and Structure** \\ \hline
1 & GO & \(\hat{a}=405.3473\), \(\hat{b}=0.01966\) \\
2 & IS & \(\hat{a}=398.5632\), \(\hat{b}=0.02049\), \(\hat{\beta}=0.02548\) \\
3 & ZFR & \(\hat{a}=48.7172\), \(\hat{b}=0.00231\), \(\hat{a}=0.04331\), \\  & \(\hat{\beta}=0.23542\), \(\hat{c}=0.17070\), \(\hat{p}=0.35612\) \\
4 & YID 1 & \(\hat{a}=205.0374\), \(\hat{b}=0.04056\), \(\hat{a}=0.01222\) \\
5 & YID 2 & \(\hat{a}=98.4888\), \(\hat{b}=0.08904\), \(\hat{a}=0.04632\) \\
6 & PZ & \(\hat{a}=27.1595\), \(\hat{b}=0.06871\), \(\hat{a}=0.02132\), \\  & \(\hat{b}=216867\), \(\hat{c}=232.0570\) \\
7 & PNZ & \(\hat{a}=40.2796\), \(\hat{b}=0.29103\), \\  & & \(\hat{a}=0.13231\), \(\hat{b}=0.05003\) \\
8 & TP & \(\hat{a}=17840.283\), \(\hat{b}=0.02703\), \(\hat{a}=0.00375\), \(\hat{b}=0.01292\), \\  & & \(\hat{c}=15.38103\), \(\hat{p}=0.60263\), \(\hat{a}=0.10827\) \\
9 & RMD & \(\hat{a}=42.38821\), \(\hat{b}=0.02091\), \\  & & \(\hat{a}=10.00201\), \(\hat{b}=0.00959\) \\
10 & Vtub & \(\hat{a}=1.82256\), \(\hat{b}=0.50401\), \(\hat{a}=0.52648\), \\  & & \(\hat{b}=12.54998\), \(\hat{N}=399.1401\) \\
11 & TC & \(\hat{a}=0.04460\), \(\hat{b}=0.84140\), \(\hat{a}=12.34453\), \\  & & \(\hat{b}=220.2973\), \(\hat{N}=2627.4388\) \\
12 & 3P & \(\hat{a}=0.09480\), \(\hat{b}=0.00000315\), \(\hat{b}=8.18127\), \\  & & \(\hat{c}=0.01147\), \(\hat{N}=706.1525\) \\
13 & DPF 1 & \(\hat{a}=209.7896\), \(\hat{b}=0.00754\), \\  & & \(\hat{c}=12.9492\), \(\hat{h}=31.45712\) \\
14 & DPF 2 & \(\hat{a}=206.8508\), \(\hat{b}=0.00316\), \\  & & \(\hat{c}=5.22432\), \(\hat{h}=28.1848\) \\
15 & UDPF & \(\hat{b}=0.01562\), \(\hat{a}=0.49790\), \\  & & \(\hat{b}=0.49731\), \(\hat{N}=472.0649\) \\
16 & DNN & Hidden layer = 4, Optimizer = [PERSON], Epoch = 100 \\
17 & RNN & Difference = 1, Optimizer = [PERSON], Epoch = 100 \\
18 & LSTM & Difference = 1, Optimizer = [PERSON], Epoch = 100 \\
19 & GRU & Difference = 1, Optimizer = [PERSON], Epoch = 100 \\
20 & Proposed model & \(\hat{b}=0.52\), Hidden layer = 4, Optimizer = [PERSON], Epoch = 100 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameter estimation and structure of model from Dataset 1. Figure 3: Prediction of all models for Dataset 1. Table 5 presents the results for the 10 criteria based on the estimated values.