# Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning
## 3 Numerical Example
### Results  
#### 3.3.1 Results of Dataset 1  
Using Dataset 1, Table 2 shows the results of the parameter estimation of the software reliability model assuming NHPP and the structure of the software reliability model using deep learning. In addition, Dataset 1 was divided into 80% and 90%, and parameter estimation and model fitting were performed on the divided dataset. The structure of the software reliability model using deep learning consisted of three hidden layers, and the number of nodes in each layer was set according to the size of the training dataset. For the software reliability model of the recurrent neural network type, two deep neural networks were added, except for the recurrent neural network hidden layer. Here, Adam was used as the optimization method. The learning rate was set to 0.005 for the DNN and 0.0001 for the RNN, LSTM, and GRU. \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Model** & **80\%** & **90\%** \\ \hline
1 & GO & \(\hat{u}=1961.771\), \(\hat{b}=0.00065\) & \(\hat{u}=8717.730\), \(\hat{b}=0.00015\) \\ \hline
2 & DS & \(\hat{u}=193.233\), \(\hat{b}=0.02730\) & \(\hat{u}=160.532\), \(\hat{b}=0.03091\) \\ \hline
3 & YID & \(\hat{u}=97.401\), \(\hat{b}=0.00523\), \(\hat{u}=0.10686\) & \(\hat{u}=6.2301\), \(\hat{b}=0.05319\), \(\hat{u}=0.34770\) \\ \hline
4 & PZ & \(\hat{u}=431.141\), \(\hat{b}=0.04005\), \(\hat{u}=16.359\), & \(\hat{u}=88.620\), \(\hat{b}=0.07843\), \(\hat{u}=9.0099\), \\  & & \(\hat{\beta}=27.421\), \(\hat{c}=57.233\) & \(\hat{\beta}=12.248\), \(\hat{c}=2.0246\) \\ \hline
5 & PNZ & \(\hat{u}=69.646\), \(\hat{b}=0.01607\), & \(\hat{u}=71.272\), \(\hat{b}=0.08070\), \\  & & \(\hat{u}=0.07992\), \(\hat{\beta}=1.0343\) & \(\hat{u}=0.00541\), \(\hat{\beta}=10.183\) \\ \hline
6 & TP & \(\hat{u}=72.610\), \(\hat{b}=0.06011\), & \(\hat{u}=177.307\), \(\hat{b}=0.08976\), \\  & & \(\hat{u}=18.8769\), \(\hat{b}=0.91909\), & \(\hat{u}=0.90527\), \(\hat{b}=3.2480\), \\  & & \(\hat{c}=13.7277\), \(\hat{p}=0.46681\), \(\hat{q}=0.09829\) & \(\hat{c}=7.5811\), \(\hat{p}=1.5258\), \(\hat{q}=0.62709\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter estimation and structure of model using Dataset 1. Table 3 lists the result values of the criteria using 80% of Dataset 1. The values for MSE, PRV, RMSPE, TS, and PC in LSTM were the lowest at 2.0069, 0.1345, 1.4168, 4.9250, and 14.5574, respectively. GRU showed the lowest values of 0.6706 and 1.1032 in PRR and PP, and the lowest values of MAE and MEOP of 1.21164 and 1.186 were observed in UDPF, respectively. The R\({}^{2}\) value was the highest for LSTM and GRU at 0.9929. In LSTM, 6 out of 10 criteria showed good results. For the predictions of 10% and 20%, the DPF model with a preSSE value of 47.645 had the lowest result among the software reliability models assuming NHPP. In contrast, most software reliability models using deep learning showed values in the 20-point range, and the LSTM model showed the smallest value of 21.570. Table 4 lists the result values of the criteria using 90% of Dataset 1. MSE, RMSPE, TS, and PC were the lowest in GRU at 1.9961, 0.9948, 1.413, 4.1991, and 16.1831, respectively, and PRV was the smallest in LSTM at 0.1251. In addition, PRR and PP in UDPF showed the lowest values of 0.6386 and 0.9113, and MAE and MEOP were 1.3714 and 1.3409 in DPF, respectively. The R\({}^{2}\) value was the largest at 0.9948. In GRU, 5 out of 10 criteria showed good results. In preSSE, the TC, DS, and Vtub models showed very small values of 6.367, 6.537, and 7.340, respectively. In the software reliability model using deep learning, RNN showed the smallest value of 11.397, followed by LSTM at 11.755. The resulting values showed good results for the software reliability model, assuming NHPP. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**No. ** & **Model** & **Model** & **80\%** & **90\%** & **90\%** \\ \hline
7 & TC & \(