# Deep-Learning Software Reliability Model Using SRGM as Activation Function
## 4 Numerical Examples
### Criteria  
Twenty SRCMs, software reliability models using deep learning and the proposed deep-learning software reliability model, were compared using 10 criteria to determine the superior model. These 10 criteria were based on the difference between the actual (\(y_{i}\)) and predicted values (\(\hat{m}(t_{i})\)). The mean squared error (MSE) is defined as the sum of the squares of the difference between the predicted number and the actual number of failures [55]. The mean absolute error (MAE) measures the difference between the predicted number of failures and the observed value, considering the number of observations as the sum of absolute values [56]. Predictive ratio risk (PRR) is measured by dividing the distance from the predicted number of failures to the actual number of failures by the predicted value with respect to model estimation [57]. Predictive power (PP) is measured as the distance from the actual number of failures to the predicted number of failures divided by the actual value [58]. R\({}^{2}\) measures the success of the fit in explaining the variation in the data [59]. Root mean square prediction error (RMSPE) estimates the closeness of the model to the predicted observations [60]. The mean error of prediction (MEOP) sums the absolute value of the deviation between the actual number of observations and the estimated curve and is defined elsewhere [61]. The Theil statistic (TS) is the ratio of the average deviation of all periods to the actual observations [62]. [PERSON]'s criterion (PC) considers the tradeoff between the uncertainty of the model and the number of parameters in the model [63]. The prediction sum squared error (preSSE) represents the sum of the differences between the predicted and actual values [64]. Table 3 shows a description of the 10 criteria.  
\begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Model** & **Mean Value Function and Structure** \\ \hline
16 & DNN & Hidden Layer = 4, Activation Function = Sigmoid, Optimizer = [PERSON] \\
17 & RNN & Difference = 1, Optimizer = [PERSON] \\
18 & LSTM & Difference = 1, Optimizer = [PERSON] \\
19 & GRU & Difference = 1, Optimizer = [PERSON] \\
20 & Proposed Model & \(m(t)=\frac{1}{1+e^{\frac{t^{2}}{H+1}}}\) \\ \hline \end{tabular}
\end{table}
Table 2: Software reliability models using deep learning and the proposed deep-learning software reliability model.  
Based on the above criteria, the existing SRGMs and software reliability models using deep learning are compared with the proposed deep-learning software reliability model. The R\({}^{2}\) value is closer to 1, and the other nine scales are closer to 0, indicating a better fit. Using R and MATLAB, we estimated the parameters of each model using the least-squares estimation (LSE) method, calculated the goodness of fit, and compared their performance. The LSE method estimates the parameters that minimize errors. We want to estimate the optimal parameters of each model through Equation (11) [65].  
\[LSE=\sum\limits_{i=1}^{n}(y_{i}-\hat{m}(t_{i}))^{2} \tag{11}\]  
The proposed deep learning software reliability model selects the parameter \(b\) through the LSE method, and the parameters of deep learning are trained by the given data. The model is trained 100 times for each selected parameter to present the best results.