# Software Reliability Growth Model with Dependent Failures and Uncertain Operating Environments
## 3 NHPP Software Reliability Growth Model (SRGM)
### Criteria  
Various criteria have been proposed to discuss the goodness of fit of a model. Table 4 shows the criteria used to compare the goodness of fit for the SRGMs, which are listed in Table 1.  
In Table 4, n is defined as the number of data, and m is defined as the number of parameters in the model (mean value function). The predicted value which means number of failures based on m(t) is presented as \(\hat{\text{m}}\)(t), and the actual data are presented as y\({}_{\text{i}}\). Most of the criteria mentioned measure the distance (or error) between predicted value and actual values. Therefore, it can be said that the shorter the distance, the better the mean value function of the model predicts the number of failures in the dataset. The MSE measures the distance of predicted value by a model from the actual data with the consideration of the number parameters in the model and the number of data. The PRR measures the distance of predicted value from the actual data with the consideration the predicted value by a model. The PP measures the distance of predicted value from the actual data with the consideration the actual data. The SAE measures the sum of absolute error, which means the distance between the predicted value and the actual data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No** & & **Criteria** \\ \hline \multirow{2}{*}{1} & Mean square error (MSE) [30] & \\  & & \(\frac{\sum\limits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t})-\text{y}_{\text{i}} )^{2}}{\hat{\text{m}}-\text{m}}\) \\ \hline \multirow{2}{*}{2} & Predictive ratio risk (PRR) [30] & \\  & & \(\frac{\sum\limits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t})-\text{y}_{\text{i}} )^{2}}{\hat{\text{m}}(\text{t})}\) \\ \hline \multirow{2}{*}{3} & Predictive power (PP) [30] & \\  & & \(\frac{\sum\limits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t})-\text{y}_{\text{i}} )^{2}}{\hat{\text{y}}_{\text{i}}}\) \\ \hline \multirow{2}{*}{4} & Sum of absolute error (SAE) [36] & \\  & & \(\sum\limits_{\text{i}=1}^{n}|\hat{\text{m}}(\text{t}_{\text{i}})-\text{y}_{\text {i}}|\) \\ \hline \multirow{2}{*}{5} & R-square (R\({}^{2}\)) [37] & \\  & & \(1-\frac{\sum\limits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t})-\text{y}_{\text{i }})^{2}}{\sum\limits_{\text{i}=1}^{n}(\text{y}_{\text{i}}-\text{y}_{\text{i}} )^{2}}\) \\ \hline \multirow{2}{*}{6} & [PERSON]’s information criterion (AIC) [38] & \\  & & \(-2\log\text{L}+2\text{m}\) \\ \hline \multirow{2}{*}{7} & Bayesian information criterion (BIC) [39; 40] & \\  & & \(-2\log\text{L}+\text{m}\log\text{n}\) \\ \hline \multirow{2}{*}{8} & Bias [41] & \\  & & \(\frac{\sum\
olimits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t}_{\text{i}})-\text{y}_{ \text{i}})}{\text{n}}\) \\ \hline \multirow{2}{*}{9} & Predicted relative variation (PRV) [42] & \\  & & \(\sqrt{\frac{\sum\
olimits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t}_{\text{i}})- \text{y}_{\text{i}}-\text{Bias})^{2}}{\text{n}-1}}\) \\ \hline \multirow{2}{*}{10} & Root mean square prediction error (RMSPE) [42] & \\  & & \(\sqrt{\text{Bias}^{2}+\text{PRV}^{2}}\) \\ \hline \multirow{2}{*}{11} & Mean absolute error (MAE) [43] & \\  & & \(\frac{\sum\
olimits_{\text{i}=1}^{n}|\hat{\text{m}}(\text{t}_{\text{i}})-\text{y}_{ \text{i}}|}{\text{n}-\text{m}}\) \\ \hline \multirow{2}{*}{12} & Mean error of prediction (MEOP) [43] & \\  & & \(\frac{\sum\
olimits_{\text{i}=1}^{n}|\hat{\text{m}}(\text{t}_{\text{i}})-\text{y}_{ \text{i}}|}{\text{n}-\text{m}+1}\) \\ \hline \multirow{2}{*}{13} & Theil statistic (TS) [43] & \\  & & \(100\sqrt{\frac{\sum\
olimits_{\text{i}=1}^{n}(\hat{\text{m}}(\text{t}_{\text{i}} )-\text{y}_{\text{i}})^{2}}{\sum\
olimits_{\text{i}=1}^{n}Y_{\text{i}}^{2}}}\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Criteria. The R\({}^{2}\) (coefficient of determination) is a measure of the fit of the regression equation. It represents the proportion of the regression sum of squares (SSR) to the total sum of squares (SST) of the model. The closer the value is to 0, the better the fit of the model.