## 2 Methods
### Bb-Ann
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model No.**} & \multirow{2}{*}{**C1**} & \multirow{2}{*}{**C2**} & \multicolumn{2}{c}{**Network Result**} \\ \cline{3-4}  & & & \multicolumn{2}{c}{\(R^{2}\)} \\ \cline{3-4}  & & & **Train** & **Test** \\ \hline
1 & 1.333 & 2.667 & 0.922 & 0.917 \\
2 & 2.667 & 1.333 & 0.929 & 0.924 \\
3 & 1.5 & 1.5 & 0.932 & 0.918 \\
4 & 2 & 2 & 0.935 & 0.931 \\
5 & 1.75 & 1.75 & 0.938 & 0.935 \\
6 & 1.5 & 1.75 & 0.930 & 0.922 \\
7 & 1.75 & 1.5 & 0.925 & 0.921 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Testing the different values of C1 and C2 with their \(R^{2}\). C1 = 1.75, C2 = 1.75, inertia weight = 0.75 and an iteration number of 1000 were used. According to the obtained results, after an iteration number of 400, there were no significant changes in the network results. Therefore, the value of 400 was selected as the maximum number of iterations of this study. Regarding the modeling process of BBO-ANN, the BBO has a number of parameters to initially set for a better ANN model performance. After an initial trial and error process for BBO parameters, the final parameter values were as follows: (i) Elitism rate: 0.3, (ii) Mutation probability: 0.015, (iii) Maximum value of immigration and emigration rate: 1. Figure 5 shows the process flow chart of the proposed BBO-ANN. Note that the different values for the number of habitats were tested in this study, as shown in Table 5. Based on this Table, model number 8 had the best performance with the highest R\({}^{2}\) value. Hence, 350 was selected as the number of habitats in the BBO-ANN modeling for this study. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model No.**} & \multirow{2}{*}{**Number of Particles**} & \multicolumn{2}{c}{**Network Result**} \\ \cline{3-4}  & & \multicolumn{2}{c}{\(R^{2}\)} \\ \cline{3-4}  & & **Train** & **Test** \\ \hline
1 & 50 & 0.909 & 0.905 \\
2 & 100 & 0.915 & 0.904 \\
3 & 150 & 0.919 & 0.917 \\
4 & 200 & 0.926 & 0.920 \\
5 & 250 & 0.929 & 0.922 \\
6 & 300 & 0.935 & 0.930 \\
7 & 350 & 0.943 & 0.935 \\
8 & 400 & 0.939 & 0.934 \\
9 & 450 & 0.934 & 0.933 \\
10 & 500 & 0.930 & 0.924 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Testing the different values of number of particles with their R\({}^{2}\). Figure 5: Process flow chart of the proposed BBO-ANN. After the trial process, the final architecture of ELM consisted of 15 hidden neurons with the sigmoid activation function for training and validation. Meanwhile, the MPMR model had an error tube width of \(\varepsilon=0.002\) and C = 0.9. To investigate the performance of the models, the root mean square error (RMSE), \(R^{2}\), Ratio of RMSE to the standard deviation of the observations (RSR), mean absolute error (MAE), and degree of agreement (\(d\)) were taken into account, which are shown in Equations (9)-(12) [42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56]:

\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\bigl{(}PPV_{a}-PPV_{p}\bigr{)}^{2}} \tag{9}\]

\[MAE=\frac{1}{n}\sum_{i=1}^{n}\bigl{|}PPV_{a}-PPV_{p}\bigr{|} \tag{10}\]

\[R^{2}=\frac{\left[\sum_{i=1}^{n}\bigl{(}PPV_{a}-PPV_{man}\bigr{)}^{2}\right] -\left[\sum_{i=1}^{n}\bigl{(}PPV_{a}-PPV_{p}\bigr{)}^{2}\right]}{\left[\sum_{ i=1}^{n}\bigl{(}PPV_{a}-PPV_{mean}\bigr{)}^{2}\right]} \tag{11}\]

\[d=1-\frac{\sum_{i=1}^{n}\bigl{(}PPV_{a}-PPV_{p}\bigr{)}^{2}}{\sum_{i=1}^{n} \bigl{(}\bigl{|}PPV_{p}-PPV_{man}\bigr{|}+\bigl{|}\bigl{(}PPV_{a}-PPV_{ mean}\bigr{)}\bigr{|}\bigr{)}^{2}} \tag{12}\]

where \(PPV_{p}\) is the predicted PPV obtained using the proposed models; \(PPV_{a}\) signifies the actual PPV; \(PPV_{mean}\) represents the average of the PPV data; and \(n\) stands for the number of data values. ## 6 Results and Discussion

The predicted PPV values obtained from all predictive models for only the testing phase is given in Table 6. In this Table, k is ratio of the actual PPV to the predicted PPV values for each dataset (k = Actual PPV/Predicted PPV).