# On the Estimation of Logistic Models with Banking Data Using Particle Swarm Optimization
## 5 Numerical Results  
The results of the parameter estimation are presented in Table 3 for commercial banks and in Table 4 for rural banks. The letters P-R, V, vB, R, G, hG, B, Tu, Ts refer to the Pearl and Reed generalization model (8), [PERSON] model (2), [PERSON] model (10), [PERSON] model (12), [PERSON] model (14), hyper-Gompertz model (16), [PERSON] model (17), [PERSON] et al. model (19), and [PERSON] model (20), respectively. In general, for the selection of the best model, the first thing to review is the smallest error between the data and the model. Usually, the model with the smallest error will be chosen, and the model with the biggest error is not selected. To provide a better visualization of the errors in Tables 3 and 4, we present the chart of MAPE and RMSE in Figure 3. First, we analyze the MAPE of the training data, because it is used as the objective \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **The Search Space** \\ \hline \(a_{1},a_{2},a_{3}\) for [PERSON] generalization model & \([0,1]\) \\ \(r\) for [PERSON], [PERSON], [PERSON], hyper-Gompertz, Blumberg, & \([0,1]\) \\ [PERSON] et al., and [PERSON] models & \([0,50]\) \\ \(r\) for von Bertalanffy model & \([0,50]\) \\ \(\alpha\) for [PERSON], [PERSON] et al., and [PERSON] models & \([0,5]\) \\ \(\beta\) for [PERSON] and [PERSON] models & \([0,3]\) \\ \(\gamma\) for hyper-Gompertz model & \([0.75,3]\) \\ \(\gamma\) for [PERSON], [PERSON] et al., and Tsoularis models & \([1,5]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some intervals as the search space for estimating the parameter of logistic models. function of the estimation parameters. From the figure, the Pearl-Reed generalization model has the smallest MAPE, and it is followed by the [PERSON] model. Meanwhile, the von [PERSON] model has the biggest MAPE, followed by the [PERSON] model and the hyper-Gompertz model. Based on the information on the MAPE values of the training and validation data, the first models to be out from consideration are the von [PERSON] model, the Gompertz model, and the hyper-Gompertz model. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline  & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) & \(K\) & **Stop** & \(\mathbf{t_{\text{inf}}}\) & \(\mathbf{y_{\text{inf}}}\) & **MAPE t.** & **MAPE v.** & **RMSE t.** & **RMSE v.** \\ \hline P-R & 0.1838 & 0.0080 & 4.00 \(\times 10^{-9}\) & & 162,036 & 57 & 8.42 & 94,498 & 1.34\% & 2.57\% & 961 & 4655 \\ \hline  & \(r\) & \(\alpha\) & \(\beta\) & \(\gamma\) & & & & & & & \\ \hline V & 0.2052 & & & 259,816 & 24 & 11.28 & 129,908 & 2.16\% & 2.82\% & 1587 & 4180 \\ vB & 8.3632 & & & 2,157,160 & 26 & 38.58 & 612,388 & 4.16\% & 0.72\% & 2597 & 1274 \\ R & 0.5296 & 3.0127 & & & 153,790 & 74 & 8.63 & 96,968 & 1.57\% & 2.97\% & 1086 & 5291 \\ G & 0.0443 & & & 1,919,658 & 24 & 33.49 & 706,203 & 2.73\% & 7.20\% & 2339 & 10,268 \\ hG & 0.0447 & & & 1 & 1,872,938 & 146 & 33.09 & 689,016 & 2.38\% & 6.51\% & 2131 & 9299 \\ B & 0.0878 & & 1.0875 & 4.2360 & 712,615 & 64 & 12.33 & 145,850 & 2.12\% & 3.69\% & 1656 & 5337 \\ Tu & 0.2165 & 2.4256 & & 1.0072 & 166,654 & 150 & 8.86 & 99,583 & 1.66\% & 1.74\% & 1111 & 3475 \\ Ts & 0.2477 & 4.3027 & 0.9657 & 4.6135 & 195,562 & 150 & 13.75 & 95,839 & 1.58\% & 2.92\% & 1117 & 5159 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Result of PSO in estimating parameters of logistic models using rural banking assets data in Indonesia. Figure 3: MAPE and RMSE of the obtained models for (**a**) commercial banks and (**b**) rural banks.