# On the Estimation of Logistic Models with Banking Data Using Particle Swarm Optimization
## 5 Numerical Results
\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 & 0.1375 & 0.0156 & \(4.28\times 10^{-10}\) & & 8,228,341 & 62 & 7.33 & 5,114,283 & 1.68\% & 5.78\% & 101,902 & 489,191 \\ \hline  & \(r\) & \(\alpha\) & \(\beta\) & \(\gamma\) & & & & & & & \\ \hline V & 0.1844 & & & & 15,071,474 & 25 & 11.22 & 7,535,737 & 3.06\% & 2.29\% & 139,536 & 203,508 \\ vB & 29.3160 & & & & 142,549,510 & 25 & 44.89 & 42,236,895 & 4.21\% & 1.93\% & 174,153 & 181,636 \\ R & 0.6557 & 4.2911 & & & 8,255,810 & 42 & 8.16 & 5,599,424 & 2.45\% & 4.82\% & 113,273 & 412,614 \\ G & 0.0431 & & & & 86,377,000 & 27 & 31.76 & 31,776,323 & 3.50\% & 5.77\% & 163,688 & 491,075 \\ hG & 0.0432 & & & 1 & 85,794,137 & 115 & 31.70 & 31,561,899 & 3.11\% & 4.67\% & 160,263 & 402,819 \\ B & 0.0718 & & 1.0683 & 4.7342 & 48,868,588 & 66 & 13.17 & 8,997,285 & 3.07\% & 3.33\% & 142,053 & 287,811 \\ Tu & 0.2014 & 2.6711 & & 1.0062 & 946,0970 & 150 & 8.54 & 5,778,078 & 2.62\% & 2.07\% & 118,677 & 182,535 \\ Ts & 0.5401 & 2.9998 & 0.9158 & 3.0812 & 13,585,176 & 150 & 14.08 & 6,091,104 & 2.77\% & 0.52\% & 122,988 & 55,416 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Result of PSO in estimating parameters of logistic models using commercial banking assets data in Indonesia. From the perspective of the MAPE of the validation data, for the commercial banking case, the Pearl-Reed generalization model has the biggest MAPE followed by the Gompertz model, the Richards model, and the hyper-Gompertz model. For the rural banking case, the Gompertz model has the biggest validation MAPE, followed by the hyper-Gompertz model. The Pearl-Reed generalization model and the Richards model have small MAPE of the training data but their MAPE of the validation data is big. Let us look at their plots versus the data in Figure 4. From the figure, those models are very good at approaching the training data, but they fail to predict the validation data. The prediction strays too far from the data; see the parts marked by a red circle. Also, as we can see from the figure, the prediction failure is caused by the values of their carrying capacity being less than the highest of the validation data (\(y_{d}(t_{155})=8\),562,974) for the commercial banking case, and are just a little bit higher than the highest of the validation data (\(y_{d}(t_{156})=149\),872) for the rural banking case. Figure 4: Plot of the Pearl–Reed generalization model versus the data of (**a**) commercial banks and (**c**) rural banks and the [PERSON] model versus the data of (**b**) commercial banks and (**d**) rural banks. The failure of the [PERSON] generalization model and the Richard model in producing an estimated carrying capacity which is far enough above the last validation data gives us insight into the carrying capacity value of the other models. Figure 5 presents the chart of the carrying capacity values of all models. From the figure, in addition to the two models mentioned, [PERSON] et al. also produce the carrying capacity value which is close to the highest of the validation data. For the reasons of the digression of the models from the data and the closeness between the carrying capacity values with the highest validation data, we take out the Pearl and Reed model, the [PERSON] model, and the [PERSON] et al. model from the consideration as the best model. Now, there are three models that need to be discussed: the Verhulst logistic model, the Blumberg model, and the Tsoularis model. From Figure 1, for the commercial banking case, the Tsoularis model has the smallest MAPE and RMSE of the training and validation data than the other two models. Therefore, we conclude that the Tsoularis model is the best logistic model to fit the Indonesia commercial banking data.