# On the Estimation of Logistic Models with Banking Data Using Particle Swarm Optimization
## 9 Discussion and Conclusions  
The use of various logistic growth models to estimate the assets of commercial and rural banks in Indonesia provides important insights into the dynamics of banking growth. Each model brings its own assumptions about growth rates, carrying capacities, and the non-linear behavior of financial assets over time. In this study, nine logistic models, including the Verhulst, Gompertz, and Tsoularis models, were applied to model the asset data of commercial and rural banks. The findings revealed that the Tsoularis model, which is known for capturing complex growth dynamics, provided the best fit for commercial banks, whereas the Verhulst model was more appropriate for rural banks. The model fitting is illustrated in Figure 7. This variation underscores the need to tailor growth models to specific segments within the banking industry, reflecting their unique growth patterns and constraints. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case Data** & **New Model** & **Tsoularis Model** & **Verhulst Model** \\ \hline \multirow{8}{*}{Commercial banks} & \(a_{1}=0.3622\) & \(r=0.5401\) & \\  & \(a_{2}=0.0932\) & \(a=2.9998\) & \\  & \(a_{3}=-0.0084\) & \(\beta=0.9158\) & \\  & \(a=2.9998\) & \(\chi=3.0812\) & \\  & \(\gamma=3.0812\) & \(\chi=13.585,176\) & \\  & \(\chi=13.585,176\) & MAPE t. = 27\% & \\  & MAPE t. = 1.68\% & MAPE v. = 0.52\% & \\  & MAPE v. = 5.29\% & \\ \hline \multirow{8}{*}{Rural banks} & \(a_{1}=0.2114\) & \multirow{8}{*}{None} \\  & \(a_{2}=0.0203\) & & \\ \cline{1-1}  & \(a_{3}=-0.0019\) & & \(K=259,811\) \\ \cline{1-1}  & \(\alpha=4.3027\) & & MAPE t. = 2.16\% \\ \cline{1-1}  & \(\beta=0.9657\) & & MAPE v. = 2.82\% \\ \cline{1-1}  & \(\chi=195,562\) & & \\ \cline{1-1}  & MAPE t. = 1.58\% & & \\ \cline{1-1}  & MAPE v. = 6.37\% & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Result of PSO in estimating parameters of the new proposed logistic model (23). The results suggest that commercial banks, with larger asset bases and broader market exposure, experience growth that is more influenced by intricate factors, as captured by the Tsoularis model. This model's ability to account for variations in growth rates and environmental limits might explain its superior fit. Conversely, the Verhulst model, which assumes a simpler logistic growth dynamic, proved effective for rural banks, where growth might be more linear and constrained by local economic conditions. These outcomes indicate that asset growth in rural banks is less volatile and tends to stabilize at lower carrying capacities compared to commercial banks, which experience more complex and potentially exponential growth before stabilizing. The PSO demonstrated higher performance with lower MAPE in both training and validation compared to the SpO algorithm used as a benchmark. The robustness of the PSO algorithm was verified in this study through parameter tuning using the Taguchi method. This study also attempted to introduce a new logistic model that incorporated a more complex structure with an increased number of parameters. This new model was found to be superior (for data fitting) but inferior (for data prediction) to the established Tsoularis and Verhulst models. The complexity introduced by the nonautonomous nonlinear differential equations in the new model posed significant challenges for numerical precision when all the parameters are estimated simultaneously. This underscores the potential trade-off between model complexity and practical applicability, suggesting that simpler models may sometimes yield more reliable outcomes in real-world scenarios where numerical accuracy and convergence are critical. Despite these findings, several research gaps remain. Notably, while this study focused on modeling growth dynamics using traditional logistic models, future work could incorporate external macroeconomic factors such as inflation rates, policy shifts, and technological advancements that influence banking growth.