## 5 Conclusions  
This study replaced PSO with FPSO. We used the control theory viewpoint to deduce the CFPSO algorithm, rewrote the system into an uncertain linear system, and proved its controllability. As the fractional value \(\lambda\) will affect the random number range, this research disassembled \(\lambda\) into 20 values and deduced them one by one at intervals of 0.1. The range obtained when \(\lambda\) is 0.3 was the least conservative. Therefore, the random number selected had a wide range. This study used CFPSO to find the best hyperparameters of the model when predicting the heart disease data set. The experimental results show that the best hyperparameters found through CFPSO were better than the hyperparameters officially recommended by XGBoost. Therefore, CFPSO is more suitable for helping physicians quickly determine whether a patient has heart disease through the learned model. In future work, variable-order fractional operators (VOFO) can be involved in this topic for coping with complicated real-world issues because VOFO can provide the robustness and flexibility characteristics more in control theory [34; 35; 36; 37; 38].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Hyperparameters** & **Best Hyperparameters** \\ \hline Learning Rate & 0.265201 \\ Max\_depth & 7 \\ Gamma & 0.194108 \\ Subsample & 0.95 \\ Colsample\_bytes & 0.63 \\ Reg\_lambda & 4.846424 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The best hyperparameters found using CFPSO.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Accuracy** & \(F_{\beta score}\) & **Custom Fitness** \\ \hline XGBoost officially recommended & & & \\ hyperparameters & 91.4\% & 89.5\% & 0.1056 \\ CFPSO best hyperparameters & 98.7\% & 98.9\% & 0.0120 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Evaluation index results of training and validation sets.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Accuracy** & \(F_{\beta score}\) & **Custom Fitness** \\ \hline XGBoost officially recommended & & & \\ hyperparameters & 91.4\% & 89.5\% & 0.1056 \\ CFPSO best hyperparameters & 98.7\% & 98.9\% & 0.0120 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Evaluation index results of the test set.  
**Author Contributions:** Conceptualization, F.-I.C. and T.-H.H.; methodology, P.-Y.Y.; software, C.-H.L.; validation, P.-Y.Y. and C.-H.L.; formal analysis, W.-H.H.; investigation, J.-H.C.; resources, J.-H.C.; data curation, T.-C.L.; writing--original draft preparation, F.-I.C.; writing--review and editing, T.-H.H.; visualization, W.-H.H.; supervision, J.-H.C.; project administration, J.-H.C.; funding acquisition, J.-H.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** Publication costs are funded by the Ministry of Science and Technology, Taiwan, under grants MOST 110-2221-E-992-091-MY3, MOST 110-2221-E-037-005, and MOST 110-2221-E-153-010. The design and part writing costs of the study are funded by NKUST-KMU JOINT RESEARCH PROJECT (#NKUSTKMU-110-KK-001) and the \"Intelligent Manufacturing Research Center\" (iMRC) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data available in a publicly accessible repositoryThe data presented in this study are openly available in UCI Machine Learning Repository at [[http://archive.ics.uci.edu/ml](http://archive.ics.uci.edu/ml)]([http://archive.ics.uci.edu/ml](http://archive.ics.uci.edu/ml)), reference number [32].  
**Conflicts of Interest:** The authors declare no conflict of interest.