# Prediction of the Yield Strength of RC Columns Using a PSO-LSSVM Model
## 4 Model Development and Verification
### Evaluation of the PSO-LSSVM Prediction Performance
#### 4.4.3 Comparative Evaluation of Model Prediction Performance  
For further elaborating the effectiveness of the PSO algorithm in terms of parameter (\(\sigma\), \(\gamma\)) optimization, other metaheuristic algorithms such as GA [30], DE [31] and ABC [32] are also applied to optimize the parameters of the LSSVM model. The identical training and testing samples are employed to make a performance comparison among standard LSSVM models without the optimized algorithm, and LSSVM optimized by GA, ABC, DE and PSO algorithms, respectively. The configuration of the standard LSSVM model is the same as that of the proposed model with the parameters of standard LSSVM and kernel set as \(\sigma^{2}=0.1\), \(\gamma=10\)[39]. The parameter settings of the GA, ABC and DE algorithms are identical to that used in the PSO, i.e., the size of the solution \(M=30\) and the maximum iteration number \(k_{\text{max}}=200\). Table 5 is the evaluation results of the LSSVM, GA-LSSVM, ABC-LSSVM, DE-LSSVM, and PSO-LSSVM models. As shown in Table 5, it is obvious that the prediction performance of the standard LSSVM is improved by adopting the optimized algorithms due to the increase in \(R^{2}\) and EV, and the decrease in RMSE and MAE. The higher \(R^{2}\) and EV, and lower RMSE and MAE imply the higher accuracy of models. Meanwhile, comparing the evaluation indices (\(R^{2}\), EV, RMSE and MAE) of testing data between the LSSVM optimized by other metaheuristic algorithms and the proposed PSO-LSSVM model, it can be found that the \(R^{2}\) of the PSO-LSSVM model in predicting yield strength (\(F_{y}\)) and yield displacement (\(\Delta_{y}\)) can be up to 0.96 and 0.97, respectively. The RMSE and MAE of the PSO-LSSVM model are both sufficiently minimal. For the prediction of \(F_{y}\), there is little difference in the prediction accuracy between GA-LSSVM, ABC-LSSVM, DE-LSSVM and PSO-LSSVM models. While for the prediction of \(\Delta_{y}\), the prediction accuracy of the PSO-LSSVM model is higher than other models. This demonstrates that the PSO-LSSVM model performs better than GA-LSSVM, ABC-LSSVM and DE-LSSVM models in terms of \(R^{2}\), EV, RMSE and MAE, indicating the proposed PSO algorithm has an outstanding advantage over the GA, ABC and DE algorithms in establishing the LSSVM model used for effectively predicting the yield properties of RC columns.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{\multirow{2}{*}{**Model**}} & \multicolumn{2}{c}{**Training**} & \multicolumn{6}{c}{**Testing**} \\ \cline{2-10}  & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) & **EV** & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) & **EV** \\ \hline \multirow{5}{*}{\(F_{y}\)} & LSSVM & 25.41 & 15.65 & 0.9859 & 98.5933 & 43.29 & 26.58 & 0.9248 & 92.6090 \\  & GA-LSSVM & 20.61 & 13.41 & 0.9070 & 99.0746 & 31.92 & 23.93 & 0.9591 & 95.9857 \\  & ABC-LSSVM & 20.54 & 13.25 & 0.9908 & 99.0802 & 31.60 & 23.38 & 0.9599 & 96.0527 \\  & DE-LSSVM & 19.90 & 12.94 & 0.9913 & 99.1371 & 31.42 & 23.17 & 0.9603 & 96.1449 \\  & PSO-LSSVM & 20.53 & 13.22 & 0.9907 & 99.0821 & 31.45 & 23.25 & 0.9601 & 96.0898 \\ \hline \multirow{5}{*}{\(\Delta_{y}\)} & LSSVM & 2.067 & 1.282 & 0.9775 & 99.7536 & 7.870 & 2.895 & 0.7848 & 78.5773 \\  & GA-LSSVM & 1.800 & 1.108 & 0.9830 & 98.2953 & 4.647 & 2.87 & 0.9249 & 92.5010 \\ \cline{1-1}  & ABC-LSSVM & 1.876 & 1.168 & 0.9815 & 98.1484 & 4.943 & 2.674 & 0.9151 & 91.5142 \\ \cline{1-1}  & DE-LSSVM & 1.831 & 1.139 & 0.9823 & 98.2360 & 4.975 & 2.686 & 0.9140 & 91.4036 \\ \cline{1-1}  & PSO-LSSVM & 1.795 & 1.140 & 0.9832 & 98.3245 & 2.798 & 1.822 & 0.9728 & 97.2999 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Evaluation results of standard LSSVM, GA-LSSVM, ABC-LSSVM, DE-LSSVM, PSO-LSSVM.