# On the Use of Neuro-Swarm System to Forecast the Pile Settlement
## 3 Model Development
### Neuro-Swarm Design  
In this part, neuro-swarm modeling procedure to predict pile settlement is presented considering the suggested ANN architecture in previous section. It is worth mentioning that neuro-swarm models were constructed using 80% of the whole database as training datasets and 20% of the whole database as testing datasets. Approaching the implement of neuro-swarm model starts by determining the number of swarm or swarm size (SS). In order to select the best SS, values of 25, 50, 75, 100, 150, 200, 250, 300, 350 and 400 were employed and through a parametric study the best one among them was selected. Table 2 tabulates the parametric study's results. In conducting this parametric study, other effective parameters of PSO were considered constant as follows (\(C_{1}=C_{2}=2\)), (max iteration = 100) and (\(w\) = 0.25). Ranking system used in Table 2, was utilized according to the study conducted by [PERSON] et al. [69]. According to this method, each performance index (i.e., herein root mean square error, RMSE and R\({}^{2}\)) is ordered in its class. For example, values of 0.749, 0.772, 0.828, 0.805, 0.818, 0.827, 0.827, 0.796, 0.844 and 0.828 were obtained for training datasets of neuro-swarm model numbers of 1-10, with rankings of 3, 4, 9, 6, 7, 8, 8, 5, 10 and 9, respectively. Therefore, as a result, model number 10 \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**No. of Hidden**} & \multirow{2}{*}{**1**} & \multirow{2}{*}{**2**} & \multirow{2}{*}{**3**} & \multirow{2}{*}{**4**} & \multirow{2}{*}{**5**} & \multirow{2}{*}{**6**} & \multirow{2}{*}{**7**} & \multirow{2}{*}{**8**} & \multirow{2}{*}{**9**} & \multirow{2}{*}{**10**} & \multirow{2}{*}{**11**} \\ \multicolumn{2}{c}{**Neuron**} & & & & & & & & & & & & \\ \hline \multirow{2}{*}{**R\({}^{2}\)**} & **Train** & 0.753 & 0.782 & 0.809 & 0.805 & 0.803 & 0.784 & 0.755 & 0.774 & 0.799 & 0.702 & 0.665 \\ \cline{2-11}  & **Test** & 0.766 & 0.743 & 0.816 & 0.812 & 0.795 & 0.757 & 0.760 & 0.789 & 0.802 & 0.741 & 0.766 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of neuro-swarm models with different swarm size (SS) values. with total ranking of 33 was chosen as the best model and 400 was selected as the best SS. Note that, the equation of RMSE is presented as follows:  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(x_{iprod}-x_{imas}\right)^{2}} \tag{4}\]  
where, \(x_{imas}\) and \(x_{iprod}\) are the measured and predicted values, respectively. \(N\) is the total No. of datasets. The second parametric study is related to determination of max iteration in predicting pile settlement. The same SS values of the previous stage were used with max iteration number of 500 as shown in Figure 4. The results of this parametric study were assessed based on RMSE values. According to Figure 4, the changes of RMSE values are very small for all SS after iteration number of 400. Hence, the max iteration number was considered as 400. The next stage of modeling, it seems that there is a need to conduct another parametric study to determine coefficients of velocity equation. [PERSON] and [PERSON] (2000) mentioned that the highest performance prediction of neuro-swarm model can be obtained when \(C_{1}=C_{2}=2\) with total summation of 4. Therefore, another parametric study with various combinations of \(C_{1}\) and \(C_{2}\) (with total summation of 4) was considered and applied in order to choose the best \(C_{1}\) and \(C_{2}\). Table 3 presents the results