## 2 Research Data and Methods
### Theoretical Backgrounds and Model Development
#### 2.2.5 Model Development and Performance Assessment  
The training dataset contains five input neurons as predictors. Initially, the ELM model was configured by the matrix of input weights and hidden biases. The output weight matrix was calculated from the input weights and biases using the basic ELM algorithm. Since the performance of the ELM is based on the number of neurons in the hidden layer and number of training epoch, a trial was conducted with 100 epochs between the number of hidden units in the single hidden layer versus the Root Mean Square Error (RMSE) to determine the best number of hidden neurons. The final structure of the developed optimized ELM model was obtained as 5 input neurons, 150 hidden neurons, and output neurons based on the number of training samples. The PSO algorithm was used to find the optimal values of the input weights matrix (\(5\times 150\)) and bias matrix (\(150\times\mathrm{m}\)) based on the minimal RMSE value. The number of decision variables for the PSO population was determined from the ELM learning parameters, which was set as 900 (\(5\times 150+150\)). These learning parameters were optimized during the training phase to obtain the optimal input weights and hidden biases. The PSO is influenced by its intrinsic parameters, such as the number of populations, acceleration coefficients, and inertia weight. After some initial trials, the swarm size was set as 30, C\({}_{1}\) = 1, C\({}_{2}\) = 2, and inertia weight = 0.9 for both PSO-ELM and PSO-ANN. In addition, the KELM (RBF-ELM) model was trained using the radial basis function (RBF) as the activation function, in which users do not need to know the number of hidden units, and the hidden layer feature mapping is not required to be known by the user [50].  
[PERSON] et al. [4] evaluated the normalized and non-normalized data for predicting M\({}_{r}\), and they found that the SVR model outperformed the LS method for the normalized data. Herein, the whole dataset was initially normalized and back transformed to obtain original values after prediction. T normalization of the data (\(Cn\)) can be performed using the following equation:  
\[Cn=\frac{(C-C_{min})}{(C_{max}-C_{min})} \tag{7}\]  
where \(C\), \(C_{max}\), and \(C_{min}\) represent the data value, maximum of used data, and minimum of used data, respectively.  
To assess the model performance of the developed models, goodness of fit statistics such as \(r^{2}\), RMSE, and mean absolute error (MAE) were used. The corresponding fitness indices were represented by Equations (8)-(10).  
\[r^{2}=\left(\frac{\sum_{i=1}^{l}\left(Mr_{E_{i}}-Mr_{\overline{E_{i}}}\right) \left(Mr_{O_{i}}-Mr_{\overline{O_{i}}}\right)}{\sqrt{\sum_{i=1}^{l}\left(Mr_{E_{ i}}-Mr_{\overline{E_{i}}}\right)^{2}\sum_{i=1}^{n}\left(Mr_{O_{i}}-Mr_{\overline{O_{i}}} \right)^{2}}}\right)^{2} \tag{8}\]  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{l}\left(Mr_{E_{i}}-Mr_{O_{i}}\right)^{2}}{l}} \tag{9}\]  
\[MAE=\frac{1}{l}\sum_{i=1}^{l}\left|Mr_{O_{i}}+Mr_{E_{i}}\right| \tag{10}\]  
where \(Mr_{E_{i}}\) denotes the \(i^{th}\) predicted resilient moduli of stabilized base aggregates; \(Mr_{O_{i}}\) represents the \(i^{th}\) observed resilient moduli of stabilized base aggregates; \(Mr_{\overline{E_{i}}}\) is the average of the predicted resilient moduli of the stabilized base aggregates; \(Mr_{\overline{O_{i}}}\) is the average of the observed resilient moduli of the stabilized base aggregates; and \(l\) is the number of observations.