## 2 Research Data and Methods
### Theoretical Backgrounds and Model Development
#### 2.2.3 Elm  
An Extreme Learning Machine (ELM) can be described as a least square-based single hidden layer feed-forward neural network (SLFN) for both classification and regression problems [24], [PERSON] et al. [35] replaced the hidden layer with large number of nodes with a kernel function in the design of an ELM. [PERSON] and [PERSON] [24] and [PERSON] et al. [35] both proposed techniques, and the following is a summary of those methods:  
ELM for training data, \(N\), hidden neurons, \(H\), and activation function _f(x)_ can be represented as follows:  
\[e_{j}=\sum_{i=1}^{H}\alpha_{i}f\left(w_{i},c_{i},x_{j}\right)\,j=1\ldots N \tag{3}\]  
where \(w_{i}\) and \(\alpha_{i}\) are the weight vectors of the connecting inputs-hidden layers (input weights), and hidden-outputs layers, respectively. \(x_{j}\) represents the input variables. \(C_{i}\) is the hidden bias of the \(i^{\text{th}}\) hidden neuron, and \(e_{j}\) is the output from ELM for the data points, \(j\). The input weights are randomly generated and are based on a continuous probability distribution [24].  
The output weights are calculated using a linear equation (Equation (4)), which can be simplified as follows:  
\[\beta=A\,\dagger\,Y \tag{4}\]  
where \(A\) is the output matrix of the hidden layer (Equation (5)), \(A\dagger\) represents the Moore-Penrose generalized inverse of \(A\), and \(Y\) represents the target values of ELM. Equation (4) can be rewritten in a compact form as \(A\alpha=Y\), where \(A\) is the hidden layer output matrix of the neural network, and \(Y\) is the output variable vectors. The three matrices in the compact form can be presented as follows:  
\[A=\left[\begin{array}{c}h(x_{1})\\ \vdots\\ h(x_{N})\end{array}\right]=\left[\begin{array}{ccc}f(w_{1},c_{1},x_{1})&..&f (w_{H},c_{H},x_{1})\\ \vdots&..&\vdots\\ f(w_{1},c_{1},x_{j})&..&f(w_{H},c_{H},x_{j})\end{array}\right]\,\,\,\,\,\, \alpha=\left[\begin{array}{c}\alpha_{1}^{T}\\ \vdots\\ \alpha_{H}^{T}\end{array}\right]\,\,and\,\,Y=\left[\begin{array}{c}y_{1}^{T }\\ \vdots\\ y_{N}^{T}\end{array}\right] \tag{5}\]  
where \(h(x)\) is the hidden layer feature mapping.  
The output of the ELM algorithm is mainly based on matrix \(A\). In the case of using the traditional solution, the neural network is used in the hidden layer, and matrix \(A\) can be solved using gradient-based  
Figure 1: A single layer architecture of ANN.  
algorithm optimization as presented in [24]. Otherwise, the kernel function (\(k\big{(}x_{i},x_{j}\big{)}\)) is used to solve the ELM, and feature mapping can be used to calculate the kernel matrix as follows [5; 24; 35]:  
\[k\big{(}x_{i},x_{j}\big{)}=h(x_{i}).h\big{(}x_{j}\big{)} \tag{6}\]  
In this study, KELM was applied to study the effect of the kernel on the \(\mathrm{M_{r}}\) prediction, and ELM was integrated with PSO to design a new model for \(\mathrm{M_{r}}\) prediction of stabilized base aggregates. The ELM has a faster learning rate, better generalization ability, and better predictive performance than traditional neural connections. Basic ELM randomly produces the values of input weights and hidden biases, and determines the weights of the output layer using the Moore-Penrose generalized inverse method [36; 37]. Figure 2 shows the SLFN with a number of input layer neurons 'n', a number of hidden layer neurons, and a number of output layer neurons'm'. For example, if the training dataset is \(\{X_{i},Y_{i}\}\), then the input dataset is \(X_{i}=[X_{i1},\,X_{i2},\,\ldots,\,X_{in}]\), and the output dataset is \(Y_{i}=[Y_{i1},\,Y_{i2},\,\ldots,\,Y_{im}]\) and \(i=1,\,2,\,\ldots,\,n\). m is the number of training samples.