## 3 The Proposed Model
### ELM Based Classification  
At this stage, the ELM classifier can be employed for the effective identification and classification of educational data. The ELM model has \(n\) input layers, \(l\) hidden layers, and \(m\) output layers. Initially, considering the training instance \(\{X,\ Y\}=\{x_{i},\ y_{i}\}(i=1,2,\ \dots,\ Q)\), and it is comprised of the input feature \(X=\big{[}x_{i1}x_{i2}\dots\ x_{iQ}\big{]}\) and matrix \(Y=\big{[}y_{1j}y_{2}\dots y_{jQ}\big{]}\) with training instance, where the matrix \(X\) and \(Y\) are expressed by [16]:  
\[X=\left[\begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1Q}\\ x_{21}&x_{22}&\cdots&x_{2Q}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{n2}&\cdots&x_{nQ}\end{array}\right], \tag{6}\]\[Y=\left[\begin{array}{cccc}y_{11}&y_{12}&\cdots&y_{mQ}\\ y_{21}&y_{22}&\cdots&y_{mQ}\\ \vdots&\vdots&\ddots&\vdots\\ y_{m1}&y_{m2}&\cdots&y_{mQ}\end{array}\right],\]  
where \(n\) and \(m\) parameters denote the dimension of input and output matrix. Next, ELM set the weights amongst input and hidden layers randomly:  
\[w=\left[\begin{array}{cccc}w_{11}&w_{12}&\cdots&w_{1n}\\ w_{21}&w_{22}&\cdots&w_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ w_{l1}&w_{l2}&\cdots&w_{ln}\end{array}\right], \tag{7}\]  
where \(w_{ij}\) denotes the weight from \(i\)th hidden and \(j\)th input layers. Figure 2 showcases the framework of ELM. Then, ELM considers the weight from output and hidden layers that can be shown below:  
\[\beta=\left[\begin{array}{cccc}\beta_{11}&\beta_{12}&\cdots&\beta_{1m}\\ \beta_{21}&\beta_{22}&\cdots&\beta_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ \beta_{11}&\beta_{12}&\cdots&\beta_{lm}\end{array}\right], \tag{8}\]  
where \(\beta_{jk}\) indicates the weight from \(j\)th hidden and ith output layers. Next, ELM set a bias of hidden layers randomly:  
\[B=\left[b_{1}b_{2}\;\cdots\;b_{n}\right]^{T}. \tag{9}\]  
After that, ELM chooses the network activation function. According to, output matrix \(T\) is characterized by:  
\[T=\left[t_{1},\;t_{2},\;\ldots,\;t_{Q}\right]_{m\times Q}. \tag{10}\]  
Figure 2: Framework of ELM.  
The column vector of output matrix \(T\) is shown in the following:  
\[t_{j}=\left[\begin{array}{c}t_{1j}\\ t_{2j}\\ \vdots\\ t_{mj}\end{array}\right]=\left[\begin{array}{c}\sum\limits_{i=1}^{l}\beta_{i 1}g\big{(}w_{i}x_{j}+b_{i}\big{)}\\ \sum\limits_{i=1}^{l}\beta_{i2}g\big{(}w_{i}x_{j}+b_{i}\big{)}\\ \vdots\\ \sum\limits_{i=1}^{l}\beta_{im}g\big{(}w_{i}x_{j}+b_{i}\big{)}\end{array} \right](j=1,2,3,\ \ldots,\ Q). \tag{11}\]  
Moreover, considering Equations (10) and (11), it is modelled in the following:  
\[H\beta=T^{\prime}, \tag{12}\]  
where \(T^{\prime}\) indicates transpose of \(T\) and \(H\) represents simulation outcomes of hidden neurons. To achieve better solution with less error, the least square model is employed for determining the weight matrix measure of \(\beta\)[17; 18].  
\[\beta=H^{+}T^{\prime}. \tag{13}\]  
To enhance the normalization ability of the system and provide stable outcomes, the regularization parameter of \(\beta\) is used. The amount of hidden layers is minimal in contrast with the amount of training samples, \(\beta\) is shown in the following [19]:  
\[\beta=\left(\frac{I}{\lambda}+H^{T}H\right)^{-1}H^{T}T^{\prime}. \tag{14}\]  
When the amount of hidden layers is maximal than the amount of training samples, \(\beta\) is denoted as follows [20]:  
\[\beta=H^{T}\bigg{(}\frac{I}{\lambda}+HH^{T}\bigg{)}^{-1}T^{\prime}. \tag{15}\]