## 3 The Proposed Model
### WELM-Based Classification  
Next, the WELM model was used for the identification and classification of cyberbullying. The WELM classification model was used to assign appropriate class labels. ELM was used for the classification of a balanced dataset, while WELM was used for the classification of an imbalanced dataset [19]. The training dataset has \(N\) distinct instances, \((x_{i},\,z_{i}),i=1,2,\ldots,N\). A single hidden layer NN with \(L\) hidden layer node was formulated by:  
\[\sum_{i=1}^{L}\beta_{i}\cdot 1\left(w_{i}\cdot x_{j}+b_{i}\right)=z_{j};j=1, \ldots,N \tag{4}\]  
whereas \(w_{i}\) characterizes a single hidden layer's input weight, \(l()\) denotes the activation functions, \(\beta_{i}\) indicates the output weights, and \(b_{i}\) signifies the bias of the individual hidden layers.  
\[S\beta=T \tag{5}\]  
where \(S\) denotes the output matrix of a single hidden layer  
\[S(w_{1},\,\ldots,\,w_{L},\,b_{1},\,\ldots,\,b_{L},\,x_{1},\,\ldots,\,x_{N})= \left(\begin{array}{cccc}1(w_{1}\cdot x_{1}+b_{1}&\cdots&1(w_{L}x_{1}+b_{L}) \\ \vdots&\ddots&\vdots\\ 1(w_{1}\cdot x_{N}+b_{1})&\cdots&1(w_{L}\cdot x_{N}+b_{L})\\ \end{array}\right)_{N\times L} \tag{6}\]  
Based on the Karush-Kuhn-Tucker concept, the Lagrangian factor was used to change the trained ELM as a dual problem [20]. The output weight \(\beta\) can be evaluated as follows:  
\[\beta=S^{T}\bigg{(}\frac{1}{C}+SS^{T}\bigg{)}^{-1}T \tag{7}\]  
Here, \(C\) denotes the regularization coefficient. Therefore, the output function of the ELM classification is formulated by:  
\[F(x)=s(x)S^{T}\big{(}\frac{1}{C}+SS^{T}\big{)}^{-1}T=\begin{bmatrix}K(x,x_{1} )\\ \vdots\\ K(x,x_{N})\\ \end{bmatrix}^{T}\bigg{(}\frac{1}{C}+\chi\bigg{)}^{-1}T \tag{8}\]where \(\chi\) denotes the kernel matrix, which is calculated by:  
\[\chi=SS^{T}=s(x_{i})s\big{(}x_{j}\big{)}=K\big{(}x_{i},\ x_{j}\big{)} \tag{9}\]  
It is obvious in (13) that the hidden layer feature maps \(s(x)\) denote the independence in classification results of ELM. The classification result is related to the kernel function, \(K(x,\ y)\). \(K(x,\ y)\) denotes an inner product, and the number of hidden layers is no outcomes in the resulting output.  
\[K(x,\ y)=\exp\ \Big{(}-\gamma\|x-y\|^{2}\Big{)} \tag{10}\]  
Therefore, the KELM classification efficacy is distributed into two variables, namely the kernel function parameter, \(\gamma\) and the penalty parameter, \(C\). The original goal of ELM is achieved in WELM by preserving the weight for distinct instances to handle imbalanced classification problems. It is calculated by the following equations:  
\[F(x)=\begin{bmatrix}K(x,x_{1})\\ \vdots\\ K(x,x_{N})\end{bmatrix}^{T}\Big{(}\frac{1}{C}+W\chi\Big{)}^{-1}WT \tag{11}\]  
\[W=diag\ (w_{ii}),i=1,2,\ \ldots,\ N \tag{12}\]  
where \(W\) denotes the weight matrix. WELM has two weightage models as follows:  
\[w_{ii}=\frac{1}{\#(z_{i})} \tag{13}\]  
\[w_{ii}=\left\{\begin{array}{ll}\frac{0.618}{\#(z_{i})},\ if\ z_{i}>\overline {z}\\ \frac{1}{\#(z_{i})},\ otherwise\end{array}\right. \tag{14}\]  
where \(\#(z_{i})\) denotes the number of instances assigned to class \(z_{i}\), \(i=1,\ldots,m\); \(m\) denotes the number of classes; and \(\overline{z}\) indicates the average instance of all the classes. Figure 2 showcases the framework of ELM.