## 3 The Proposed Model
### Image Classification Using the WELM Model  
In this study, the WELM model is exploited for lung nodule classification [20]. An extreme learning machine (ELM) is employed for the classification of a balanced dataset, whereas WELM is applied for the classification for imbalanced dataset. The training dataset has \(N\) distinct instances, \((x_{i},\ z_{i}),i=1,2,\ldots,N\). A single hidden layer (HL) and a neural network (NN) with \(L\) HL nodes are expressed herewith:  
\[\sum_{i=1}^{L}\beta_{i}\cdot 1\big{(}w_{i}\cdot x_{j}+b_{i}\big{)}=z_{j},j=1, \ldots,N \tag{5}\]  
Whereas \(l()\) indicates the activation function, \(w_{i}\) signifies a single HL input weight, \(\beta_{i}\) denotes the output weight, and \(b_{i}\) denotes single HLs bias, which are described below.  
\[S\beta=T \tag{6}\]  
Here, \(S\) denotes the resultant matrix of single HL.  
\[S(w_{1},\ \ldots,\ w_{L},\ b_{1},\ \ldots,\ b_{L},\ x_{1},\ \ldots,\ x_{N})= \left(\begin{array}{cccc}l(w_{1}\cdot x_{1}+b_{1})&\cdots&l(w_{L}\cdot x_{1} +b_{L})\\ \vdots&\ddots&\vdots\\ l(w_{1}\cdot x_{N}+b_{1})&\cdots&l(w_{L}\cdot x_{N}+b_{L})\end{array}\right)_{ N\times L} \tag{7}\]  
Based on the Karush-Kuhn-Tucker concept, the Lagrangian factor is identified in changing the trained ELM as many problems are being handled. The resultant weight \(\beta\) is evaluated by the following equation:  
\[\beta=S^{T}\bigg{(}\frac{1}{C}+SS^{T}\bigg{)}^{-1}T \tag{8}\]  
Now, \(C\) indicates the regularization coefficient. Henceforth, the output function of ELM classification is expressed below:  
\[F(x)=s(x)S^{T}(\frac{1}{C}+SS^{T})^{-1}T=\left[\begin{array}{c}K(x,x_{1})\\ \vdots\\ K(x,x_{N})\end{array}\right]^{T}\bigg{(}\frac{1}{C}+\chi\bigg{)}^{-1}T \tag{9}\]  
whereas \(\chi\) represents the kernel matrix that is evaluated, as given below.  
\[\chi=SS^{T}=s(x_{i})s\big{(}x_{j}\big{)}=K\big{(}x_{i},\ x_{j}\big{)} \tag{10}\]  
It is obvious that the HL feature maps \(s(x)\) denote the independence in classification outcome of ELM. Figure 2 demonstrates the infrastructure of ELM. The classification outcome was compared with the kernel function, \(K(x,\ y)\). \(K(x,\ y)\) considers an inner product process as follows:  
\[K(x,\ y)=\exp\ \Big{(}-\gamma\|x-y\|^{2}\Big{)} \tag{11}\]Therefore, the kernel extreme learning machine (KELM) classification efficacy was distributed as to two parameters, namely the kernel function parameter, \(\gamma\), and the penalty parameter, \(C\). The original benefit of ELM is accomplished in WELM by preserving the weight for distinct instances to handle imbalanced classification problems. It is defined through the following equation:  
\[F(x)=\left[\begin{array}{c}K(x,x_{1})\\ \vdots\\ K(x,x_{N})\end{array}\right]^{\mathsf{T}}\left(\frac{1}{C}+W_{X}\right)^{-1}WT, \tag{12}\]  
\[W=diag\;(w_{ii}),i=1,2,\;\ldots,\;N \tag{13}\]  
Here, \(W\) characterizes the weighted matrix. WELM has two weightage models as given herewith:  
\[w_{ii}=\frac{1}{\#(z_{i})}, \tag{14}\]  
\[w_{ii}=\left\{\begin{array}{c}\frac{0.618}{\#(z_{i})},\;ifz_{i}>\Xi\\ \frac{1}{\#(z_{i})},\;otherwise\end{array}\right. \tag{15}\]  
Now, \(\#(z_{i})\) denotes the count of instances, going to class \(z_{i}\), \(i=1,\ldots,m\). \(m\) indicates the count of classes. \(\Xi\) denotes the average of complete instance.