## 3 Methodology
### Classification  
The GRU is an updated version of the Long Short Term Memory (LSTM) network that integrates forget and input gate into a single gate named the \"update gate\" and further, the GRU model includes an additional gate named the \"reset gate\". Compared to the LSTM network, the GRU model is simple; therefore, it is becoming increasingly popular. Firstly, the GRU modulates the feature information inside the unit without using a memory cell. In the GRU model, the activation function \(h_{t}^{j}\) is a linear interpolation between the previous activation \(h_{t-1}^{j}\) and candidate activation \(\widetilde{h}_{t}^{j}\) at the time state \(t\), which is mathematically specified in Equation (18) [29; 30]. \[h_{t}^{j}=\left(1-z_{t}^{j}\right)\,h_{t-1}^{j}+z_{t}^{j}\widetilde{h}_{t}^{j} \tag{18}\]  
where \(z_{t}^{j}\) represents the update gate that decides the number of units updating its activation and \(\widetilde{h}_{t}^{j}\) states candidate activation. The mathematical expressions of the update gate and the candidate activation are defined in Equations (19) and (20). \[z_{t}^{j}=\sigma(w_{z}A_{t}+U_{z}h_{t-1})^{j} \tag{19}\]  
\[\widetilde{h}_{t}^{j}=tanh(wA_{t}+U_{r}(r_{t}^{j}\times h_{t-1}))^{j} \tag{20}\]  
where \(r_{t}^{j}\) states reset gates and \(tanh\) states hyperbolic tangent function. The \(r_{t}^{j}\) is mathematically calculated using Equation (21). \[r_{t}^{j}=\sigma(w_{r}A_{t}+h_{t-1})^{j} \tag{21}\]  
where \(w\) states parameter or weight, and \(\sigma\) indicates sigmoid function. In this scenario, the update gate \(z_{t}^{j}\) controls the prior states, where the long-term dependency units are called active update gates and the short-term dependency units are called active reset gates. The Stochastic Gradient Descent (SGD) optimization algorithm  
Figure 5: Flowchart of the binary cuckoo search algorithm. is applied in the GRU model for optimizing the stochastic objective functions based on the lower-order moments. The iterative algorithm: SGD initially starts with the random point of the gradient curve, and then it slants in the slope with the help of a user-defined learning rate until the gradient curve reaches the minimum value. In this study, the SGD optimization algorithm updates the weight or parameter \(w\) utilizing the gradient value \(\partial L/\partial w\), and then the corresponding gradient value is multiplied by the learning rate \(\alpha\). Therefore, the updated reset gate is mathematically defined in Equation (22). \[r_{t}^{j}=\sigma(w_{r+1}+h_{t-1})^{j} \tag{22}\]  
where \(w_{r+1}=w_{r}-\alpha\partial L/\partial w_{r}\) and the term \(\partial L/\partial w_{r}\) states gradient loss function \(L\) that reduces \(w\). If any decimal values occur, the GRU model approximately rounds off the respective decimal values into complete values. The architecture of the GRU model is specified in Figure 6.  
The parameter settings of the GRU model are listed as follows: lambda loss amount is 0.0015, the number of hidden units are 32, the learning rate is 0.0025, and the number of iteration is 100. To resolve the un-constrained non-linear optimization issues, a BFGS algorithm is integrated with the GRU model. The BFGS algorithm uses a gradient descent function \(\mu\) to further reduce the gradient value to the local minimum. The gradient descent function \(\mu\) is mathematically defined in Equation (23). \[\mu=argmin\{\Psi(\mu)\} \tag{23}\]  
where, \(\Psi(\mu)\) states non-convex function. The point \(\mu^{k+1}\) is computed in the next iteration \(k\) using the point \(\mu^{k}\), as mentioned in Equation (24). \[\mu^{k+1}=\mu^{k}+\vartheta_{k}d_{k} \tag{24}\]  
where \(d_{k}\) states search direction and \(\vartheta_{k}\) denotes step size, and the minimizer \(\vartheta_{k}\) is mathematically defined in Equation (25). Additionally, the search direction is specified in Equation (26).