## 2 Methodology
### Soft-Sensor Design
Levenberg-Marquardt Training Algorithm  
The artificial neural network (ANN) training algorithm formulates estimation error as an objective function to be minimised through every training iteration (Figure 6). It distributes the total estimation error among the nodal connection weights. Learning and decay rates are defined to control the upper and lower limits, respectively, of these weights. The optimization algorithm minimises the estimation error over successive training epochs. The transformation function sums up the products of input nodes and respective weights, and yields the output value in conjunction with a bias factor and an activation function. The weighted sums are scaled up to the values of the input nodes by the bias factor to guarantee that the input node values are preserved for further calculations. The activation function ensures that at least the minimal threshold of the calculated nodal outputs is contributed to the next layer [51]. The Levenberg-Marquardt algorithm employs a Jacobian matrix in conjunction with the gradient vector and expresses the loss function as the sum of squared errors (\(f\)). \[f=\sum_{r=0}^{N}e_{r}^{2} \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Training Method** & **Levenberg-Marquardt** \\ \hline Sample division & as shown in Table 4 \\ Activation function (\(f_{1}\)) & Sigmoid function \\ Activation function (\(f_{2}\)) & Linear function \\ Initial weights and bias & zero \\ Input layer nodes & 2 \\ Output layer nodes & 1 \\ Hidden layer nodes & 10 \\ Performance index & Mean squared error (MSE) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Two-layer feed-forward neural-network features. Figure 6: Supervised learning flow. where \(N\) is the number of data set instances, and \(e\) is the difference between derived and predicted SOC values [52; 53]. The Jacobian matrix of this loss function is composed of the error derivatives having \(N\) by \(P\) dimensions. \[\mathbf{J}_{i,j}=\frac{\partial e_{i}}{\partial\mathbf{W}_{j}} \tag{13}\]  
where \(i=1,2,\cdots,N\) is the number of data set instances, \(J=1,2,\cdots,P\) is the number of neural network parameters and \(\mathbf{W}_{j}\) is the weight vector corresponding to a particular input. The gradient vector is determined for this loss function as follows. \[\
abla f=2\mathbf{J}^{T}\cdot e \tag{14}\]  
The Hessian matrix is approximated as follows. \[\mathbf{Hf}\approx 2\mathbf{J}^{T}\cdot\mathbf{J}+\mu\mathbf{I} \tag{15}\]  
where \(\mathbf{I}\) is the identity matrix, and \(\mu\) is the Hessian damping factor. Lastly, loss function parameters are improved by the algorithm as per the following expression [54]. \[\mathbf{W}^{i+1}=\mathbf{W}^{i}-(\mathbf{J}^{(i)T}\cdot\mathbf{J}^{(i)}+\mu^{ i}\mathbf{I})^{-1}\cdot(2\mathbf{J}^{(i)T}\cdot e^{j}) \tag{16}\]  
The network training process is terminated on the basis of the algorithm meeting any one of the stopping criteria depicted in Table 6.  
For comparative analysis, the SOC, SOE, and PL of the FSEV considered in the current study were also modelled using linear regression, nonlinear regression, and parametric structures such as ARX, ARMAX, OE, and BJ models. Table 7 depicts the generalized forms of these parametric model structures widely used in the system identification of various systems. The prediction accuracy of all developed models was compared on the basis of R-squared/FIT% and mean squared errors (MSE).