## 6 Regression-Analysis-Based Hybrid Approach  
Figure 1 is considered again, and the relevant information was already presented in Table 1. In this approach, first, \(H_{fcs}\) with a smaller \(\Delta\)g were obtained and correlated with the transfer functions at fewer selected positions, obtaining the CCvs at those locations. The relationship between the CCvs and line lengths (with a smaller possible \(\Delta\)g) was learned to construct a regression model. This model was then used for fault location prediction on new CCvs (obtained by correlating the selected transfer functions with the fault occurrence stage transfer function). Therefore, fewer \(H_{fcs}\) needed to be stored instead of all of the \(H_{fcs}\) and, in this way, ultimately relieving the memory burden of the data storage devices. The MATLAB regression learner app (RLA) was used for the regression analysis [29]. In this application, the training data were imported and trained with the available regression models. Based on their statistical properties, the best-fit models were selected, which were then evaluated on new data and and exported for future use as standalone applications. The available regression models in RLA are linear regression, decision trees, Gaussian process regression, support vector machines, and ensembles of tree. Five locations were selected along the line length of 50 km as the storage locations, e.g., \(x_{g}\) (SL) = [10 m, 12,500 m, 25,000 m, 37,500 m, 49,990 m]. The transfer functions corresponding to these positions are known as storage transfer functions (\(H_{STF}\)). These \(H_{STF}\) were then correlated with \(H_{fcs}\) (obtained at \(\Delta\)g = 10 m), and the CCvs were computed utilising Equation (6), mathematically written as follows:  
\[CCvs(x_{g}[SL])=\frac{\left\langle H_{STF}(x_{g}[SL])H_{fcs}(x_{g}[\Delta g=10 \text{ m}])\right\rangle}{\sigma(x_{g}[SL])\sigma(x_{g}[\Delta g=10\text{ m}])} \tag{20}\]  
These CCvs served as the independent variable, and a line length of 50 km with a spatial step of \(\Delta\)g = 10 m was taken as the dependent variable. This training dataset was imported into MATLAB RLA to learn the relationship, employing all the available models. Table 6 displays the statistical outcomes of these models. It is important to note that, for the sake of simplicity, only models with substantial outputs are listed in the table, where MSE = mean-squared error, RMSE = root-mean-squared error, MAE = mean absolute error, and R\({}^{2}\) = determination coefficient, which are mathematically defined as [30]  
\[MSE=\frac{\sum\limits_{i=1}^{N}\left(y_{i}-\hat{y}_{i}\right)^{2}}{N-(q+1)}\ ;\ \ RMSE=\sqrt{MSE}\]  
\[MAE=\frac{100}{N}\sum\limits_{i=1}^{N}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}} \right|\ ;\ R^{2}=1-\frac{\sum\limits_{i=1}^{N}\left(y_{i}-\hat{y}_{i}\right)^{2}}{ \sum\limits_{i=1}^{N}\left(y_{i}-\hat{y}_{i}\right)^{2}} \tag{21}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Transmission** & & & & & \\
**Line Length** & **Frequency** & & & & \\ \hline  & & & **1000 m** & **500 m** & **100 m** & **10 m** \\ \hline  & & & & **Memory Space Required** & \\ \hline
50 km & DC-1 MHz & 73 MB & 146 MB & 733 MB & **7.40 GB** \\  & DC-5 MHz & 370 MB & 741 MB & 3.67 GB & **35.25 GB** \\
100 km & DC-1 MHz & 147 MB & 295 MB & 1.47 GB & **14.9 GB** \\  & DC-5 MHz & 748 MB & 1.5 GB & 7.35 GB & **74.5 GB** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Existing correlation estimator: memory space required for different spatial steps in the fault estimation stage. \(y_{i}\), \(y_{i}\), and \(y_{i}\) are the mean, observed, and fit values of the dependent variable, respectively, whereas \(N\) and \(q\) are the total No. of observations and predictors, respectively. On the basis of these parameter values, the performance of the regression models was evaluated and the best-fit model was selected. As can be seen in Table 6, the Gaussian process regression (GPR) models outperformed the other regression models.