## 6 Regression-Analysis-Based Hybrid Approach
In addition, GPR exponential has the lowest RMSE value = 18 and a determination coefficient value (R2) of \"1\". Although the other regression models also had a determination coefficient value of \"1\", the RMSE is considered the best criterion for the decision; moreover, the model that gave the required results would be the best choice [30]. The response plot at one of the storage locations for GPR exponential is shown in Figure 18. It is clear that there was a perfect agreement between the regression estimation and the training sample data. After training the data, the regression models were tested on new data, which were obtained by correlating \(H_{STF}\) with \(H_{fos}\) of some random fault locations along the line (e.g., 20 m, 370 m, 2750 m, 7320 m, 18,690 m, 23,980 m, 34,110 m, and 42,540 m), mathematically defined as  
\[CCvs(new)=\frac{\left<H_{fos}(f)H_{STF}(f;x_{g}[SL])\right>}{\sigma_{fos} \sigma_{STF}(x_{g}[SL])} \tag{22}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Regression Model** & **RMSE** & **R2** & **MAE** & **Training Time (s)** \\ \hline Tree Fine & 159 & 1 & 108 & 5.49 \\ Tree Medium & 170 & 1 & 112 & 3.2 \\ SVM Fine Gaussian & 1787 & 0.98 & 1654 & 69.75 \\ SVM Medium Gaussian & 1292 & 0.99 & 1147 & 14.8 \\ Ensemble Bagged Trees & 178 & 1 & 49 & 24.6 \\ GPR Squared Exponential & 72 & 1 & 18 & 837.26 \\ GPR Matern 5/2 & 27 & 1 & 8 & 1218 \\
**GPR Exponential** & **18** & **1** & **5.11** & **1991** \\ GPR Rational Quadratic & 35 & 1 & 12 & 2450 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical outputs of regression models. Figure 18: Response plot of GPR exponential at one of the storage locations, i.e., 37,500 m.  
It was observed that the GPR models provided more precise and consistent fault location estimation. As shown in Figure 19, the exponential model produced more promising outcomes within the GPRs, with the error near zero. The absolute error is mathematically denoted as  
\[Absolute\ Error\ (m)=\left|x_{f}-x_{g}\right| \tag{23}\]  
As the objective of this presented analysis was to reduce the amount of memory space required, the case was made to show that a fault can be computed accurately with just a few transfer functions and a regression model. Table 7 compares the memory requirements of the hybrid technique and the existing correlation estimator (with a smaller spatial step). It is obvious that only tens or hundreds of megabytes were required, as opposed to allotting gigabytes (GBs) of memory for \(H_{fes}\). For instance, for the case of line length = 100 km and frequency spectrum = DC-5 MHz, while employing the existing correlation estimator, the memory space requirement was 75 GB, whereas, the same job could be performed with 80 MB by taking advantage of the machine learning algorithms. As discussed in Section 3, two separate copies of \(H_{fes}\) are required for low- (\(Z_{f}\) = 10 \(\Omega\)) and high-impedance (\(Z_{f}\) = 1000 \(\Omega\)) fault scenarios to locate the fault position accurately. This will, however, use more memory space; therefore, a hybrid approach can be applied to both scenarios independently to alleviate the large memory requirement for the data storage devices. Consequently, the fault location processes can be stated as follows: whenever a fault occurs, \(H_{fos}\) will be correlated with \(H_{STF}\) of low- and high-impedance faults. The regression model and \(H_{STF}\) for fault location prediction will be selected based on the maximum CCv of the two scenarios. For instance, in Figure 20, it can be seen that the \(H_{fos}\) due to \(x_{f}\) = 10 km with \(Z_{f}\) = 0 \(\Omega\) was correlated with \(H_{STF}\) of low- and high-impedance