## 4 Nonlinear Regression of the Type-A Grounding System Effective Length
A Breusch-Pagan test was performed for the regression function given by Equation (2) with parameters given in Table 2, and the \(p\)-values obtained were 1.2231 \(\times\) 10\({}^{-58}\) for the training set and \(9.7\times 10^{-7}\) for the test set, which are well below the usual cut-off value of 0.05. Attempts to increase these \(p\)-values using the usual methods, such as input data transformation, yielded little to no effect, although this was to be expected considering the simplicity of the regression function in question. Moving on to the numerical regression quality metrics mentioned before--these metrics confirm the previous visual analysis (Table 3). The \(R^{2}\) value of the regression function is high for both the training and test sets, which is to be expected given that the results cluster around the perfect score. The average error, measured by the RMSE, is consistent across both sets, approximately 4 m. In conclusion, this simplest regression function will yield moderately satisfactory results. Figure 5: Actual vs. predicted values obtained using Equation (2): (a) Training set, (b) Test set. Figure 6: Residuals vs. predicted values obtained using Equation (2): (a) Training set, (b) Test set. In addition to this simplest regression formula, we also attempted to fit the input data using a slightly more complex model function with hopes of obtaining more accurate results:  
\[\ell_{eff}(\rho,T_{1},d)=\beta_{1}\cdot\rho^{\beta_{2}}\cdot d^{\beta_{3}}\cdot T _{1}^{\beta_{4}}+\beta_{5}\cdot\rho^{\beta_{6}}\cdot d^{\beta_{7}}\cdot T_{1}^{ \beta_{8}} \tag{3}\]  
As before, we fitted the selected model function to the input data from the training set using a nonlinear least-squares regression algorithm and obtained optimized values of the unknown coefficients. These values, along with their standard errors, are given in Table 4.  
Figure 7a,b depict the actual vs. predicted values obtained using Equation (3) for the training set and the test set, respectively. Figure 8a,b depict the residuals vs. predicted results obtained using Equation (3) for the training set and the test set, respectively. It is immediately noticeable that by introducing a sum of two products of power functions, more accurate predicted results were obtained. Results are clustered much closer to the perfect score line for both the training set and the test set (Figure 7a,b). This is also seen when observing the residuals plot (Figure 8a,b), where we can see that the maximum residual values peak at cca 4 m. Note that in this case, heteroscedasticity is still present in the model, which is observable from previous figures and from the \(p\)-values of the Breusch-Pagan test--\(3.4499\times 10^{-38}\) for the training set and \(0.000237\) for the test set. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Coefficient** & **Coefficient Value** & **Standard Error** \\ \hline \(\beta_{1}\) & \(0.97352081\) & \(0.01888089\) \\ \(\beta_{2}\) & \(0.54254976\) & \(0.00229135\) \\ \(\beta_{3}\) & \(-0.03689075\) & \(0.00221832\) \\ \(\beta_{4}\) & \(0.44419153\) & \(0.01415724\) \\ \(\beta_{5}\) & \(-0.0331507\) & \(0.01011605\) \\ \(\beta_{6}\) & \(0.5364613\) & \(0.00815618\) \\ \(\beta_{7}\) & \(-0.02284202\) & \(0.00807003\) \\ \(\beta_{8}\) & \(1.54295113\) & \(0.09497008\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimated values of coefficients for the regression function given by Equation (3). Figure 7: Actual vs. predicted values obtained using Equation (3): (**a**) Training set, (**b**) Test set. Again, the regression quality metrics confirm this visual analysis (Table 5)--the \(R^{2}\) is closer to 1, and the RMSE is almost three times lower for both the training and test sets. Similarly, other metrics also reflect this level of approximation quality.