## 2 Materials and Methods
### Validation
Since a decreasing behavior is observed and there are only isolated points, the overall trend is not affected. From the figure of the data after the regularization, Figures 7-9, it can be seen that this regularization technique modifies only the critical points as expected. Figure 7: Dataset after the data regularization. Data measured at fixed load condition of \(R=100\)\(\Omega\), for three different distance coil conditions, with \(z_{0}=0\), 4 and 8 mm. For this dataset, a very simple SR model is used, with 5 iterations and 5 populations, and only \"+\" and \"*\" as binary operators, and no unary operators. Despite its simplicity, this model achieves good results. It can still predict the trends of the curves and provide good estimates, even if they are not very accurate. This is explained in more detail in the results section. On the other hand, a more complex model is studied, to predict the more sophisticated behavior of the functions. For this reason, 500 iterations with 100 populations are used, with \"+\", \"*\" and \"/\" as binary operators, and exponential and logarithmic functions as unary operators, to introduce nonlinear behavior. This complex model obtains only a few percentage points in improved accuracy compared to the simpler one, detecting very similar formulas. In addition to the simpler model, it also detects the dependencies on the load. Table 1 is provided to summarize the various models and their respective parameters. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameters** & **Experimental NFC** & **Simulated NFC** & **Simple Model Phase** & **Complex Model Phase** \\ \hline INPUT data & \(Z\)-parameter & \(S\)-parameters & phase, \(R\), \(z_{0}\) & phase, \(R\), \(z_{0}\) \\ \hline OUTPUT data & Distance & Distance & Phase & Phase \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SR models used with their parameter settings. Figure 8: Datasetafter the data regularization. Data measured at fixed load condition of \(R=1000\)\(\Omega\), for three different distance coil conditions, with \(z_{0}=0\), 4 and 8 mm. Figure 9: Datasetafter the data regularization. Data measured at fixed coil distance of \(z_{0}=4\) mm, for six different load conditions. On the other hand, the prediction of the configuration does not yield accurate results, as the formulas found have no particular physical significance. The same results are obtained with classical NNs for both the experimental and simulated datasets. This is likely due to insufficient information in the input parameters. Therefore, the dataset can be extended with other sensors, such as magnetic sensors, to add more information and derive complete formulas. As a first approach of this method to a WPT system, very promising results are obtained, with prediction accuracy comparable to that of NNs. Although the formulas derived from symbolic regression are accurate, their complexity makes direct transcription impractical and less informative. Therefore, we focus on a qualitative analysis of the functional relationships discovered, emphasizing the dependencies and trends that offer valuable insights into the system's behavior. Specifically, for the input impedance (real part, \(R_{in}\), and imaginary part, \(X_{in}\)), the symbolic regression model identifies a \(1/R_{in}\) dependence on distance and a parabolic trend for \(X_{in}\). For the scattering parameters, the model reveals parabolic and logarithmic dependencies. Figure 11: Predicted distance values and original ones using \(Z_{in}\) parameters for the experimental measurement dataset, for the case 1. Figure 12: \(S\) parameters at different distance values. These are the data of the simulation dataset taken at a fixed configuration. The \(S\) parameter values reported here are in unitary (dimensionless) form. Conversion to decibels (dB) can be achieved using the formula \(S_{dB}=20\log_{10}(|S_{unitary}|)\).