## 4 Evaluation of the Adopted Regression Model and Discussion  
In this section, an evaluation of the adopted regression model is performed and some practical aspects of its application are discussed. The simplest method of evaluation assumes the use of existing Python tools and plotting substitution \(t\) values, retrieved using numerical methods, against the values coming from the polynomial regression model. An example of such a diagram, covering the whole set of generated input data, is shown in Figure 9.  
Figure 8: Graphical interpretation of relationship amongst the substitution \(t\), turn angle \(\theta_{\mathrm{S}}\), and \(R/d\) ratio. It is evident from the diagram in Figure 9 that the polynomial regression function fits the numerically calculated values of the substitution \(t\) very well. A certain discrepancy occurs for the substitution \(t\) values of less than 0.10, which corresponds to turn angles \(\theta_{S}\) smaller than 40\({}^{\circ}\). To evaluate the potential limitations and pre-suppositions for its use, robustness analyses of the adopted polynomial regression model were conducted. Standard methods and their corresponding procedures, briefly described in Table 4, were applied to test the robustness of the established fourth-degree polynomial regression model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No** & **Methods** & **Brief Description of Steps to Test the Robustness of the Selected Method** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{K-Fold Cross-Validation} & The procedure has a single parameter called k that refers to the number of groups that a given data sample is to be split into. As such, the procedure is often called k-fold cross-validation. \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{Repeated Train/Test Splits} & Perform multiple train/test splits with different random seeds and evaluate the model on each split. Calculate and compare the performance metrics (e.g., R\({}^{2}\), RMSE) across these splits. \\ \hline \multirow{2}{*}{3} & \multirow{2}{*}{Robustness to Noise} & Add different levels of noise to the input features and observe changes in performance metrics. For example, Gaussian noise can be added to analyzed features to see how the model’s predictions and accuracy are affected. \\ \hline \multirow{2}{*}{4} & \multirow{2}{*}{Outlier Analysis} & Introduce synthetic outliers into the dataset and evaluate the model’s robustness by observing changes in performance metrics. \\ \hline \multirow{2}{*}{5} & \multirow{2}{*}{Generalization to Unseen Data} & If researchers have access to an external dataset that was not used in training, they can evaluate the model on this dataset to check its generalization capabilities. \\ \hline \multirow{2}{*}{6} & \multirow{2}{*}{Stress Testing with Extreme Values} & Test the regression model with extreme or edge-case values for input features to evaluate how it handles these unusual cases. \\ \cline{3-4}  & & \\ \end{tabular}
\end{table} 
Table 4: Overview of standard methods to test robustness of a regression model. Figure 9: Graphical interpretation of the relationship between the substitution \(t\) values obtained from numerical calculation and the polynomial regression model. K-fold cross-validation was used for the RMSE and R\({}^{2}\) metrics' robustness testing. Herein, the parameter k was equal to 5, which means that the sample dataset was divided into five parts that were (roughly) equal in size. The k-fold cross-validation results are listed below:  
* Cross-validation RMSE scores: [0.00050482 0.00051121 0.00050779 0.00050810 0.00052693];
* Cross-validation mean RMSE score: 0.00051177;
* Cross-validation R\({}^{2}\) Scores: [0.99987991 0.99988177 0.9998805 0.99987864 0.99987698];
* Cross-validation mean R\({}^{2}\) score: 0.99987956. The produced cross-validation RMSE scores and the corresponding mean show that there is no RMSE value above the established benchmark for the desired level of accuracy (Equation (29)).