## 4 Evaluation of the Adopted Regression Model and Discussion
Also, the obtained cross-validation R\({}^{2}\) scores and their mean negligibly differ from the R\({}^{2}\) value which was initially achieved for the whole generated dataset. The methods \"repeated train/test Splits, robustness to noise, outlier analysis, and generalization to unseen data\" cannot be directly implemented because polynomial regression function was established by using the whole dataset. In addition, the used dataset with different combinations of parameters \(R\), \(d\) and \(\theta\) was artificially created, while the previously listed methods, particularly robustness to noise and outlier analysis, are more suitable for the robustness analysis of experimentally collected data, i.e., data acquired from measurements in the field. Generalization to unseen data requires the introduction of an external dataset that was not used in model training, which is not relevant in this case--because predetermined combinations of the key input parameters \(R\), \(d\), and \(\theta\) were used to generate the whole dataset. The stress testing with extreme values method gave the following results:  
* MSE on the high-value subset (HVS) range: 2.75711\(\times 10^{-7}\);
* RMSE on the HVS range: 0.000403;
* MSE on the low-value subset (LVS) range: 6.017358\(\times 10^{-7}\);
* RMSE on the LVS range: 0.000595. HVS includes predicted values of substitution \(t\) in the range: \(t>\text{MDN}_{t}\);  
LVS includes predicted values of substitution \(t\) in the range: \(t\leq\text{MDN}_{t}\). It can be seen that higher mean square error (MSE) and RMSE values were obtained below the median (MDN) of predicted substitution \(t\) values, which is MDNt = 0.188780. MDNt closely matches to the combination of the turn angle \(\theta\) of 75\({}^{\circ}\) and the \(R/d\) ratio of 1.40845. This means that the adopted regression model has a slightly lower accuracy for turn angles \(\theta\) smaller than 75\({}^{\circ}\) and \(R/d\) ratios below 1.408451 or \(d=0.71\cdot R\). Robustness analyses of the fourth-degree polynomial regression model revealed new pre-suppositions for its use. In addition to the precondition \(R>d\), originally imposed by the ICAO, and the numerically calculated possible range for the substitution \(t\) values \(0<t<1\) (Equation (28)), robustness analyses pointed out that the regression model has a negligibly lower prediction accuracy for the turn angles of \(\theta<75^{\circ}\) and \(R/d\) ratios in the range of \(R/d<1.408451\). The accuracy of the adopted regression model was thoroughly evaluated by comparing substitution \(t\) values calculated both numerically and by the fourth-degree polynomial regression function (29). In addition to the substitution \(t\) values, corresponding steering angles \(\beta_{d}\) and distances \(F_{OT_{max}}\) determining the vehicle maximum transient offftracking positions were calculated and mutually compared. As expected, the biggest absolute differences between the parameters \(\beta_{d}\) and \(F_{OT_{max}}\) calculated by the two compared methods appear only in a limited number of cases for the combinations of input parameters at the beginning of the generated input dataset. These critical combinations of input parameters refer to the ones with the smaller turn angles \(\theta_{S}\), usually less than 35\({}^{\circ}\), which is in accordance with the previous findings acquired from the diagram displayed in Figure 9 and robustness analyses. Moreover, the results of the calculation methods' comparison displayed in Table 5 should be discussed from the road designers' perspective as well. Although the 0.20 m difference between the compared values \(F_{OT_{max}}\) at first seems like a serious error in the prediction of the regression model, it should be highlighted that it is actually an error in determining the position of the vehicle's datum point in the \"longitudinal sense\".