## 3 Criteria of Response Function Selection
### Step Three
The \(RMSE_{\text{mod}}\) criterion made it possible to obtain satisfactory accuracy of the probabilistic moments \(\max\lambda_{\text{rel,m}}\) for the discrete data generated by the functions based on polynomials of various \(POb\) regardless of the possible accuracy of the values of the coefficients \(\Delta_{\text{rel,c}}\) (Table 5) being the result of the number of significant digits rounding of the trial points. The validity of using the criterion was also confirmed by the graphs showing the indicated approximations in relation to the reference functions and the discrete data generated (Figure 12). Figure 7:0 Diagrams of approximations (maximizing \(RMSE_{\text{mod}}\)) in relation to the reference functions and the discrete data generated: (**a**) A results, (**b**) B results, (**c**) C results. The \(RMSE_{\text{mod}}\) rounded value above which--for \(\alpha_{\text{max}}*(x)=0.30\)--all the obtained a posteriori errors are below 1% is 36 (Figure 11a). It is worth noting that this number was almost independent--it visually imperceptibly decreased--from the interval width of the input coefficient of variation in which the accuracy of the approximation was tested--equivalent to \(\alpha_{\text{max}}\)*. With its decrease, the relative error value \(\max\Delta_{\text{rel,m}}\) decreased (Figure 11a-f). It should be noted that the least squares method was also based on the condition of minimizing the criterion similarly to the \(RMSE\). In the opinion of the author, the abovementioned conclusions confirm that the proposed criterion is correct. The limit value of the adopted criterion in relation to the form of the penalty term resulted in that only the approximations based on the base polynomial of the maximum eighth degree could exceed it. In the case of the ninth degree, despite the minimum values of the \(RMSE\) and \(RMSEw\) components (but not less than \(10^{-20}\)--the number of digits used in the experiment by the MAPLE system when performing calculations using the floating-point numbers was 20), the final value of the \(RMSE_{\text{mod}}\) coefficient was still lower than the limit value equal to 36. This limitation seems rational if we take into account the number of trial points (\(n=11\)) and the desire to avoid the overfitting phenomenon. It is worth mentioning that all the considered criteria determining the quality of approximation fit--linear correlation coefficient, root-mean-square error, fitting variance, coefficient of determination--used both individually and in pairs did not lead to a marked dependence of the maximum value of the a posteriori error on the criterion value. Adoption of the approximation for which their extreme value is achieved in most cases also led to results of unsatisfactory accuracy--points with the ordinate above \(10^{-2}\) in Figure 7:1  
In the abovementioned experiment, a total of 174,469 approximations were analyzed, and 17,660 of them (10.1%) had error values less than 1%. Only 1003 of them (7.91% from the previous group) had the \(RMSE_{\text{mod}}\) value exceeding the established limit. This was due to the desire to ensure--apart from the required accuracy--that the result was independent of the adopted weighting scheme and counteract the phenomenon of overfitting. The approximation that would meet the criterion limit value was found not for all the sets of trial points in the experiment. Often, none of the 480 approximations for a given series of discrete data met the error condition--points with an ordinate greater than \(10^{-2}\) in Figure 14a. It could be caused by (1) the insufficient number of analyzed functions, (2) the insufficient number of trial points or (3) too narrow interval of their examination. The first reason is justified by considering only polynomial approximation. The best results obtained (Figure 14b) were significantly worse than without this restriction (Figure 14a). Additionally, in most cases, the assumed threshold of 1% of the relative error \(\max\Delta_{\text{rel,m}}\) was exceeded (Figure 14b).