## 3 Criteria of Response Function Selection
### Step Two--Accuracy of the Probabilistic Moments  
In the formula for the Cornell reliability index, there are only the expected value and the variance. However, the description of the distribution only with their use is not sufficient because the values of these parameters may be similar for extremely different data, as in the case of the Anscombe quartet [74] (Figure 5). Taking into account the additional values of the third and fourth central moments--and the skewness and kurtosis based on them--allowed indicating numerical differences between such data (Table 4). It should also be noted that reliability can be ultimately described by indicators whose formula takes into account not only the expected value and the variance [4; 5; 6; 7; 8]. This is another confirmation of the statement that the compliance of values should also apply to some additional parameters describing the distribution. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Base Polynomial Type** & **Polynomial Form** & \(\boldsymbol{\Delta_{\text{rel},\text{c}}}\) \\ \hline reference & \(1.000000\cdot 10^{1}+1.00000\cdot 10^{-2}x-1.000000\cdot 10^{-2}x^{2}\) & – \\ approximation for \(y_{1}\) & \(1.000024\cdot 10^{1}+\ 0.954347\cdot 10^{-2}x-0.977856\cdot 10^{-2}x^{2}\) & 4.57 \\ approximation for \(y_{22}\) & \(1.000579\cdot 10^{1}-0.153796\cdot 10^{-2}x-0.425345\cdot 10^{-2}x^{2}\) & 115.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results of the first part of the experiment assuming \(D1=7\) and \(POb=POa=2\) on the example of dependences for which the greatest relative errors of the coefficients were obtained. In the further part of the experiment, four probabilistic moments were considered: the first raw moment--marked as \(m1\)--equivalent to the expected value is given in Equation (1), while the central moments from the second upwards--marked from \(m2\) to \(m4\)--are described in Equation (3). The approximation of the discrete data to \(D1=7\) significant digits was adopted because it is sufficient for groups of functions based on the \begin{table}
\begin{tabular}{c c c c c} \hline  & **Set I** & **Set II** & **Set III** & **Set IV** \\ \hline
**Expected value of \(\xi\)** & & & 9.0 (identical) & \\
**Variance of \(\xi\)** & & & 11.0 (identical) & \\
**Expected value of \(\eta\)** & & & 7.50 (with the accuracy of two decimal places) & \\
**Variance of \(\eta\)** & & & \(4.125\pm 0.003\) & \\
**Linear regression equation** & & \(\eta=0.500\xi+3.00\) (with an accuracy of three and two decimal places, respectively) & \\
**Linear correlation** & & & 0.816 (three decimal places match) & \\
**coefficient of** & & & 0.666 (three decimal places match) & \\
**determination** & & & \\
**Third central moment of \(\eta\)** & \(-0.406\) & \(-8.207\) & \(11.553\) & 9.384 \\
**Skewness of \(\eta\)** & \(-0.048\) & \(-0.979\) & \(1.380\) & \(1.121\) \\
**Fourth central moment of \(\eta\)** & \(30.7\) & \(42.3\) & \(72.1\) & 61.7 \\
**Kurtosis of \(\eta\)** & \(-1.199\) & \(-0.514\) & \(1.240\) & \(0.629\) \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical parameters for the four datasets from the Anscombe quartet (the most important differences are underlined). Figure 5: Graphical representation of the datasets from the Anscombe quartet: (**a**) Set I, (**b**) Set II, (**c**) Set III, (**d**) Set IV. linear base polynomials (max\(\Delta_{\text{rel,c}}\) below 1%), not sufficient for the second-order base polynomials (max\(\Delta_{\text{rel,c}}\) even above 100%, Table 5) and definitely insufficient for the fifth-order base polynomials (max\(\Delta_{\text{rel,c}}\) even above \(5\times 10^{8}\)). Therefore, in the further part of the experiment, only the results obtained on the basis of base polynomials with the abovementioned degrees were considered to verify whether, regardless of the possible accuracy of the coefficient values--which is also the resultant of the significant digits number--it is possible to obtain satisfactory accuracy of probabilistic moments measured by their relative error \(\Delta_{\text{rel,m}}\).