## 3 Criteria of Response Function Selection  
### Assumptions and Selection of Reference Functions for Numerical Examples  
It was decided to consider here both the reference dependences and approximations having the following polynomial form:  
\[Y=w(X)=\sum_{i=0}^{n}D_{i}\cdot X^{i} \tag{10}\]  
where it was assumed that the quantities used above are functions of single variables, e.g., in order to use the LSM method:  
\[X=\varphi(x,y)\equiv\varphi(x) \tag{11}\]  
and  
\[Y=\psi(x,y)\equiv\psi(y) \tag{12}\]  
This leads to the following dependence:  
\[y=\psi^{-1}(w(\varphi(x))) \tag{13}\]  
where the function \(w\) is hereinafter called the base polynomial and the argument is a random variable \(x\) with the truncated Gaussian probability density distribution, which is characterized by the input coefficient of variation \(a(x)\)\(\in\)[0.0, 0.30] to explore a fairly wide three-sigma range (\(0.1E[x]\), \(1.9E[x]\)). The use of as mathematically simple as possible selected elementary functions used in empirical formulas in Equations (11) and (12) led to obtaining 48 groups of dependences based on polynomials. Their list is included in Table 1.  
Ultimately, they are to constitute performance functions describing extreme magnitudes of structures, e.g., deflections, displacements or stresses, with respect to a random parameter. This means that they only have to take positive values, which is the first limitation when it comes to choosing the reference base polynomials for numerical examples:  
\[\underset{x_{\max}\geq x\geq x_{\min}}{\forall}y>0 \tag{14}\]  
where  
\[x_{\min}=(1-3a_{\max}(x))E[x] \tag{15}\]  
and  
\[x_{\max}=(1+3a_{\max}(x))E[x] \tag{16}\]  
according to the three-sigma rule. It should be emphasized that the sought reference base polynomials are assumed to be used for each dependence from Table 1, therefore, taking into account all forms of the functions, Inequality (14) is reduced to the following:  
\[\underset{x_{\max}\geq x\geq x_{\min}}{\forall}w(\varphi(x))>1 \tag{17}\]  
which, due to the properties of the considered functions \(\varphi(x)\), is additionally simplified in the case of \(x>0\):  
\[\underset{x_{\max}\geq X\geq X_{\min}}{\forall}w(X)>1 \tag{18}\]  
where  
\[X=\varphi(x,y)\equiv\varphi(x) \tag{11}\]0and  
\[X=\varphi(x,y)\equiv\varphi(x) \tag{11}\]1  
Finding the exemplary coefficients for which Inequality (18) is satisfied was facilitated by narrowing down the interval \((X_{\min},X_{\max})\) in which the base polynomial was considered. Its smallest width occurs at \(E[x]\approx 1.14\), while in further calculations \(E[x]=x^{0}=1.0\) was assumed. With such a value, the variable \(x\) is considered in the interval (0.1, 1.9), which satisfies the constraint imposed by Inequality (18) so that \(x>0\). In order to be able to also easily assess the accuracy of the approximation visually, all coefficients values of the sought base polynomials are assumed to have the form \(1D_{i}\,|=10^{k}\), where \(k\in\mathbb{Z}\) and \(i\in\{0,\ldots,\,n\}\). Moreover, their value should be such that after substituting for all groups of functions, it should be possible to further use the generated discrete data in the computer algebra system so that they do not exceed the limits imposed on the numbers, which was not easy to achieve, especially for the dependency \(y_{30}\). It was decided to analyze the base reference polynomials up to the 6 th degree inclusive in order to obtain the appropriate level of their \"complexity\" and at the same time leave the possibility of approximation using the functions based on the base polynomials of much higher orders--up to the 10 th degree inclusive. Additionally, the possibility of obtaining a polynomial in which the influence of components derived from higher powers of the variable would clearly decrease--the polynomials only theoretically of a given degree--was avoided. The assumption was made that it should be similar to what can be schematically written as follows:  
\[X=\varphi(x,y)\equiv\varphi(x) \tag{11}\]2  
where \(j\in\{1,\ldots,\,n-1\}\).