## 3 Results
### Non-Linear Regression  
Only two of the graphs of the dependence of the \(A_{U}\) variable (floor area of the building) on individual independent variables (Figure 2a,b) reveal a non-linear character. Based on these, the forms of mathematical functions \(f_{i}\) for the given explanatory variable were selected, from which a multivariate additive non-linear regression model was built. The forms of the tested \(f_{i}\) functions, building the model (2), are given in Table 3. Polynomial functions of degrees 1, 2 and 3 and the exponential function were tested. For the bivariate variable (\(S_{N}\)--number of storeys), a linear function was the only one that could be considered. The initial acceptable form of the additive multivariate non-linear regression model took the Formula (11):  
\[ \begin{split}& A_{U}=a_{0}+b_{1}\cdot X_{1}+b_{2}\cdot X_{2}+c_{1} \cdot X_{3}+c_{2}\cdot X_{3}^{2}+\\ &+d_{1}\cdot X_{4}+d_{2}\cdot X_{4}^{2}+d_{3}\cdot X_{4}^{3}+b_{3 }\cdot X_{5}+b_{4}\cdot X_{6}+b_{5}\cdot X_{7}\end{split}  \tag{11}\]  
where \(A_{U}\)--the explained variable, usable floor area of the building;  
\(a_{0}\)--the constant term;  
\(X_{i}\)--independent variables, listed in Table 3;  
\(b_{i}\), \(c_{i}\), \(d_{i}\)--model parameters. As with the linear model, parameter estimation of the non-linear model was carried out several times, gradually removing outliers or irrelevant variables. Table 4 shows \begin{table}
\begin{tabular}{c c} \hline \hline
**Independent Variable** & **Function Type** \\ \hline Covered area—\(A_{C}\) & linear function \\ Building’s height—\(H\) & linear function \\ Garage area—\(G_{A}\) & exponential function \\  & polynomial 2° \\ Boiler room—\(B\) & linear function \\  & polynomial 3° \\ Knee–wall’s height—\(h\) & polynomial 3° \\ Number of storeys—\(S_{N}\) & linear function \\  & foot slope —\(R_{S}\) & linear function \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Function types to describe the dependence of the usable area on individual exogenous variables. Figure 2: Relationship between the usable area (A\({}_{\text{U}}\)) variable and selected explanatory variables: garage area (a) and boiler room (b). the results of parameter estimation of the final form of non-linear multiple regression, considered optimal, built on the basis of 217 observations, of Formula (12):  
\[A_{U}=a_{0}+b\cdot A_{C}+c\cdot H+d_{1}\cdot G_{A}+d_{2}\cdot G_{A}^{2}+g\cdot S_ {N}+h\cdot R_{S} \tag{12}\]  
Twenty-eight cases (11%) were considered to be the outliers, half of which appeared as outliers in the linear model, as well as thus supporting the conclusion that they are truly incompatible with the set analysed. The model predicts the value of a building's floor area based on variables \(A_{C}\), \(H\), \({G_{A}}^{2}\), \(G_{A}\), \(S_{N}\), and \(R\)s. An almost identical set of variables was found to be significant in our previous study [1]. Then, however, instead of the \(R_{S}\) variable, the set of significant variables included the \(h\) variable. The value of the \(R^{2}\) is 0.893; its adjusted value is slightly smaller and equals 0.890. This means that as much as 89% of the variance in the response variable is explained by the model. A small difference between \(R^{2}\) and adjusted \(R^{2}\) confirms that the model remains stable even after taking into account the relationship between the amount of data and the number of variables. The fit of the model to the data is not weaker than that of the linear model. The final model includes explanatory (exogenous) variables with the parameters being statistically significant (last two columns of Table 4). The model shows that most of the buildings' characteristics have high predictive power (statistically significant parameters at \(p\)-value \(<\) 0.05), except for the variable \(S_{N}\), where the parameter can also be considered significant (but at \(p\)-value of 0.10). In the remainder of this paper, the model is referred to as model \(D_{N}\).