## 2 Materials and Methods
### Linear and Non-Linear Modelling  
During the first and second phases of the analysis, the parameters of both multiple linear and non-linear regression models were estimated. Linear models took on the form:  
\[A_{U}=a_{0}+\sum_{i=1}^{m}a_{i}\cdot X_{i} \tag{1}\]  
while non-linear models were of the form:  
\[A_{U}=a_{0}+\sum_{i=1}^{m}f_{i}(X_{i}) \tag{2}\]  
where \(A_{U}\)--the explained variable, usable floor area of the building;  
\(a_{0}\)--the constant term in the equation;  
\(a_{i}\)--linear regression model coefficients;  
\(X_{i}\)--explanatory variables of the model, listed in Table 1;  
\(m\)--the number of explanatory variables in the model;  
\(f_{i}\)--a function of the dependence of \(A_{U}\) on \(X_{i}\), selected on the basis of the scatterplot of \(A_{U}\) vs. \(X_{i}\). In model (1), it is hypothesised that each explanatory variable has a constant effect on the explanatory variable \(A_{U}\). However, in order to optimally select the form of the non-linear model in the first step of the procedure for estimating its parameters, a series of scatter plots was made, showing the dependence of \(A_{U}\) on each explanatory variable. Similar to the article [1], the objective of this step was to build a non-linear model (2) which would approximate the relationships between the buildings' features more accurately than a linear regression model does. The detailed procedure can be found in [43]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature** & **Variable** & **Design Projects** & **Existing Buildings (Koszalin)** \\ \hline Usable area & \(A_{U}\) & 48.12–302.42 m\({}^{2}\) & 80.71–407.51 m\({}^{2}\) \\ Covered area & \(A_{C}\) & 69.8–334.68 m\({}^{2}\) & 100.5–394.71 m\({}^{2}\) \\ Number of storeys & \(S_{N}\) & 1–2 & 1–2 \\ Height & \(H\) & 5.7–9.57 m & 5.7–10.17 m \\ Knee wall’s height & \(h\) & 0–1.8 m & No data \\ Garage area & \(G_{A}\) & 0–56.55 m\({}^{2}\) & 0–66.25 m\({}^{2}\) \\ Number of rooms & \(R\) & No data & 4–12 \\ Number of roof surfaces & \(R_{N}\) & No data & 4–15 \\ Presence of a boiler room & \(B\) & 0; 1 & No data \\ Presence of an in-built garage & \(G_{P}\) & No data & 0; 0.5; 1 \\ (0.5–attached or partially in-built) & \(R_{S}\) & 23.8–40\({}^{\circ}\) & No data \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Features of 436 residential buildings from architectural offices and features of 48 existing buildings with their respective ranges of values in the dataset. The quality of the model's fit to the data was verified using the coefficient of determination. This coefficient was then tested for its significance and corrected for the number of degrees of freedom of the model. The presence of outliers, which have the capacity to reduce the model's fit and distort the true relationships, was also analysed. It was assumed that the number of such observations should not exceed approximately 10% of the total set. The next step in strengthening the model was to progressively eliminate the statistically insignificant components of the multivariate model, starting with the least significant, while monitoring the increasing coefficient of determination and changing the significance level of the remaining components. Among the non-significant explanatory variables were \(h\) (knee-wall height) and \(B\) (boiler room) in both types of models. Following the elimination of the variables \(h\), \(B\) and 26 outlier observations (10%) in the linear model, there was a substantial improvement in the model's fit: the adjusted coefficient of determination increased from the original 74% to 88%. In the case of the non-linear model, 28 outliers (11%) and 4 of the 10 model components had to be removed, which eliminated the same two explanatory variables (\(h\) and \(B\)), to achieve an 89% fit of the model to the data, after the initial 75%. The need to remove a larger number of outlier cases in the non-linear model indicates its slightly poorer fit compared to the linear model, thus confirming the conclusions of earlier analyses reported in [1].