# Linear and Nonlinear Modelling of the Usable Area of Buildings with Multi-Pitched Roofs
## 2 Materials and Methods
The first stage of modelling was focused on estimating the parameters of the linear multiple regression model, as (1):  
\[\mathbf{A_{U}}=\mathbf{a_{0}}+\sum_{t=1}^{m}\mathbf{a_{i}}\cdot\mathbf{X_{i}} \tag{1}\]  
where \(A_{U}\) is the dependent variable, usable area of the building, \(a_{0}\) is the intercept of the linear regression model, \(a_{i}\) are linear regression model parameters, \(X_{i}\) are independent variables (Table 1), \(m\) is the number of independent variables. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature** & **Variable** & **Design Projects** &
\begin{tabular}{c} **Existing Buildings** \\ **(Koszalin)** \\ \end{tabular} \\ \hline Usable area & \(A_{U}\) & 92.9–302.42 m\({}^{2}\) & 80.71–407.51 m\({}^{2}\) \\ Covered area & \(A_{C}\) & 86.65–290 m\({}^{2}\) & 100.5–394.71 m\({}^{2}\) \\ Number of storeys & \(S_{N}\) & 1–2 & 1–2 \\ Height & \(H\) & 6.64–9.57 m & 5.7–10.17 m \\ Knee wall’s height & \(h\) & 0–1.8 m & No data \\ Garage area & \(G_{A}\) & 0–44.63 m\({}^{2}\) & 0–66.25 m\({}^{2}\) \\ Number of rooms & \(R\) & No data & 4–12 \\ Number of roof surfaces & \(R_{S}\) & 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) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Features of 191 residential buildings from architectural offices and 48 existing houses in Koszalin with a range of values in the dataset. In this model (1), it was assumed that each of the analyzed variables have uniform effects on \(\mathbf{A_{U}}\). The quality of the model fit was then verified using a coefficient of determination adjusted for the number of degrees of freedom. The presence of outliers that could negatively affect model fit was also analyzed, distorting the disclosure of real relationships. The next step was to gradually eliminate outliers, starting with the most diverse, while controlling for the increasing coefficient of determination. At the same time, variable \(B\) (boiler room), which was to be statistically insignificant for the model, was removed. The model fit was significantly improved after removing this variable and 4% of outlier observations (8 cases from the database of architectural offices, describing 191 buildings). The adjusted coefficient of determination increased from the original 87% to 95%, marking a significant improvement in the quality of the model's fit. A series of scatterplots were then created to illustrate the dependence of \(\mathbf{A_{U}}\) on each independent variable. The plots include lines estimated using the least squares method, which show the relationship between individual independent variables and \(A_{U}\). These figures allow visualization of the shape of the impact of individual variables on the usable area of the building and influence the decision to select the mathematical function \(f_{i}\), which describes the dependence of \(A_{U}\) on individual independent variables. The aim of this process, described in detail in [41], was to create a nonlinear model in an additive form (2) that would reflect the relationships between variables better than linear regression. \[\mathbf{A_{U}}=\mathbf{a}_{0}+\sum_{i=1}^{m}\mathbf{f}_{i}(\mathbf{X}_{i}) \tag{2}\]  
where \(A_{U}\) is the dependent variable, usable area of the building, \(\mathbf{a_{0}}\) is the intercept of the nonlinear regression model, \(X_{i}\) are the independent variables of the model (Table 1), \(m\) is the number of independent variables of the model, \(f_{i}\) is the dependency function \(A_{U}\) from \(X_{i}\) selected on the basis of a scatterplot \(A_{U}(X_{i})\). The parameters of the nonlinear model were then estimated. As in the case of linear models, an analysis of outliers was performed during the estimation. Gradually, outliers and variables that were irrelevant were removed individually, controlling the quality of the model fit at each stage.