# Linear and Nonlinear Modelling of the Usable Area of Buildings with Multi-Pitched Roofs
## 3 Results
### Nonlinear Regression Based on Building Designs  
The construction of a nonlinear model began with the preparation of a series of scatterplots of the relationships between the dependent variable \(A_{U}\) (usable area) and the independent variables (Figure 3a-f). A regression line, estimated using the least squares method, was added on each of the graphs (except for the two-valued variable \(S_{N}\)) to illustrate the approximate type of \(A_{U}\) dependence on a particular independent variable. Based on the regression lines, the mathematical functions for the independent variables were selected, from which a total and subsequently a final nonlinear regression model was built. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(N\) = 183 & \(\beta\) & \(\sigma(\beta)\) & \(a\) & \(\sigma(a)\) & \(t(177)\) & \(p\)-Value \\ \hline intercept & & & \(-95.083\) & 9.043 & \(-10.514\) & \(<0.001\) \\ \(\mathrm{A_{C}}\) & 1.105 & 0.024 & 1.097 & 0.024 & 45.565 & \(<0.001\) \\ \(\mathrm{H}\) & 0.149 & 0.020 & 8.546 & 1.137 & 7.516 & \(<0.001\) \\ \(\mathrm{G_{A}}\) & \(-0.235\) & 0.025 & \(-0.549\) & 0.059 & \(-9.275\) & \(<0.001\) \\ \(\mathrm{h}\) & 0.073 & 0.019 & 5.909 & 1.530 & 3.863 & \(<0.001\) \\ \(\mathrm{S_{N}}\) & 0.200 & 0.021 & 13.058 & 1.368 & 9.542 & \(<0.001\) \\  & & \(R=0.974\)\(R^{2}=0.949\) Adj. \(R^{2}=0.947\); & & & \\  & & \(F(5,177)=658.15\)\(p<0.001\) SE: 6.9336 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Multiple linear regression estimates for the building designs dataset. Source: own calculation. Figure 2: Shares of independent variables in explaining the usable area of a building. Source: own calculation. The tested functions that build the model (2) are shown in Table 3. Polynomial functions of degrees 1, 2, and 3 were tested. For a binary variable, the only possible function was a linear function. Finally, the additive multivariate model of nonlinear regression was formed (6):  
\[ \begin{split} A_{U}=a_{0}&+b_{1}\cdot A_{C}+b_{2} \cdot A_{C}+c_{2}\cdot A_{C}^{2}+d_{2}\cdot A_{C}^{3}+b_{3}\cdot G_{A}+c_{3} \cdot G_{A}^{2}\\ &+b_{4}\cdot B+c_{4}\cdot B^{2}+d_{4}\cdot B^{3}+b_{5}\cdot h+c_{ 5}\cdot h^{2}+d_{5}\cdot h^{3}+b_{6}\cdot S_{N}\end{split}  \tag{6}\]  
Figure 3: (**a–f**) Dependence between usable area (\(A_{U}\)) and independent variables. The blue circles indicate the values of the variables. Source: own calculation. (**a**) \(A_{U}\) vs. \(A_{C}\) (covered area). (**b**) \(A_{U}\) vs. \(H\) (height). (**c**) \(A_{U}\) vs. \(G_{A}\) (garage area). (**d**) \(A_{U}\) vs. \(B\) (boiler room). (**e**) \(A_{U}\) vs. \(h\) (knee wall height). (**f**) \(A_{U}\) vs. \(S_{N}\) (number of stores). where \(A_{U}\) is the dependent variable, usable area of the building, \(a_{0}\) is the intercept of the nonlinear regression model, \(A_{C}\), \(H\), \(G_{A}\), \(B\), \(h\), \(S_{N}\) are the independent variables of the model (Table 3), \(b_{i}\), \(c_{i}\), \(d_{i}\) are model parameters. In the case of the nonlinear model, the analysis was performed similarly to the linear model. Parameter estimation was performed several times, which gradually eliminated the outliers and irrelevant independent variables. Table 4 presents the estimation of the parameters of the final form of nonlinear multiple regression considered to be optimal, which was based on 183 observations. Eight cases were considered to be the strongest outliers, seven of which also occurred as the strongest outliers in the linear model, which confirms their incompatibility with the analyzed dataset. The model estimates the value of the usable area of the building based on several independent variables (\(A_{C}\), \(H\), \(G_{A}\)2, \(G_{A}\), \(h\), \(S_{N}\)). The value of the coefficient of determination \(R^{2}\) was 0.951 and its adjusted value was slightly lower (0.949). This means that 95% of the variability of the dependent variable is reliably explained by the model, indicating the very high fit of the model.