# Linear and Nonlinear Modelling of the Usable Area of Buildings with Multi-Pitched Roofs
## 3 Results
### Linear and Nonlinear Models Based on the Existing Buildings in Koszalin
#### 3.3.4 Linear Models with LASSO Regularization  
We have also performed a LASSO regularization in our analysis. The method is applied to linear models (cf. Formula (5)). The results are summed up in Tables 9 and 10, for the design dataset and for the existing houses, respectively. The method allows for a selection of exogenous variables and one of our best regularized models indeed has the list of explanatory variables shortened to only three positions. \begin{table}
\begin{tabular}{c c c c c c|c c c} \hline \multicolumn{6}{c|}{**Linear Models**} & \multicolumn{6}{c}{**Nonlinear Models**} \\  & \multicolumn{1}{c}{_Model A_} & \multicolumn{1}{c}{_Model A_\({}^{+}\)**} & \multicolumn{1}{c}{_Model B_} & \multicolumn{1}{c}{_Model B_\({}^{+}\)**} & \multicolumn{1}{c}{_Model A_\({}_{n}\)**} & \multicolumn{1}{c}{_Model B_\({}_{n}\)**} & \multicolumn{1}{c}{_Model C_\({}_{n}\)**} & \multicolumn{1}{c}{_Model D_\({}_{n}\)**} \\ \hline MAE & 26.99 & 23.19 & 30.74 & 25.31 & 29.30 & 30.23 & 26.58 & 26.31 \\ RMSE & 34.39 & 30.22 & 37.22 & 32.24 & 35.63 & 35.27 & 32.72 & 32.38 \\ \hline \end{tabular} where _Model A_ included variables \(A_{C}\), \(H\), \(G_{A}\), \(B\), \(I_{H}\); _Model A_\({}^{+}\) added variable \(S_{N}\); _Model B_ included variables like in Model A with one outlier removed; _Model B_\({}^{+}\) was variable B substituted with variable \(S_{N}\); eight outliers removed; _Model A_\({}_{n}\) included variables \(A_{C}\), \(H^{2}\), \(H\), \(G_{A}\), \(B\), \(I_{H}^{+}\); _Model B_\({}_{n}\) included variables \(A_{C}\), \(H^{2}\), \(G_{A}\), \(I_{H}\), five outliers removed; _Model C_\({}_{n}\) included variables \(A_{C}\), \(H\), \(G_{A}\)\({}^{2}\), \(G_{A}\), \(I\), \(S_{N}\), eight outliers removed; _Model D_\({}_{n}\) included variables \(A_{C}\), \(H\), \(G_{A}\)\({}^{2}\), \(G_{A}\), \(S_{N}\), eight outliers removed.  
\end{table} 
Table 7: Quality of the estimation of the usable area for the validation test (\(N=10\)) for the design data. Source: own calculation. \begin{table} \begin{tabular}{c c c|c c} \hline \multicolumn{3}{c|}{**Linear Models**} & \multicolumn{3}{c}{**Nonlinear Models**} \\  & _Model A_ & _Model B_ & _Model C_ & _Model A_\({}_{n}\) & _Model B_\({}_{n}\) \\ \hline MAE & 23.60 & 28.20 & 20.95 & 21.54 & 26.23 \\ RMSE & 29.70 & 39.32 & 27.92 & 29.95 & 38.02 \\ \hline \end{tabular}  where _Model A_ included variables \(A_{C}\), \(A_{R}\), \(S_{N}\), \(H\), \(R_{S}\), \(G_{P}\); _Model B_ included variables like in Model A, three outliers removed; _Model C_ included variables \(A_{C}\), \(S_{N}\), six outliers removed; _Model A_\({}_{n}\) included variables \(A_{C}\), \(A_{C}\), \(G_{A}\)\({}^{2}\), \(G_{A}\), \(R^{2}\), \(R\), \(S_{N}\), \(H\), \(R_{S}\)\({}^{3}\), \(R_{S}\), \(G_{P}\)\({}^{2}\), \(G_{P}\); _Model B_\({}_{n}\) included variables \(A_{C}\), \(G_{A}\)\({}^{2}\), \(S_{N}\)As we see in Table 9, the best predictions were obtained from the model minimizing the cross-validation error (\(\lambda_{min}=0.25\)), at least according to the \(RMSE\) measure. Hence, this model can be considered the best in his class. It is given by  
\[A_{U}=-134.98+1.06\cdot A_{C}+14.89\cdot H-0.68\cdot G_{A}-0.38\cdot B+8.20 \cdot h+8.44\cdot S_{N} \tag{9}\]  
After comparing with the previous models, we can conclude that the best regularized model for the design dataset is worse in terms of prediction accuracy than Model A\({}^{+}\) described above. Similarly, we estimated regularized models for the existing buildings dataset. Table 10 suggests that the best of the models obtained is not the model minimizing the penalty nor the model with \(\lambda_{min}\) or \(\lambda_{1 se}=\lambda_{min}+\sigma_{\lambda}\), but it is built with a larger value of \(\lambda\).