## 3 Results
### Prediction of Usable Area in a Test Set for Linear, Non-Linear, and Regularised Models  
In the majority of applications of regression models, the most significant issue is their ability to accurately predict the value of the dependent variable for cases that did not occur in the sample used to estimate the model parameters. Consequently, the ultimate evaluation of the quality of the aforementioned models, which also serves as a comparison with other predictive models (see below), will be the results of the usable area modelling for a sample of 25 new observations. The characteristics of the models are summarised in Table 6. The table also presents the results of linear models with regularisation. The results of the machine learning (ML) models are presented in Table 7. Finally, the results obtained by hybrids of linear, non-linear and ML models are summarised in Table 8. The hybrid approach yielded the most accurate estimates of the usable floor area of the buildings under study. In the creation of predictive models for the other two datasets, a similar approach was adopted. Models for 191 building designs from the work [1] are compared with machine learning (ML) models on a sample of 10 new observations (see Table 9). Models for 48 existing buildings that were derived from the same source are compared with ML \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Linear Models**} & & & & **Non-Linear Model** \\ \multicolumn{2}{c}{_Model D_} & \multicolumn{1}{c}{_LASSO_} & \multicolumn{1}{c}{_Ridge_} & \multicolumn{1}{c}{_Elastic net\({}^{\star}\)_} & \multicolumn{1}{c}{_LASSO+_} & \multicolumn{1}{c}{_ModelD\({}_{N}\)_} \\ \multicolumn{2}{c}{_Model D_} & \multicolumn{1}{c}{_\(\lambda\) = 0.41\({}^{\star}\)_} & \multicolumn{1}{c}{_\(\lambda\) = 0.37_} & \multicolumn{1}{c}{_Elastic net\({}^{\star}\)_} & \multicolumn{1}{c}{_LASSO+_} & \multicolumn{1}{c}{_ModelD\({}_{N}\)_} \\ \hline MAE & 12.82 & 13.33 & 13.52 & 13.39 & 13.04 & 12.12 \\ RMSE & 15.09 & 16.15 & 16.23 & 16.20 & 15.67 & 14.09 \\ \(\max\lvert y-\hat{y}\rvert\) & 27.73 & 35.08 & 35.05 & 35.19 & 34.12 & 23.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Quality of the estimation of the usable area in the test group (N = 25) of the design data—best linear and non-linear models. Source: own calculation. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**ML Model**} & \multicolumn{1}{c}{**Neural Networks**} \\ \multicolumn{2}{c}{_XGBoost_} & \multicolumn{1}{c}{_NN6-(10-6)_} & \multicolumn{1}{c}{_NN5-(10-5)_} \\ \hline MAE & 12.61 & 11.89 & 11.15 \\ RMSE & 15.64 & 15.19 & 15.15 \\ \(\max\lvert y-\hat{y}\rvert\) & 34.09 & 33.83 & 37.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Quality of the estimation of the usable area in the test group (N = 25) of the design data—best ML (XGBoost and NN) models. Source: own calculation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Technique**} & \multicolumn{1}{c}{**Model Type**} & \multicolumn{1}{c}{**Design Data**} & \multicolumn{1}{c}{**Design Data**} & \multicolumn{1}{c}{**Existing Buildings**} \\ \hline Machine & XGBoost & XGBoost & XGBoost _ads_ & XGBoost _eb_ \\ learning & Neural networks & NN6-(10-6) & _NN5-(10-5)_ & _NN 6-(10-6)_ & _NN 6-(10-6)_ \\ \hline \multirow{6}{*}{Hybrid models} & & _LASSO A180D\({}_{N}\)_ & & \\  & & _XGB A180D_ & & \\  & & _XGB A180D_ & & \\  & & _XGB A180D_ & & _C+ LASSO_ \\  & & _NN5 90D\({}_{N}\)A180D_ & & \\  & & _NN6 90 A180D_ & & \\  & & _XGB + NN6_ & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._models obtained in the following subsections (see Table 10). This comparison was made on a sample of five observations due to the considerable difficulty in collecting new, reliable data with the same set of variables. The best of the linear models--model \(D\)--performed reasonably well in determining the usable floor area for new house designs.