## 3 Results
### Machine Learning for Buildings' Designs
#### 3.5.2 Hybrid Models--Combining Regression with ML  
The best results in estimating the usable area were obtained using hybrid models, which combine features of the various models described above. This is not surprising, as the idea behind their use is to take advantage of specific, desirable features of the individual models. The models developed as part of this study were created with the following observations in mind:  
1. Values of the explanatory variable below 90 m\({}^{2}\) are very rare, so a threshold can be set below which the estimated values of the \(A_{U}\) variable will be replaced by a threshold value. 2. The baseline linear \(D\) and non-linear \(D_{N}\) models handle observations closer to the extreme values of the domain better than the regularised and ML models. In particular:
3. Both \(D\) and \(D_{N}\) models predict well the usable area values in the area of the domain referred to above (values lower than 90 m\({}^{2}\)), so it is possible to replace the estimated values of the \(A_{U}\) variable that are below the threshold with the values estimated by model \(D\) or \(D_{N}\),
4. Models \(D\) and \(D_{N}\) predict usable area values for buildings with a large covered area quite well; thus, a threshold can be found above which the ML model estimate will be replaced by the \(D\) or \(D_{N}\) model estimate (two thresholds were tested: 180 and 200 m\({}^{2}\))  
A hybrid model that outperforms its component models within specific domain subsets is obtained by defining the conditions for applying each component model according to the above list of observations. These conditions depend on the variant/value of selected features or the value of the explained variable. Additionally, a combination of a regression model (other than \(D\) and \(D_{N}\)) and a machine learning model can be introduced to achieve  
Figure 3: SHAP values of the individual features for XGBoost model in explaining the usable area of buildings. Source: own calculation, visualisation in SHAPforxgboost [51]. the desired properties. Several of such hybrids have been tested, and this approach yields promising results. Consequently, it should be explored in future research. As an example to illustrate the construction of this type of hybrid model, Figure 4 presents a schematic of the NN6 90 A180 DN model, which combines the best neural network model, the DN baseline model and a threshold value. In this part of the study, numerous combinations of models were tested. The most important results are summarised in Table 8. All the models presented here estimate the usable floor area more accurately than those previously described--at least according to some criteria (cf. Tables 6 and 7). The first of the hybrid models, and the only representative of the group of ordinary regression models, is a hybrid of the best model with LASSO regularisation (\(LASSO\)\(\lambda\) = 0.41) and the \(D_{N}\) model for observations where the building area (\(A_{C}\)) exceeds 180 m\({}^{2}\), denoted as \(LASSO\)\(A180D_{N}\). This hybrid of simple regression models has estimated usable areas in the test set more accurately than all previously presented models. Two models, based on the best gradient-boosted trees (which themselves did not receive the highest ratings), combined with models \(D\) and \(D_{N}\) in a domain part where houses have large covered areas (denoted as \(XGB\)\(A180D\) and \(XGB\)\(A180D_{N}\), respectively), provide highly accurate estimates of the usable floor area. The second model produced test set predictions that outperformed all other models, including hybrid models, except for the hybrid combining the best neural network and the non-linear \(D_{N}\) model for the same buildings, with an area covered exceeding 180 m\({}^{2}\). In the latter case, the \(NN6\)\(90\)\(A180D_{N}\) hybrid model achieved an average absolute error (MAE) of 8.73, meaning it made errors that were, on average, 28% smaller than those of the \(D_{N}\) base model, while inheriting its maximum error (cf. Table 7).