# Linear and Nonlinear Modelling of the Usable Area of Buildings with Multi-Pitched Roofs
## 4 Discussion  
In the case of the architectural design dataset, both linear and nonlinear models showed a very similar degree of fit and simultaneously eliminated the same number of outliers (4%) in subsequent iterations of parameter estimation. Nonlinear models offer advantages in the analysis of more complex dependences. However, in this case, the nonlinear model, although theoretically more flexible, did not bring clear benefits. By analyzing the results, one can determine the importance of proper data processing before training the models. Atypical buildings, which are few in the random sample, cause a significant reduction in the accuracy of usable area estimates. Eliminating the knee wall variable in a nonlinear and linear model resulted in a reduction in the accuracy of predictions by only about 0.2%. This information is beneficial for property appraisers estimating the usable area of a building, as accurate measurement of this variable without entering the building is challenging and prone to significant error. However, on the other hand it may be crucial to know other parameters of the last storey instead, e.g., the slope of the roof planes can contribute to proper estimation of the overall usable area and it may be easier to collect. This will be further investigated in our future works. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Linear Models with LASSO Regularization**} \\ \multicolumn{1}{c}{\multirow{2}{*}{_Model_}} & \multicolumn{1}{c}{\multirow{2}{*}{_Model_}} & \multirow{2}{*}{_Model_} & \multirow{2}{*}{_Model_} & \multirow{2}{*}{_Model_} & \multirow{2}{*}{_Model_} \\ \cline{3-3} \cline{6-7}  & \(\lambda=0.1\) & & & & & \\ \hline MAE & 29.71 & 28.95 & 28.14 & 24.40 & 21.98 & 20.44 \\ RMSE & 41.99 & 40.00 & 37.99 & 30.20 & 25.80 & 24.16 \\ \hline \hline \end{tabular}
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Table 10: Quality of the estimation of the usable area for the validation test (\(N=5\)) for the existing buildings data. Source: own calculation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Linear Models with LASSO Regularization**} \\  & \(Model\) & \(Model\) & \(Model\) & \(Model\) & \(Model\) & \(Model\) \\  & \(\lambda=0.05\) & \(\lambda_{min}=0.25\) & \(\lambda=0.5\) & \(\lambda=1\) & \(\lambda_{1 se}=1.95\) & \(\lambda=5\) \\ \hline MAE & 27.04 & 27.40 & 27.87 & 28.73 & 29.56 & 30.71 \\ RMSE & 33.71 & 33.58 & 33.49 & 33.60 & 33.81 & 34.13 \\ \hline \hline \end{tabular}
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Table 9: Quality of the estimation of the usable area for the validation test (\(N=10\)) for the design data. Source: own calculation. In the case of the analysis of existing buildings in Koszalin, the linear and nonlinear models predicted the usable area similarly. The nonlinear multiple regression model, which was estimated after removing five outliers (10%), achieved a reliable value of \(R^{2}\) = 0.933. The best linear model was based on a set without six outliers (12.5%) and included only two significant variables (\(A_{c}\) and \(S_{N}\)), which explained 91.5% of the variability in usable area. The difference in fit (<2%) is not substantial, and both models demonstrated a high and dependable alignment with the empirical data. In a nonlinear model, there is a greater number of variables that were important. In addition to \(A_{c}\) and \(S_{N}\), \(G_{A}{}^{2}\), \(G_{P}{}^{2}\), and \(G_{P}\) were also significant. From a practical perspective, the linear model is more advantageous because it is inherently simpler and here it additionally operates with a smaller number of variables. The simplicity of this model is advantageous in that it allows for easier interpretation of the results and estimation of the usable area in practice. We have stated above that if the nonlinear model is only slightly better fitted than the linear one, we prefer and advise the use of the linear model. However, whether the linear model will prove to be better than the nonlinear one cannot be reliably predicted beforehand and that is why we do evaluate nonlinear models as well.