# Linear and Nonlinear Modelling of the Usable Area of Buildings with Multi-Pitched Roofs
## 2 Materials and Methods  
Reliable and complete data are necessary to estimate acceptable functional, linear, and nonlinear models or to effectively train machine learning models to estimate the usable area of buildings. Therefore, to build our models, we used data available online from the Lipinscy [37] and Archon [38] architectural offices. These datasets provide full access to building designs and detailed interior plans, which allows for accurate analysis and minimization of possible errors in models. The standard used to calculate the usable area of buildings is also known. This allows us to focus more precisely on residential spaces, which are key in property valuation. Using available designs makes it easier to understand potential problems with building performance, while also helping to prevent these errors. Data from architectural offices included 191 residential buildings. The analysis did not exclude any areas, e.g., boiler rooms or garages, even though some areas were not included in the usable area of the building. The design parameters were adopted without changes. The parameters of buildings from design offices are presented in Table 1.  
The second dataset includes residential buildings in Koszalin, a city located in northern Poland, about 15 km from the Baltic Sea. The selected buildings were built between 2020 and 2023 in different architectural styles. The building parameters are also presented in Table 1. The parameters of these buildings were provided by the office of the District Inspector of Building Supervision, which supervises compliance of the parameters of the constructed building with the design. After the construction is completed, authorized surveyors measure the building and prepare a report on its geometric parameters, which they send to the District Building Supervision Inspector in a given district. In the absence of data on the usable area of the building in the REPR, property appraisers in Poland can use the BDOT10K database when estimating the value of a building [28]. This database contains information from airborne LIDAR laser scanning. Data from the Geoportal for the city of Koszalin was downloaded in the CityGML 2.0 standard and accessed using QGIS [39]. Additionally, Google Street View was used to provide the missing information about the number of floors in the building. LiDAR scanning in Poland was performed at two levels of detail (LoD1 and LoD2). LoD1 scanning was performed with a measurement point density of 4 points/m\({}^{2}\), omitting the geometry of the roof, and containing only building blocks. LOD2 scanning provided a much higher density of measurement points (12 points/m\({}^{2}\)). LOD2 includes roof structures and simple additional developments, such as garages and various additions to the building. The lower accuracy of LOD1 is a limitation, as possible errors in estimating the height of a building can reach up to 50% [35]. The accuracy of LiDAR scanning for the whole Poland is binary; Western Poland (including Koszalin) has accuracy at the LOD1 level, and Eastern Poland has accuracy at the LOD2 level [40]. Our study consisted of three parts. In part one we used linear regression, in part two we used nonlinear regression, and in part three we additionally used linear regression with LASSO-type regularization. In each part, we looked for the best model to predict the value of the floor area variable for both sets of data described above. The modelling results in each part were then compared to identify the best model found for each of the datasets. The model parameters obtained were appropriate for the datasets used and, arguably, for other data, the results may differ significantly. However, the approach itself seems general enough to be a universal proposal for modelling missing data in databases describing properties.