# Linear and Nonlinear Modelling of the Usable Area of Buildings with Multi-Pitched Roofs
## 3 Results  
### Linear Regression Based on Building Designs  
The first stage of modelling was the estimation of linear regression parameters for estimating the usable area of buildings based on a dataset from architectural offices. The initial dataset consisted of 191 one- and two-storey residential buildings with multi-pitched roofs. Figure 1 shows the relationship between the predicted and observed values for the dependent variable: the usable area (AU) of the building, for the last form of the linear model, considered to be optimal, after rejecting 8 outliers and resigning from the most insignificant variable--the boiler room (B). The goodness-of-fit of this model to the data is at the level of 95%.  
Each blue point represents a specific observation, with the horizontal axis (X) showing the usable area values predicted by the model and the vertical axis (Y) showing the actual observed values. The red solid and dashed lines are the regression line and the boundaries of the 95% confidence interval, respectively. Overlapping of the points and the regression line indicates the high reliability of the model. Most of the points are close to the regression line, which visually confirms that the model fits well with the data. The red line has a slope of about 45 degrees, which confirms that the predicted values are close to the observed values. The greater the deviation of the points from the regression line, the greater the estimate's error.  
Table 2 presents detailed results of multiple regression analysis for a sample of the final 183 observations, described by five independent variables (selected from Table 1). The values confirm the model being well fitted to the data (the coefficient of determination R2  
Figure 1: Values observed in relation to the estimated usable area—circles. Red line is the trend line. The dashed line indicates the 95% confidence interval. Source: own calculation.  
with value 0.949 and adjusted R2 with value 0.947) and indicate that 95% of the variability of the usable area of the building is explained by the five independent variables. The variables covered area (\(A_{C}\)), number of storeys (\(S_{N}\)), and height (\(H\)) have a positive and significant impact on the usable area, which means that an increase in these values results in an increase in the usable area. The \(\beta\) weight (standardized regression coefficient), corresponding to the covered area, is as high as 1.105 and is the strongest estimator of the projected value of the usable area. The independent variable garage area (\(\mathrm{G_{A}}\)) has a negative effect on \(A_{U}\), which may be due to the fact that GA is not considered a usable area. The last two columns of Table 2 present the results of the significance test of the regression model parameters, namely the values of the test function \(t\) and the \(p\)-value; these confirm the high significance of all variables.  
Figure 2 illustrates the relative contribution of five different factors on a building's usable area. The covered area is the most dominant factor (72%), which is consistent with the results presented in Table 2.