# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 3 Results
### Heating Load Models
#### 3.1.2 Reduced Models  
Separated Models by Number of Floors  
A detailed revision of the model obtained in Section 3.1.1 shows two points that should be considered more in detail. First, the plots of Figure 1a-d,h-j,n,o,s,w-y show some kind of relationship between the involved variables. Considering that the vertical interfaces of the buildings are of floor or roof type (and given that the build unit is a cube, both variables must have the same value, resumed by _Floor_), the total surface can be written as  
\[S=Wall+2\cdot Floor, \tag{63}\]  
Also, given the definition of the relative compactness defined as in reference [60] taking the cube as elemental compactness measure,  
\[RC=6\cdot\frac{V^{\frac{2}{3}}}{S}, \tag{64}\]  
Equation (63) and Equation (64) introduce two constraints that jointly to the constant volume condition imposed by the algorithm of design of the buildings, reduce in two the independent or basic variables related with the building's geometry to be considered. In this study, these two variables will be the wall and the floor areas. Moreover, the overall height can be eliminated from the variable set because, given that all the basic cubic elements will have one lower side, if floor surface equals \(18\cdot 12.25=220.5\) m\({}^{2}\), overall height must be 3.5 m.  
Figure 6: Errors over independent variables. (**a**) Relative compactness; (**b**) Surface area; (**c**) Wall area; (**d**) Roof area; (**e**) Overall height; (**f**) Orientation; (**g**) Glazing area; (**h**) Glazing area distribution. Figure 7: Regression error characteristic (REC) curve for the heating load full restricted and null (Mean) models with \(h=\omega=0.033\). Selection of variables have been done, by example, in [28], where different models are compared for the sets of variables (roof area, overall height), (relative compactness, roof area, overall height), (relative compactness, surface area, wall area, roof area, overall height, glazing area) and the full dataset. In reference [18], the considered variables are (surface area, wall area, roof area, overall height, glazing area), while reference [35] uses (relative compactness, surface area, wall area, overall height, glazing area) to model the cooling problem. The importance of each variable is studied in [9] using ANOVA with the result of (relative compactness, surface area, wall area, overall height, glazing area) and (relative compactness, wall area, roof area, overall height, glazing area) as the most important variables for the heating and cooling problems, respectively. However, any extension to a more general case should consider if any of the constraints remain or can be ignored, introducing additional independent descriptors. So, the reduced set of variables considered in the modelling with the new methodology are those presented in Table 5.  
A second detail to consider is the results of Figure 5, especially the plot (e), which shows different behavior for the two values of the height, that physically would correspond to buildings with one and two floors. Moreover, form Figure 2, plots (d) and (e) show two groups for the values of the dependent variable, being this behaviour more remarkable in the case of the dependent variable height, corresponding to buildings with one and two floors. So, a separate study of two different models called 1_floor and 2_floors defined by the number of floors will be carried out (Table 6). In the case of one floor buildings, the roof area corresponds to 220.5 m 2, as was commented previously, so this variable can also be omitted. The results for the 1_floor normalised model are shown in Figure 8, for different values of h-parameters. The different behaviors of the relative error and mean absolute error (MAE) recommends the use of an intermediate value \(h=0.02\) for the selected model, whose results are shown on Figure 9.  
The 2_floors model results can be seen in Figure 10.