# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 3 Results
### Heating Load Models  
#### 3.1.1 Initial Dataset Variables  
The variables in reference [8] have been used in previous studies, but some considerations can be done with respect them previously to their inclusion in the proposed model. A more detailed analysis will be done in Section 3.1.2. The relationships between the variables can be seen in Figure 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model Type**} & \multicolumn{2}{c}{**Heating Load**} & \multicolumn{2}{c}{**Cooling Load**} \\ \cline{2-5}  & **MAE (kW)** & **MAPE (\%)** & **MAE** & **MAPE** \\ \hline LR & 2.09(0.122) & 21.43(3.9) & 2.27(1.64) & 6.94(4.6) \\ MLP & 0.35(0.25) & 1.62(1.20) & 0.72(0.39) & **2.34**(1.65) \\ ANN & 0.56(0.085) & 8.05(2.36) & 1.47(0.57) & 6.37(3.82) \\ k-NN & 1.94(1.75) & - & 1.88(1.17) & 4.95(**1.55**) \\ SVR & 0.56(0.236) & 2.26(1.13) & 1.76(0.59) & 3.47(2.65) \\ RBFN & 0.51(0.35) & 21.5(2.76) & 1.15(0.99) & - \\ RF and TE & 0.51(0.19) & 2.18(1.35) & 1.3(0.41) & 4.19(2.3) \\ ELM & **0.19**(**0.018**) & 13.65(9.56) & 0.98(0.24) & 17.87(8.02) \\ GBRT & 0.21(0.18) & - & **0.4**(0.31) & - \\ MARS & 0.53(0.08) & 2.57(2.20) & 1.12(0.15) & 11.66(4.09) \\ GPR & 0.35(0.18) & 11.97(1.40) & 0.93(0.45) & - \\ Hybrid & 0.62(0.037) & 2.28(1.56) & 0.68(**0.13**) & 3.1(2.45) \\ Genetic & 0.83(0.38) & **0.96**(**0.43**) & 0.97(0.52) & 3.4(1.92) \\ Other & - & 1.51(1.1) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Median and minimum MAE and MAPE for different Machine Learning techniques. Bold value represent the smallest value of each model’s quality parameter. Figure 1: _Cont._  
Figure 1: Cont. Figure 1: _Cont._  
Figure 1: Relationship between the independent variables for the cooling load problem. (**a**) Surface vs. Compactness; (**b**) Wall vs. Compactness; (**c**) Roof vs. Compactness; (**d**) Height vs. Compactness; (**e**) Orientation vs. Compactness; (**f**) Glazing area vs. Compactness; (**g**) Glazing distribution vs. Compactness; (**h**) Wall vs. Surface; (**i**) Roof vs. Surface; (**j**) Height vs. Surface; (**k**) Orientation vs. Surface; (**l**) Glazing area vs. Surface; (**m**) Glazing distribution vs. Surface; (**n**) Roof vs. Wall; (**o**) Height vs. Wall; (**p**) Orientation vs. Wall; (**q**) Glazing area vs. wall; (**r**) Glazing distribution vs. Wall; (**s**) Height vs. Roof; (**t**) Orientation vs. Roof; (**u**) Glazing area vs. Roof; (**v**) Glazing distribution vs. Roof; (**w**) Orientation vs. Height; (**x**) Glazing area vs. Height; (**y**) Glazing distribution vs. Height; (**z**) Glazing area vs. Orientation; (**aa**) Glazing distribution vs. Orientation and (**bb**) Glazing distribution vs. Glazing area. With respect to the dependent variable, the relationship between the heating load and the independent descriptors can be seen in Figure 2.  
An analysis of the data using octahedric regression using different values of h and \(\omega\) parameters gives a result that can be seen in Figure 3:  
Figure 2: Heating load values depending on (**a**) relative compactness; (**b**) surface area; (**c**) wall area; (**d**) roof area; (**e**) overall height; (**f**) orientation; (**g**) glazing area; (**h**) glazing area distribution. The selection of parameters h and d \(\omega\) can be done from the results shown in Figure 3. However, the total process can be accelerated diminishing the number of models to evaluate using a simple relationship between the parameters \(\omega\) and \(h\), as \(\omega=h\). This model is called normalized octahedric regression and increases the errors in a small magnitude from the best case of free selection for \(\omega\) and \(h\), but the model selection is easier, as can be seen in Figure 4.  
As was previously observed from Equation (61), full model tends to overfitting when \(h\to 0\), while restricted models tend to overestimate the stochastic errors.