# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 3 Results  
The fit of a model can be measured using different parameters. If \(y_{i}\) and \(\overline{y_{i}}\) are the observed and estimated values,  
1. Mean squared error: \(MSE=\frac{1}{p}\cdot\sum_{i=1}^{p}(y_{i}-\overline{y_{i}})^{2}\)
2. Mean absolute error: \(MAE=\frac{1}{p}\cdot\sum_{i=1}^{p}\bigl{|}y_{i}-\overline{y_{i}}\bigr{|}\)
3. Mean absolute percentage error: \(MAPE=\frac{100}{p}\cdot\sum_{i=1}^{p}\Bigl{|}\frac{y_{i}-\overline{y_{i}}}{y_ {i}}\Bigr{|}\)
4. Regression error characteristic (REC) curve: It is a curve obtained plotting the error tolerance on the X-axis versus the percentage of points predicted within the tolerance on the Y-axis.  
As a case study, the energy efficiency dataset [8] from the UCI Machine Learning Repository [7] has been selected. The data corresponds to the input values of two simulations generated by the software Ecotect [58] from Autodesk, San Rafael (CA) US, representing the heating and cooling loads necessary to achieve comfortable indoor conditions. The dataset variables are represented in Table 2.  
The studied buildings are generated using 18 cubes with a side length of 3.5 m, forming 12 different building forms with equal volume but different areas and dimensions. Each side can act as different elements (wall, floor, roof, or window) with different thermodynamic properties. The different combinations of values allowed for the model can be consulted with more detail in reference [8]. As a result of the process, 768 buildings are simulated, and the results for both simulations (heating and cooling cases) are presented in the dataset.  
The quality parameters obtained for the learning techniques used in reference [8] are shown in Table 3. Iteratively reweighted least squares (IRLS) is a method used to diminish the effect of outliers in classical linear regression [59], while RF stands for random forest.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Input Variables** & **Output Variables** \\ \hline Relative compactness & Heating load \\ Surface area & Cooling load \\ Wall area & - \\ Roof area & - \\ Overall height & - \\ Orientation & - \\ Glazing area & - \\ Glazing area distribution & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input and output variables for the energy efficiency problem.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Model** & **MAE** & **MSE** & **MAPE** \\ \hline \multirow{2}{*}{Heating load} & IRLS & \(2.14\pm 0.24\) & \(9.87\pm 2.41\) & \(10.09\pm 1.01\) \\  & RF & \(0.51\pm 0.11\) & \(1.03\pm 0.54\) & \(2.18\pm 0.64\) \\ \multirow{2}{*}{Cooling load} & IRLS & \(2.21\pm 0.28\) & \(11.46\pm 3.63\) & \(9.41\pm 0.80\) \\ \multirow{2}{*}{} & RF & \(1.42\pm 0.25\) & \(6.59\pm 1.56\) & \(4.62\pm 0.70\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of quality for models of energy efficiency dataset in reference [8].  
Given the huge development of research references based on this dataset, the median and minimum value for each applied algorithm of machine learning is shown on Table 4, where the bold text represents the best case of each quality parameter.