# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 2 Materials and Methods
From Equation (29),  
\[c_{*}\left(x_{0}\pm\frac{h}{2}\cdot e_{i}\right)\cdot W^{0}\!\left(x_{0}\pm\frac{ h}{2}\cdot e_{i}\right)=\int_{\Omega}f(x)\cdot\Phi\!\left(\|x-(x_{o}\pm\frac{h}{2} \cdot e_{i})\|,\omega\right)dx, \tag{33}\]  
Developing \(W^{0}\) and \(\Phi\) around \(x_{o}\) up to second order in \(h\),  
\[W^{0}\!\left(x_{o}\pm\frac{h}{2}\cdot e_{i}\right)=W^{0}(x_{o})\pm\partial_{i} W^{0}(x_{o})\cdot\frac{h}{2}+\frac{h^{2}}{8}\cdot\partial_{i}^{2}W^{0}(x_{o})+O \!\left(h^{3}\right)\!, \tag{34}\]  
\[\Phi\!\left(\|x-x_{o}\mp\frac{h}{2}\cdot e_{i}\|,\omega\right)=\Phi\!\left(\|x -x_{o}\|,\omega\right)\mp\partial_{i}\Phi\!\left(\|x-x_{o}\|,\omega\right) \cdot\frac{h}{2}+\frac{h^{2}}{8}\cdot\partial_{i}^{2}\Phi\!\left(\|x-x_{o}\|, \omega\right)+O\!\left(h^{3}\right)\!, \tag{35}\]  
Equation (33) takes the form  
\[ \begin{split}\left[W^{0}(x_{o})\pm\partial_{i}W^{0}(x_{o})\cdot \frac{h}{2}+\frac{h^{2}}{8}\cdot\partial_{i}^{2}W^{0}(x_{o})\right]\cdot c_{*} \!\left(x_{o}\pm\frac{h}{2}\cdot e_{i}\right)=\\ =\int_{\Omega}f(x)\Bigg{[}\Phi\!\left(\|,x-x_{o}\|,\omega \right)\mp\partial_{i}\Phi\!\left(\|,x-x_{o}\|,\omega\right)\cdot\frac{h}{2}+ \frac{h^{2}}{8}\cdot\partial_{i}^{2}\[PERSON]\!\left(\|,x-x_{o}\|,\omega\right) \Bigg{]}dx,\end{split}  \tag{36}\]  
That is,  
\[ \begin{split}\left[W^{0}(x_{o})\pm\partial_{i}W^{0}(x_{o})\cdot \frac{h}{2}+\frac{h^{2}}{8}\cdot\partial_{i}^{2}W^{0}(x_{o})\right]\cdot c_{*} \!\left(x_{o}\pm\frac{h}{2}\cdot e_{i}\right)=\int_{\Omega}f(x)\cdot\[PERSON]\! \left(\|x-x_{o}\|,\omega\right)dx\mp\frac{h}{2}\cdot\\ \int_{\Omega}f(x)\cdot\partial_{i}\Phi\!\left(\|x-x_{o}\|,\omega \right)dx+\frac{h^{2}}{8}\cdot\int_{\Omega}f(x)\cdot\partial_{i}^{2}\Phi\! \left(\|x-x_{o}\|,\omega\right)dx,\end{split}  \tag{37}\]  
Summing negative and positive components of the support on dimension \(i\),  
\[ \begin{split} W^{0}(x_{o})\cdot\left[c_{*}\!\left(x_{o}+\frac{h}{ 2}\cdot e_{i}\right)+c_{*}\!\left(x_{o}-\frac{h}{2}\cdot e_{i}\right)\right]+ \frac{h}{2}\cdot\partial_{i}W^{0}(x_{o})\cdot\left[c_{*}\!\left(x_{o}+\frac{h }{2}\cdot e_{i}\right)-c_{*}\!\left(x_{o}-\frac{h}{2}\cdot e_{i}\right)\right] \\ +\frac{h^{2}}{8}\cdot\partial_{i}^{2}W^{0}(x_{o})\cdot\left[c_{*} \!\left(x_{o}+\frac{h}{2}\cdot e_{i}\right)+c_{*}\!\left(x_{o}-\frac{h}{2} \cdot e_{i}\right)\right]=\\ =2\cdot c_{*}(x_{o})\cdot W^{0}(x_{o})+\frac{h^{2}}{4}\cdot\int_ {\Omega}f(x)\cdot\partial_{i}^{2}\Phi\!\left(\|,x-x_{o}\|,\omega\right)dx, \end{split}  \tag{38}\]  
Summing now for every dimension index \(i\) and dividing by \(2\cdot d\):  
\[ \begin{split} W^{0}(x_{o})\cdot\hat{z}(x_{o})+\frac{h}{4\cdot d} \cdot\sum_{i}^{d}\partial_{i}W^{0}(x_{o})\cdot\left[c_{*}\!\left(x_{o}+\frac{h }{2}\cdot e_{i}\right)-c_{*}\!\left(x_{o}-\frac{h}{2}\cdot e_{i}\right)\right]+ \frac{h^{2}}{16\cdot d}\\ \cdot\sum_{i}^{d}\partial_{i}^{2}W^{0}(x_{o})\cdot\left[c_{*} \!\left(x_{o}+\frac{h}{2}\cdot e_{i}\right)+c_{*}\!\left(x_{o}-\frac{h}{2}\cdot e _{i}\right)\right]\\ =c_{*}(x_{o})\cdot W^{0}(x_{o})+\frac{h^{2}}{8\cdot d}\cdot\sum_ {i}^{d}\int_{\Omega}f(x)\cdot\partial_{i}^{2}\Phi\!\left(\|x-x_{o}\|,\omega \right)dx\end{split}  \tag{39}\]  
Therefore, the octahedric regression is  
\[ \begin{split}\hat{z}(x_{o})=c_{*}(x_{o})-\frac{1}{4\cdot d\cdot W ^{0}(x_{o})}\cdot\left[\ln\cdot\sum_{i}^{d}\partial_{i}W^{0}(x_{o})\cdot\left[c _{*}\!\left(x_{o}+\frac{h}{2}\cdot e_{i}\right)-c_{*}\!\left(x_{o}-\frac{h}{2} \cdot e_{i}\right)\right]+\frac{h^{2}}{4}\\ \cdot\sum_{i}^{d}\partial_{i}^{2}W^{0}(x_{o})\cdot\left[c_{*}\! \left(x_{o}+\frac{h}{2}\cdot e_{i}\right)+c_{*}\!\left(x_{o}-\frac{h}{2} \cdot e_{i}\right)\right]-\frac{h^{2}}{2}\cdot\int_{\Omega}f(x)\cdot\sum_{i}^{d} \partial_{i}^{2}\Phi\!\left(\|,x-x_{o}\|,\omega\right)dx\end{split}