# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 2 Materials and Methods
 \tag{40}\]Taking as radial function given by Equation (26),  
\[\[PERSON](\|x\|,\omega)=e^{-\frac{\|x\|}{\omega}}\rightarrow\partial_{i}\[PERSON](\|x\|, \omega)=-\frac{1}{\omega}\cdot\[PERSON](\|x\|,\omega)\cdot\partial_{i}\|x\|=-\frac{1 }{\omega}\cdot\[PERSON](\|x\|,\omega)\cdot\frac{x^{i}}{\|x\|}, \tag{41}\]  
So, for points far from hypercube's boundary, by symmetry,  
\[\partial_{i}W^{0}(x)=\partial_{i}\int_{\Omega}\[PERSON](\|t-x\|,\omega)dt=\int_{ \Omega}\partial_{i}\[PERSON](\|t-x\|,\omega)dt=\frac{1}{\omega}\cdot\int_{\Omega} \[PERSON](\|t-x\|,\omega)\cdot\frac{(t-x)^{i}}{\|t-x\|}dt=0, \tag{42}\]  
Equation (40) becomes  
\[ \begin{split}\hat{z}(x_{0})=c_{*}(x_{o})+\frac{h^{2}}{16\cdot d \cdot W^{0}(x_{o})}\cdot\left[2\cdot\int_{\Omega}f(x)\cdot\sum\
olimits_{i}^ {d}\partial_{i}^{2}\Phi(\|x-x_{o}\|,\omega)\,dx-\sum\
olimits_{i}^{d}\partial _{i}^{2}W^{0}(x_{o})\cdot\\ \cdot\left[c_{*}\left(x_{o}+\frac{h}{2}\cdot e_{i}\right)+c_{*} \left(x_{o}-\frac{h}{2}\cdot e_{i}\right)\right]\right],\end{split}  \tag{43}\]  
That is, octahedric regression is a correction of order \(h^{2}\) to the weigthed average for central objective points. As a consequence, when \(h\to 0\), both values tend to coincide \(\hat{z}(x_{o})\to c_{*}^{s}(x_{o})\). The following definition and propositions are related with the behaviour of the estimations with relation to the experimental error distributions, considered as normal and uncorrelated over the problem domain. **Definition 8**. : _Given a function \(f(x)\) defined on a domain \(\Omega\) and a random field \(e(x)\sim N(0,\sigma(x))\) defined on \(\Omega\), an experimental realisation of associated with a sample \(e^{S}(x)\) of \(e(x)\) is a function \(y^{s}:\Omega\rightarrow\mathbb{R}\), given by_  
\[y^{s}(x)=f(x)+e^{S}(x). \tag{44}\]  
**Proposition 1**. : The expected value (denoted as,\(E[\ldots]\)) of the weighted averaged regression corresponding to the values \(y^{s}\) in any point \(x_{o}\) is  
\[E[c_{*}^{s}(x_{o})]=c_{*}(x_{o}). \tag{45}\]  
**Proof. ** Following the Equation (27), the regression of the experimental realisation is  
\[\int_{\Omega}[c_{*}^{s}(x_{o})-y^{s}(x)]\cdot\Phi(\|x-x_{o}\|,\omega)dx=0, \tag{46}\]  
Using Equation (44), the last expression can be written as  
\[\int_{\Omega}[c_{*}^{s}(x_{o})-f(x)]\cdot\[PERSON](\|x-x_{o}\|,\omega)\,dx=\int_{ \Omega}e^{s}(x)\cdot\[PERSON](\|x-x_{o}\|,\omega)\,dx. \tag{47}\]  
Therefore, by Equation (27) and using the definition (Equation (29)) of \(W^{0}(x_{o})\),  
\[(c_{*}^{s}(x_{o})-c_{*}(x_{o}))\cdot W^{0}(x_{o})=\int_{\Omega}e^{s}(x)\cdot \[PERSON](\|x-x_{o}\|,\omega)\,dx. \tag{48}\]  
Under the conditions of definition 8, \(E\Big{[}e^{S}(x)\Big{]}=0\), and the expected value of Equation (48) is  
\[E\big{[}c_{*}^{s}(x_{o})-c_{*}(x_{o})\big{]}\cdot W^{0}(x_{o})=0. \tag{49}\]So,  
\[E[c_{*}^{\mathcal{S}}(x_{o})]=c_{*}(x_{o}). \tag{50}\]  
\(\Box\)  
**Proposition 2. ** Let \(e(x)\) a distribution not correlated for different points of \(\Omega\), \(E\big{[}e^{\mathcal{S}}(x)\cdot e^{\mathcal{S}}(y)\big{]}=o^{2}(x).\delta(x-y)\). Then, the variance of the weighted averaged regression corresponding to the values \(y^{\mathcal{S}}\) is given by  
\[Var(c^{\mathcal{S}}(x_{o})-c_{*}(x_{o}))\cdot\big{(}W^{0}(x_{o})\big{)}^{2}= \int\limits_{\Omega}o^{2}(x)\Phi^{2}(\|x-x_{o}\|,\omega)dx. \tag{51}\]  
**Proof. ** Taking Equation (48) to the square and calculating expected values,  
\[E\big{[}(c_{*}^{\mathcal{S}}(x_{o})-c_{*}(x_{o}))^{2}\big{]}\cdot \big{(}W^{0}(x_{o})\big{)}^{2} \tag{52}\] \[=\int\limits_{\Omega}\int\limits_{\Omega}E\Big{[}e^{\mathcal{S}}( x)\cdot e^{\mathcal{S}}(y)\Big{]}\cdot\Phi(\|x-x_{o}\|,\omega)\cdot\Phi(\|y-x_{o}\|, \omega)dxdy\] \[E[c_{*}^{s}(x_{o})]=c_{*}(x_{o}). \tag{45}\]0  
That is,  
\[E[c_{*}^{s}(x_{o})]=c_{*}(x_{o}). \tag{45}\]1  
\(\Box\)  
Now, the computational algorithm is presented with some considerations relative to its implementation.