# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 2 Materials and Methods  
**Definition 1**. : _A parametrised radial function [56; 57], is a function \(\Phi:\mathbb{R}_{+}x\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) characterized by a parameter \(\omega\) that accomplishes the conditions_  
\[\left. \begin{array}{l}\forall\omega\
eq 0\lim_{\tau\rightarrow\infty}\Phi(r, \omega)=0\\ \forall r\
eq 0\lim_{\omega\to 0}\Phi(r,\omega)=0\end{array}  \right\}, \tag{25}\]An example of parametrised radial function is the exponential  
\[\Phi(r,\omega)=e^{-\frac{r}{\omega}}, \tag{26}\]  
The following definitions introduce the basic tools used afterwards to develop the methodology. **Definition 2**. : _Given a function \(f(x)\) defined on a domain, \(\Omega=\left[0,1\right]^{d},\) the weighted average regression of \(f(x)\) in a point \(x_{o}\), \(c_{*}(x_{o}),\) is a number implicitly defined as_  
\[\int_{\Omega}[c_{*}(x_{o})-f(x)]\Phi(\|x-x_{o}\|,\omega)dx=0, \tag{27}\]  
where \(\Phi(.,. )\) is a parametrised radial function. **Definition 3**. : _The J-th moment of the radial function \(\Phi\) is defined as_  
\[W^{l}(x_{0})=\int_{\Omega}\|x-x_{0}\|^{l}\cdot\Phi(\|x-x_{0}\|,\omega)dx, \tag{28}\]  
Using this expression, Equation (27) can be written as  
\[c_{*}(x_{o})\cdot W^{0}(x_{0})=\int_{\Omega}f(x)\cdot\Phi(\|x-x_{0}\|,\omega)dx, \tag{29}\]  
Definitions 4 and 5 introduce the concept of support of an interpolation to be used in the following definitions. **Definition 4**. : _Given a point \(x_{o}e\Omega\) and a set of \(Q\)\(d\)-dimensional vectors \(\{\zeta_{r}\}_{r=1}^{Q}\), with \(Q\geq d=dim(\Omega)\), the set of points obtained by the combinations \(\{x_{o}+\zeta_{r}\}_{r=1}^{Q}\subset\Omega\) is called a support for interpolation around \(x_{o}\)._  
**Definition 5**. : _An octahedric support for interpolation of size \(h\) around \(x_{0}\) is the support for interpolation around \(x_{0}\) given by the \(2\cdot d\) vectors_  
\[\left\{\frac{h}{2}\cdot e_{1},\frac{h}{2}\cdot e_{2},\ldots,\frac{h}{2}\cdot e _{d},-\frac{h}{2}\cdot e_{1},-\frac{h}{2}\cdot e_{2},\ldots,-\frac{h}{2}\cdot e _{d}\right\}, \tag{30}\]  
Definitions 6 and 7 present the main objective of the research in form of an interpolation of several radial function-based estimations. **Definition 6**. : _Given an octahedric support for interpolation of size \(h\) around a point \(x_{o}\) and a function \(f(x)\), the interpolated function \(\hat{f}(x)\) is defined as_  
\[\hat{f}(x)=\frac{1}{2d}\cdot\sum\
olimits_{i=1}^{d}\left[f\left(x_{o}+\frac{ h}{2}\cdot e_{i}\right)+f\left(x_{o}-\frac{h}{2}\cdot e_{i}\right)\right], \tag{31}\]  
**Definition 7**. : _Given a function defined on \(\Omega\), and a point \(x_{o}\in\Omega\), the octahedric estimation \(\hat{z}(x)\) of width \(h\) of \(f(x)\) at \(x_{o}\) is defined as the interpolation of the weighted averages on its octahedric support of size \(\mathrm{h}\), given by_  
\[\hat{z}(x_{o})=\frac{1}{2d}\cdot\sum\
olimits_{i=1}^{d}\left[c_{*}\left(x_{o} +\frac{h}{2}\cdot e_{i}\right)+c_{*}\left(x_{o}-\frac{h}{2}\cdot e_{i}\right)\right], \tag{32}\]  
where \(c_{*}(. )\) are the numbers defined in Definition 2. By similitude with the methods introduced in Section 1, the value of \(\frac{1}{h}\) is called pseudo-complexity. To study the meaning of the definition 7, let consider the case when \(h\) is small.