# An Octahedric Regression Model of Energy Efficiency on Residential Buildings
## 1 Introduction
\begin{table}
\begin{tabular}{c c} \hline \hline
**Machine Learning Techniques** & **Papers** \\ \hline Artificial Neural Networks (ANN) & [2; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19] \\ Decision Trees (DT) & [20] \\ Support Vector Regression (SVR) & [11; 13; 14; 19; 21; 22; 23] \\ Random Forest (RF) and Trees Ensemble & [8; 13; 14; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29] \\ Multi-Layer Perceptron (MLP) & [14; 20; 23; 27] \\ Gaussian Mixture Model (GMM) & [30] \\ Gradient Boosted Regression Trees (CBRT) & [24; 31] \\ Extreme Learning Machine (ELM) & [10; 32; 33] \\ Linear Regression (LR) & [10; 13; 21; 23; 27; 34; 35] \\ Radial Basis Function Networks (RBFN) & [10; 12; 19; 23; 27] \\ Hybrid models & [10; 11; 19; 22; 28; 36; 37] \\ Multivariate Adaptive Regression Splines (MARS) & [10; 19; 23] \\ Gaussian Processes Regression (GPR) & [18; 21; 22; 27; 38] \\ k-Nearest Neighbor (k-NN) & [12; 23] \\ Genetic and Evolutionary Techniques (GT) & [14; 17; 18; 39; 40] \\ Others & [11; 17; 18; 21; 27; 28; 35; 41; 42; 43; 44; 45] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Machine learning (ML) techniques used to model the reference [8] dataset. Given a differential equation defined by a differential operator, \(D\),  
\[D(f)=v, \tag{5}\]  
where \(f,vlev,\)\(V\) being a function space, the finite element method replaces \(V\) by a finite dimensional subspace \(V_{h}\subset V\), which is composed by continuous piecewise polynomial functions of degree K (Sobolev space), associated with a division of the domain, \(\left[0,1\right]^{d}\), where the problem is defined in \(N_{e}\) parts called elements \(\Omega_{i}\). \[\left[0,1\right]^{d}=\cup_{i=1}^{N_{e}}\Omega_{i}, \tag{6}\]  
The problem to solve is now  
\[D(f_{h})=v_{h}\text{where}f_{h},v_{h}eV_{h}, \tag{7}\]  
Let us consider a basis for the functions of \(V_{h}\), with \(\text{dim}V_{h}=Q\)  
\[V_{h}=\left\langle\varphi_{1}(x),\varphi_{2}(x),\ldots\ldots,\varphi_{Q}(x) \right\rangle, \tag{8}\]  
satisfying the next conditions at a set of Q points called nodes \(\left(\zeta_{1},\zeta_{2},\ldots,\zeta_{Q}\right)\)  
\[\varphi_{i}\left(\zeta_{j}\right)=\delta_{ij}, \tag{9}\]  
The functions of the basis are called shape functions and they are used to interpolate in points different from the nodes. Thus, any function \(f\) can be developed in terms of the basis with components \(u_{i}\) taking the form  
\[f_{h}(x)=\sum\
olimits_{i=1}^{Q}u_{i}\cdot\varphi_{i}(x), \tag{10}\]  
According to Equation (9), the sum is restricted to the nodes that determine the element which contains the point \(x\). Several conditions can be defined to get the optimum values of \(u_{i}\) as an approximation for the solution to the differential equation. The selection of the nodes is a crucial point for the precision of the results. The operation that constructs the set of nodes and elements is called meshing. The resulting mesh can be locally characterized by a parameter related to the size of each element and the error of its approximated solution. The numerical regression model for the relation Equation (3) consists in the set formed by the nodal values \(\left\{u_{i}\right\}\), which is called a representation model for the system. Then, the value of the relationship at any point can be estimated using the expression  
\[f(x)\cong f_{h}(x)=\sum\
olimits_{i=1}^{Q}u_{i}\cdot\varphi_{i}(x), \tag{11}\]  
For simplicity in the subsequent calculations, let us use a mesh formed by regular hyper-cubic elements with edge length \(h\). Introducing the complexity \(c\) as the number of elements in each dimension,\(h=1/c\), the total number of elements is \(c^{d}\), and the total number of nodes is \(\left(c+1\right)^{d}\). The key point in FEM-based methods is the determination of the values \(\left\{u_{i}\right\}\) considering the definition of an error function and its minimization. That problem is equivalent to solve a linear system of size, \(\left(c+1\right)^{d}\).