# Predicting Heating Load in Energy-Efficient Buildings Through Machine Learning Techniques
## 3 Model Development
### Multi-Layer Perceptron Regressor (MLPr)  
The MLP is a widely-used and well-known predictive network. Accordingly, MLPr aims to coordinate the best potential of regression between a set of data samples (shown here in terms of S). The MLPr divides the S into both of the set training and testing databases. An MLP involves several layers of computational nodes. Similar to many previous MLPr-based studies, a single hidden layer  
\begin{table}
\begin{tabular}{l|l|l|l|l|l|l|l|l|l} \hline \hline  & \multicolumn{3}{c|}{**Data Layers Used as Input**} & \multicolumn{3}{c|}{**Main**} \\ \cline{2-10}  & \multicolumn{1}{c|}{**Relative Compactness**} & \multicolumn{1}{c|}{**Surface Area**} & \multicolumn{1}{c|}{**Wall Area**} & \multicolumn{1}{c|}{**Roof Area**} & \multicolumn{1}{c|}{**Overall Height**} & \multicolumn{1}{c|}{**Orientation**} & \multicolumn{1}{c|}{**Glazing Area**} & \multicolumn{1}{c|}{**Glazing Area**} & \multicolumn{1}{c|}{**Heating Load**} \\  & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} \\ \hline Used label & X\({}_{1}\) & X\({}_{2}\) & X\({}_{3}\) & X\({}_{4}\) & X\({}_{5}\) & X\({}_{6}\) & X\({}_{7}\) & X\({}_{8}\) & Y\({}_{1}\) \\ \hline No. of data & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} \\ \hline Minimum & 0.6 & 514.5 & 245.0 & 110.3 & 3.5 & 2.0 & 0.0 & 0.0 & 6.0 \\ \hline Maximum & 1.0 & 808.5 & 416.5 & 220.5 & 7.0 & 5.0 & 0.4 & 5.0 & 43.1 \\ \hline Average & 0.8 & 691.7 & 318.5 & 176.6 & 5.3 & 3.5 & 0.2 & 2.8 & 22.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The statistical description details in term of energy-efficient design.  
Figure 3: Schematic view of some of the output data layers (i.e., heating load).  
was used. This is because even with a single hidden layer and increasing the number of nodes in the hidden layer an excellent rate of prediction can be achieved. Figure 4 shows a common MLP structure. The optimum number of neurons in each of the hidden layer are obtained after a series of trial and error processes (i.e., sensitivity analysis) as shown in Figure 5. Noteworthily, only one hidden layer was selected since the accuracy of a single hidden layer was found to be high enough to not make the MLP structure more complicated.  
Each node generates a local output. In addition, it sets the local output to the subsequent layer (the next nodes in a further hidden layer) until reaching the nodes of output, i.e., the nodes placed in the layer of output. Equation (1) shows the normal operation carried out considering a dataset of \(N\) groups of records by the \(j\)th neuron to compute the predicted output:  
\[O_{j}=F\left(\sum\
olimits_{n=1}^{N}I_{n}W_{nj}+b_{j}\right), \tag{1}\]  
where \(I\) symbolizes the input, \(b\) denotes the bias of the node, \(W\) is the weighting factor, and \(F\) signifies the activation function. Tansig (i.e., the tangent sigmoid activation function) is employed (Equation (2)). Note that we can have several types of activation functions (e.g., (i) sigmoid or logistic; (ii) Tanh--Hyperbolic tangent; (iii) Relu--rectified linear units) and that their performances are best suitable for different purposes. In the specific case of the sigmoid, this function (i) is real-valued and  
Figure 4: Multi-layer perceptron regressor (MLPr) neural network typical architecture.  
Figure 5: Sensitivity analysis based on number of neurons in a single hidden layer.  
differentiable (i.e., to find gradients); (ii) has analytic tractability for the differentiation operation; and (iii) is an acceptable mathematical representation biological neuronal behavior.  
\[Tansig(x)=\frac{2}{1+e^{-2x}}-1 \tag{2}\]