## 2 Materials and Methodology
### Artificial Intelligence Method
\[\varphi_{\mathrm{ij}}=\exp\left\{\frac{-\left(U_{\mathrm{i}}-m_{\mathrm{ij}} \right)^{2}}{\sigma_{\mathrm{ij}}^{2}}\right\}\mathrm{i}=1,\ldots,N_{1}; \mathrm{j}=1,\ldots,N_{2}, \tag{1}\]  
The input number is \(N_{1}\); \(N_{2}\) is the total number of nodes in the intermediate layer; the center of the membership function is shown as \(m_{\mathrm{ij}}\); \(\sigma_{\mathrm{ij}}\) is the width of the membership function, \(U_{\mathrm{i}}\) is the i-th input, \(\varphi_{\mathrm{ij}}\) is the output neuron of the i-th input and the output connected to the j-th node of the intermediate layer. Figure 4: Structure of the GNF system. Figure 3: Flow chart of methodology. Equation (2) shows the outputs for the second layer, which corresponds to the rule system. Equation (3) is the general output. \[\epsilon_{\rm i}=\min\bigl{|}\varphi_{\rm ij},\ \varphi_{\rm 2j},\ldots,\varphi_{\rm N _{\rm ij}}\bigr{|}\,{\rm j}=1,\ldots,{\rm N}_{2}, \tag{2}\]  
\[{\rm Y_{k}}=\frac{\sum_{\rm j=1}^{\rm N_{2}}s\vartheta_{\rm jk}\epsilon_{\rm j} }{\sum_{\rm j=1}^{\rm N_{2}}\epsilon_{\rm j}}{\rm k}=1,\ldots,{\rm N}_{3} \tag{3}\]  
where \({\rm N}_{3}\) is the number of GNF outputs, and \(s\vartheta_{\rm jk}\) is the estimated value of the k-th output from the j-th node. As Equations (1)-(3) show, there is a correlation between the width and center of the membership functions, the estimated system outputs and the number of nodes of the intermediate layer. To establish these parameters, a learning process is developed in three phases, where in the first two phases the initial values for the parameters are established and the number of nodes in the hidden layer is optimized. Finally, in the third phase these values are modified to obtain better results. In the first phase, known as the Unsupervised Learning Phase, a [PERSON]'s self-organizing [24] map provides the initial values for \({\rm m_{ij}}\) and \(s\vartheta_{\rm jk}\). The inputs are shown in Equation (4). \[{\rm V}=\bigl{[}{\rm U_{1}}\ldots{\rm U_{N_{1}}}{\rm Y_{1}}\ldots{\rm Y_{N_{ 3}}}\bigr{]} \tag{4}\]  
The vector \([{\rm U_{i}}]\) represents the input values and \([{\rm Y_{k}}]\) is the desired output of the GNF system. At the end of this unsupervised learning phase, the winning node is found and the center of the membership functions (\({\rm m_{ij}}\)) is obtained, thus allowing the system outputs (\({\rm s\vartheta_{jk}}\)) to be estimated. This stage represents the first assignment, since these values will be improved during the ensuing learning phases. Once \({\rm N_{2}}\), \({\rm m_{ij}}\), and \(s\vartheta_{\rm jk}\) are established, in the second phase a GNF is built in order to fix values for the parameters \(\sigma_{\rm ij}\). Furthermore, an optimization process is also carried out with the purpose of minimizing the number of rules, that is, of reducing the number of nodes in the hidden layer. This phase uses the genetic process [21; 22; 25] inspired by the biology prototypical of genetic evolution. In this analogy, there is an individual with elementary data (in this case, a vector), and there are genes (in this research, vector components) that represent the hidden nodes by a Boolean parameter and the width of the membership functions. Next, a fitness function is determined by considering the variation between the real outputs and the individual one. Once this stage is concluded, adequate individual values for \(\sigma_{\rm ij}\), and a satisfactory set of rules (nodes on the hidden layer) are determined. The last phase, known as the Supervised Learning Phase, has the purpose of improving the values set for the \({\rm m_{ij}}\), \(\sigma_{\rm ij}\), and \(s\vartheta_{\rm jk}\) parameters. As was pointed out previously, the system proposed in this paper is comparable to a three-layer neural network; therefore, the same equation to describe the input nodes is used as for the neurons in a radial basis neural network [26]. Additionally, the least mean squared learning algorithm is also employed.