## 3 Structure
### Classification
#### SOM (Self-Organizing Maps)  
A Self-Organizing Map (_SOM_) model has two layers of neurons. The input layer, with (51*S2) neurons, one for each input data, and the output layer, formed by \(\rho\) neurons. The output layer stores the information and forms the feature map. The information is propagated from the initial or input layer to the final layer or output data. Each neuron \(i\) of the initial layer is connected through the weights \(\Omega_{i,j}\) with each of the neurons \(\eta^{*}\xi\) of the final layer or output. This product will be developed in the remainder of this section. In this type of neural network, we use competitive learning, which means that the layer of neurons modifies its weights so that these weights are increasingly more _similar_ than the input data. The competitive learning concept presents the adaptive weight strategy, allowing one to incorporate knowledge at a specific neuron and the related modification at the surrounding neurons. These weights are called BMU (best matching unit), which, in our case, will be stored in the matrix \(\Omega\). Each neuron in the output layer has as input the data vector modified by the weight vector \(\Omega\). Once this is completed, _competence_ consists of comparing the weight vectors with the input data \(\Delta_{1,1}^{x}\) and verifying which will be the closest among them using the algebraic expression:  
\[{{dif}_{i,j}=\parallel\Omega_{i,j}-\Delta_{1,1}^{x}\parallel} \tag{7}\]  
We will concretely develop this equation concerning the work followed in previous sections. One of the fundamental characteristics in generating the map is the approximation of neighboring neurons to the winning neuron, but on a two-dimensional basis, which, in our case, is represented by a Gaussian equation, where \(c_{i,j}^{2}\) is the related variance. \[{{h}_{i,j}=exp^{-\frac{{dif}_{i,j}^{2}}{{\sigma}_{i,j}^{2}}}} \tag{8}\]  
\[{{\phi}_{i,j}=\sqrt{{{h}_{i,j}}}} \tag{9}\]  
Therefore, concerning the modification of weights seen as a learning stage, a classical equation is proposed for the modification of weights. \[{{\Omega}_{i,j}=\Omega_{i,j}-\alpha*{{\varphi}_{i,j}}*((\psi_{i,l}^{x},\sigma_ {i,l}^{x})-\Omega_{i,j})} \tag{10}\]  
where:  
\(\alpha\) is the learning rate or learning factor, which will be a value in the interval [0, 1];  
\(\Omega\) is the weight matrix;  
\(\Delta\) is the training matrix that is the product of the formation of the Approximations and the Details seen in the previous sections;  
\(i\), \(j\) corresponds to the iterations concerning the indices of the initial formation of the _information cube_ based on Equation (1). Since the multidimensional integration of the information in this processing mechanism is on an _information cube_ that expands into several cubes depending on the number of levels generated by the _wavelets_ and the decomposition from a local sliding window, it is of interest to have the tracking of the index as a consequent plane given in \(\theta\), as well as the indices expressed in Equations (A6)-(A8) and (3) (Appendixs B and C). In this way, each data point backed up in \(\sigma\) and \(\psi\) has the following indexation:  
\[\sigma_{i,j}^{\kappa} \tag{11}\] \[\psi_{i,j}^{\kappa} \tag{12}\]  
where:  
\(\kappa\): the index of deepening planes based on Equation (1);  
\(j\): index of each data point in the vector resulting from the processing of the _Pooling_;  
\(i\): index on the column in the current plane resulting from processing the _Pooling_. This results in the following equation, which is in the operation of each scalar in \(\Lambda\)  
\[dif_{i,j}=\|\Omega_{i,j}-\Delta_{1,1}^{\kappa}\|=\sqrt{(\omega_{i,j,1}-\psi_{ 1,1}^{\kappa})^{2}+(\omega_{i,j,2}-\sigma_{1,1}^{\kappa})^  
\[1\leq i\leq\eta\] \[1\leq j\leq\psi\] \[1\leq i\leq S1\] \[1\leq l\leq S2\]  
Now, to implement the algorithm called SOM, in this case, we build a tuple of values for the dimension of the cube to be processed and incorporate the data into these for the training processing.