## 3 Tree-Based Entropy Structured Self-Organizing Maps--Tessom
### TESSOM Result Interpretation
To assess the participation of each variable during training, we can use the graph in Figure 5. In this figure, it is noted that in the case of neuron 4, there were equal contributions from the two variables. For the other neurons, the contribution varied, with _Attribute2_ having the greatest influence. Using the heat map shown in Figure 6, it is possible to identify which range of values of each variable contributed to each neuron. Neuron 4 was more influenced by values for _Attribute1_ equal to or above 7, and for _Attribute2_, the values were equal to or above 3. This represents that objects whose values are in this range belong to Class 3 only. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Neurons** & **2** & **\% 2** & **3** & **\% 3** & **Total** & **Entropy** \\ \hline
1 & 15 & 83.333\% & 3 & 16.667\% & 18 & 0.650022 \\
2 & 24 & 58.537\% & 17 & 41.463\% & 41 & 0.97887 \\
3 & 11 & 37.931\% & 18 & 62.069\% & 29 & 0.957553 \\
4 & 0 & 0.000\% & 12 & 100.000\% & 12 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Entropies and numbers of objects—level 1. Figure 4: Object overlap per neuron and Voronoi representation—Synthetic database. The analysis proceeds to neuron 2, identified as the neuron with the greatest overlap according to the entropy calculated in Table 1. It is also possible to verify by the percentages presented that this neuron has much overlapping. The analysis was performed in the same way as for the first model. The neuron 2 data subset allowed the generation of a new SOM model, which produced the values presented in Table 2. In this model, neuron 4 had the lowest entropy, presenting one object from the minority class and five from the majority class. However, since Equation (1) considers the relationship between the classes, the value remains high, at 0.650022. Neuron 2 showed the highest level of entropy, at 0.985228. The distribution map of the objects in the figure and the Voronoi diagram of Figure 7 demonstrate the distribution of the objects in this model. For the Voronoi diagram, the initial \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Neurons** & **2** & **\% 2** & **3** & **\% 3** & **Total** & **Entropy** \\ \hline
1 & 7 & 63.636\% & 4 & 36.364\% & 11 & 0.94566 \\
2 & 8 & 57.143\% & 6 & 42.857\% & 14 & 0.985228 \\
3 & 8 & 80.000\% & 2 & 20.000\% & 10 & 0.721928 \\
4 & 1 & 16.667\% & 5 & 83.333\% & 6 & 0.650022 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Entropies and volumes of objects—level, 2 neuron 2. Figure 5: Attribute distribution—synthetic database. Figure 6: Variable heatmap—synthetic database. scale was kept between 0 and 1 to make the comparison between the images more visual, and as SOM does, it identified the neuron objects through vector proximity, through Euclidean distance, from the neighborhood with the identified prototypes. When analyzing the contributions of the variables in the neurons of this model (Figure 8), we see something different from what happened with the level 1 model. The neuron that presented the lowest entropy obtained the approximation of the objects mainly via combination between high values for _\"Attribute1\"_ and low values for _\"Attribute2,\"_ and the variable with higher entropy had similar participation by both variables. The heat maps of this model (Figure 9) indicate that, for neuron 4, values above 6.2 for \"Attribute1\" and less than 2.7 for \"Attribute2\" were the values that made the entropy of this neuron small. As for neuron 2, what contributed were values above 6.2 for the variable _\"Attribute1\"_ and between 2.8 and 2.9 for the variable _\"Attribute2.\"_  
Figure 8: Variable distribution—synthetic database, level 2, neuron 2. Figure 7: Object overlap per neuron and Voronoi representation—synthetic database, level 2, neuron 2. The analysis can continue in this way in all neurons, according to the pattern or behavior that one wants to identify. In larger databases, it is possible to identify features in smaller groups that would otherwise not be identified by other classification algorithms.