## 4 Test Results and Discussion
Of the 82 classified samples, all were correctly assigned to their corresponding outputs (75 samples for normal operation--class 1, 4 samples for thermal failure--class 2, and 3 samples for electrical failure--class 3), as illustrated in Figure 10b and in the confusion matrix presented in Figure 11b. The results regarding the training and validation of the PRN network applied to 100% of the samples are shown in Figure 10c, as well as in the confusion matrix presented in Figure 11c. It is observed that, when using 10 neurons in the hidden layer, the network recorded only 14 errors, occurring exclusively in the training phase, resulting in a success rate of 97.8%. Overall, the accuracy rate achieved was 98%, slightly surpassing the 97.8% achieved with five neurons in the hidden layer. Both configurations proved to be effective as automatic models for classifying the operating conditions of power transformers based on the gases present inside them. Tables 3-5 present the weights of the connections between the input and hidden layers, from the hidden layer to the output, and the respective bias weights of the hidden and output layers for the network with 10 neurons in the hidden layer. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & \multicolumn{10}{c}{**Neurons Hidden Layer**} \\ \cline{2-11}  & & \(\mathbf{m_{1}}\) & \(\mathbf{m_{2}}\) & \(\mathbf{m_{3}}\) & \(\mathbf{m_{4}}\) & \(\mathbf{m_{5}}\) & \(\mathbf{m_{6}}\) & \(\mathbf{m_{7}}\) & \(\mathbf{m_{8}}\) & \(\mathbf{m_{9}}\) & \(\mathbf{m_{10}}\) \\ \cline{2-11}  & \(\mathbf{R_{1}}\) & \(0.0322\) & \(-0.3067\) & \(0.5928\) & \(1.3855\) & \(-0.4574\) & \(-1.2926\) & \(-0.9642\) & \(0.6219\) & \(5.0458\) & \(1.1364\) \\ \cline{2-11}  & \(\mathbf{R_{2}}\) & \(-1.2657\) & \(0.5935\) & \(0.3621\) & \(-0.3653\) & \(0.1491\) & \(1.3524\) & \(0.4590\) & \(0.9758\) & \(1.1697\) & \(-0.8334\) \\ \cline{2-11}  & \(\mathbf{R_{3}}\) & \(1.4485\) & \(-0.4754\) & \(3.0243\) & \(-1.4813\) & \(-1.6088\) & \(-0.6485\) & \(1.2424\) & \(1.9435\) & \(0.7289\) & \(-1.2262\) \\ \cline{2-11}  & \(\mathbf{R_{4}}\) & \(0.2304\) & \(1.2848\) & \(0.1588\) & \(0.7570\) & \(-1.5067\) & \(3.0534\) & \(1.4099\) & \(0.3570\) & \(1.6211\) & \(-0.2550\) \\ \cline{2-11}  & \(\mathbf{R_{5}}\) & \(1.0501\) & \(-1.5996\) & \(-1.4915\) & \(0.8923\) & \(0.2894\) & \(-0.2318\) & \(-0.0893\) & \(-0.5393\) & \(-0.6183\) & \(1.2611\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Weights of the connections between the neurons of the input layer and the hidden layer (\(\mathbf{W_{Rm}}\)). Figure 9: Confusion matrix for classifying the three classes (1—normal operating condition, 2—thermal faults, 3—electrical faults) obtained by PRN with 5 neurons in the hidden layer, (a) training phase with 90% of samples (733), (b) validation phase with 10% of samples (82), (c) 100% of samples (815). Figure 10: **MSE** performance for PRN with 10 neurons in hidden layer, (**a**) training phase with 90% of samples (733), (**b**) validation phase with 10% of samples (82), (**c**) 100% of samples (815). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Neurons Output Layer**} \\ \hline \hline  & & \(\mathbf{i_{1}}\) & \(\mathbf{i_{2}}\) & \(\mathbf{I_{3}}\) \\ \cline{2-5}  & \(\mathbf{m_{1}}\) & \(-0.1062\) & \(0.8985\) & \(0.1317\) \\ \cline{2-5}  & \(\mathbf{m_{2}}\) & \(-0.1235\) & \(-0.3878\) & \(-0.8237\) \\ \cline{2-5}  & \(\mathbf{m_{3}}\) & \(-2.2868\) & \(0.5225\) & \(2.9425\) \\ \cline{2-5}  & \(\mathbf{m_{4}}\) & \(0.2307\) & \(1.4119\) & \(-0.8567\) \\ \cline{2-5}  & \(\mathbf{m_{5}}\) & \(-0.3885\) & \(0.2586\) & \(-0.9823\) \\ \cline{2-5}  & \(\mathbf{m_{6}}\) & \(-2.1606\) & \(2.8833\) & \(-1.0345\) \\ \cline{2-5}  & \(\mathbf{m_{7}}\) & \(-0.7728\) & \(0.0791\) & \(0.2054\) \\ \cline{2-5}  & \(\mathbf{m_{8}}\) & \(-0.5972\) & \(-0.3729\) & \(-0.3257\) \\ \cline{2-5}  & \(\mathbf{m_{9}}\) & \(-8.1766\) & \(3.4204\) & \(3.5715\) \\ \cline{2-5}  & \(\mathbf{m_{10}}\) & \(-0.5427\) & \(0.4388\) & \(-1.2068\) \\ \hline \hline \end{tabular}
\end{table}