## 4 Results and Discussion  
### On Different Scenarios for Dealing with the Unbalanced Data Set  
Several scenarios were considered to assess the effectiveness of the proposed model. In all these scenarios, we all used an RF with 1000 trees for classification. The different scenarios identified as follows: RF-i denotes a random forest trained with the unbalanced dataset described by Table 2; RF-b2 denotes a random forest trained with a subsampled balanced dataset with only 140 observations per condition; RF-GAN and RF-GAN2 stands rand forests trained with data sets that have the real 14,000 healthy observations and 14,000 faulty observations, the only difference between these scenarios is that RF-GAN uses the technique described in Section 2.2 to select the best current model for generating samples while RF-GAN2 uses the model obtained in the last iteration (i.e., iteration 20,000) of training process to generate the samples. RF-GAN1 is similar to RF-GAN with the difference that only 13,860 synthetic faulty states were generated while the remaining 140 are the original faulty states presented in the training data set. For comparison purposes, we also considered a random forest \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fault Id** & **Training Set** & **Test Set** \\ \hline A & 14,000 & 6000 \\ B & 140 & 6000 \\ C & 140 & 6000 \\ D & 140 & 6000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of observations in the training and test sets used for hold out validation. The training set is clearly unbalanced. Figure 9: Vibration signals acquired in the following fault conditions: (**a**) Healthy state; (**b**) Pitting in Sun gear 1; (**c**) Broken tooth in Sun gear 1; and (**d**) Cracking in Planetary gear 2.  
trained with a data set previously processed by SMOTE, a popular oversampling method for dealing with unbalanced data sets. In all GAN based models, the generators are multi-layer-perceptrons with a 64:1014:128 fully connected topology while discriminators are also multi-layer-perceptrons with 128:1024:2048:1 fully connected topology. These were selected empirically after some preliminary tests. The Adam optimizer is used with its key parameter settings of \(\beta_{1}\) = 0.9, \(\beta_{2}\) = 0.999 and \(\epsilon\) = 1 \(\times\) 10\({}^{-8}\). The learning rate for the generators is set to 1 \(\times\) 10\({}^{-5}\), while, for the discriminators, is set to 1 \(\times\) 10\({}^{-4}\). The \(\alpha\) and \(\lambda\) are set to 1 \(\times\) 10\({}^{-4}\) and 1.0, respectively. A maximum number of 10,000 iterations was set for training. Figure 10 shows the distribution of the obtained accuracy for each scenario using boxplots. A boxplot summarizes a data distribution stressing five of its characteristic values: minimum, lower quartile, median, upper quartile, and maximum value. The red line denotes the median value. The distribution pertains to 20 independent repetitions. The results presented in Figure 10 were analyzed by the Friedman test, a non parametric statistic test of hypotheses to evaluate whether or not there is a statistically significant difference between the results (boxplots) of the different scenarios. The [PERSON] null hypothesis is that there is no statistically significant difference between the results of the different scenarios. Given a significant level, \(\alpha\), this hypothesis cannot be rejected whenever the \(\text{p}_{\text{\[PERSON]}}\), the \(p\)-value generated by the test, satisfies \(\text{p}_{\text{\tiny [PERSON]}}>\alpha\). The null hypothesis is rejected otherwise, meaning that there is a statistically significant difference between the analyzed scenarios. In such a case, we can detect which of the scenario is responsible for such difference resorting to a pairwise _posthoc_ test. A ranking can be obtained by counting the number of times that a method was a winner in the pairwise comparison. See [46; 47] for further details. Here, we are using the usual \(\alpha=0.05\) and the Wilcoxon test as _posthoc_.