# Bearing Fault Diagnosis Approach under Data Quality Issues
## 4 Results and Discussion
### Fractal Characteristics of Vibration Signals
At the short-time scale, the scaling  
Figure 5: Fluctuation functions (\(F(n)\)) of bearing vibration signals (48,000 samples/s): (**a**) normal (black), (**b**) minor (green), and severe (red) ball faults, (**c**) minor (green) and severe (red) inner race faults, (**d**) minor (green) and severe (red) outer race faults. exponents showed a shift from anticorrelation in normal bearings to long-range correlation in faulted bearing. A promising application of our proposed method is the detection of different bearing faults from the vibration signals by estimating the scaling exponents (\(\alpha_{1}\)) at the short-time scale. In the literature, DL methods have been used increasingly in fault detection and diagnosis due to their high efficiency and accuracy. Specifically, several DL methods have been proposed to diagnose bearing faults based on the COWU data set [36]. The most common methods include auto-encoders, convolutional neural networks (CNN), deep belief networks (DBN), and generative adversarial networks (GAN). As shown in Table 3, based on the selected classifier, these methods can diagnose bearing faults at different accuracy rates: auto-encoders (83-99.8%), CNN (96.8-100%), DBN (84-99.6%), and GAN (96-99.9%). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Method** & **Hyperparameter** & **Accuracy** \\ \hline
[25] & auto-encoders & Gath-Geva (GG) & 83.4\% \\ \hline
[37] & auto-encoders & softmax & 99.83\% \\ \hline
[38] & CNN & rectified linear unit, softmax, and sub-sampling layer & 96.75\% \\ \hline
[39] & CNN & batch normalization, maxpool, rectified linear unit, dropout, and softmax & 100\% \\ \hline
[26] & DBN & sigmoid and rectified linear unit & 84.2\% \\ \hline
[40] & DBN & sigmoid & 99.55\% \\ \hline
[41] & GAN & sigmoid and softmax & 96\% \\ \hline
[42] & GAN & rectified linear unit and leaky rectified linear unit & 99.93\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Diagnostic performance of different deep learning methods. Figure 6: Scaling exponents of bearing vibration signals (48,000 samples/s) at short- and long-time scales. In this paper, the scaling exponent (\(\alpha_{1}\)) was proposed as the input of two classification approaches. In the first one, we aimed to classify the bearings as either healthy or faulty based on scaling exponents from several 1000-sample vibration signals. Secondly, we classified the severity of the fault into either minor or severe fault. The performance of the scaling exponent (\(\alpha_{1}\)) as a binary classifier can be assessed using the confusion matrix and receiver operating characteristic (ROC) curve. Specifically, we chose the overall accuracy, as shown in Equation (4), to evaluate performance of the proposed method. The accuracy quantified the performance by calculating the ratio of correctly predicted samples with respect to the total number of samples. \[\text{Accuracy }=\frac{TP+TN}{TP+TN+FP+FN} \tag{4}\]  
_TP_ and _TN_ are the number of correctly predicted positive and negative samples, respectively. _FP_ represents the number of positive samples that was classified incorrectly by the method. In addition, _FN_ denotes the number of negative samples that was misclassified. In the first approach, we calculated the scaling exponents (\(\alpha_{1}\)) from the vibration signals collected from healthy and faulted bearings. Moreover, each scaling exponent was calculated from a vibration signal of 1000 samples. The scaling exponent distributions of all the vibration signals are shown in Figure 7a. Using ROC curve, we evaluated the performance of our classification method by calculating the area under curve (AUC). In general, the AUC values range from 0.5 (random prediction) to 1 (perfect prediction). As shown in Figure 7b, the AUC was equal to 1, which indicates that our method can identify healthy and faulted bearings correctly without any misclassifications. In Figure 7c, we have the overall accuracy of the proposed method at different cutoff values.