# Bearing Fault Diagnosis Approach under Data Quality Issues
## 4 Results and Discussion
### Fractal Characteristics of Vibration Signals
It seems that the highest accuracy, 100%, was achieved at scaling exponent cutoff of 1.63. In the second approach, we focused on the classification of bearings with minor or severe faults based on the scaling exponent (\(\alpha_{1}\)). Similarly, the scaling exponents were calculated over several 1000-sample vibration signals. As shown in Figure 8, we computed the scaling exponent distributions of the faulted bearings with minor or severe faults. Based on ROC and accuracy curves, the method had high prediction capability of identifying fault severity with AUC of 0.9. Moreover, the overall accuracy of this approach was 85% and it was achieved at \(\alpha_{1}=1.91\). Figure 7: (**a**) Scaling exponent (\(\alpha_{1}\)) distributions of bearing vibration signals (1000 samples each) collected from healthy and faulted bearings. (**b**) Receiver operating characteristic (ROC) curve for the proposed method. (**c**) Accuracy vs. cutoff plot. For this paper, we focused our analysis on vibration signals with high sampling rate, 48,000 samples/s. Clearly, the high sampling rate provided better visibility of the vibration signal dynamics. Therefore, our approach is capable of diagnosing bearing faults with high accuracy rates, 85% and 100%, as shown in Table 4. The other CWRU data sets had lower sampling rate, 12,000 samples/s, recorded during the three bearing faults. As a result of low sampling rate, the fluctuation functions had smaller linear regions at the short- and long-time scales, as shown in Figure 9. Therefore, it was not possible to estimate the scaling exponents (\(\alpha_{1}\) and \(\alpha_{2}\)) accurately. Moreover, it is clear from Figure 10 that the classification of different bearing faults was more challenging and less accurate. In Figure 11, we evaluate the diagnostic performance of the proposed method by calculating the scaling exponent (\(\alpha_{1}\)) from vibration signals at sampling rate of 12,000 samples/s. Using ROC and accuracy curves, it is shown that accuracy of diagnosing minor and severe faults was 65% at scaling exponent (\(\alpha_{1}\)) equal to 1.33. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Classification** & **Sampling Rate (Samples/s)** & **Cutoff** & **AUC** & **Accuracy** \\ \hline Healthy and faulted bearings & 48,000 & 1.63 & 1 & 100\% \\ \hline Bearing faults into minor and severe faults & 48,000 & 1.91 & 0.9 & 85\% \\ \hline Bearing faults into minor and severe faults & 12,000 & 1.33 & 0.66 & 65\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Diagnostic performance of the proposed method using detrended fluctuation analysis (DFA). Figure 8: (**a**) Scaling exponent (\(\alpha_{1}\)) distributions of bearing vibration signals (1000 samples each) collected from faulted bearings. (**b**) ROC curve for the proposed method. (**c**) Accuracy vs. cutoff plot. Figure 9: Fluctuation functions (\(F(n)\)) of bearing vibration signals (12,000 samples/s): (**a**) minor (green) and severe (red) ball faults; (**b**) minor (green) and severe (red) inner race faults; (**c**) minor (green) and severe (red) outer race faults.