## 2 The ProbPlot via IR-AVPCA Technique for Bearing Fault Detection and Diagnosis (BFDD)
### ProbPlot via IR-AVPCA to Online Fault Detection and Diagnosis Under Variable Speed  
The \(m\) features were also generated in Matlab as an image of 100 dpi and of 800 \(\times\) 600 pixels. Then, each feature was projected to the AVPCA subspace to generate the training databases that contain the healthy base vector and the faulty base matrix. The training databases and the generated eigenfaces for each four faulty cases under the variable speed condition can be seen in Figure 13(a),b, respectively.  
As for the constant speed environment, the vibration signal that was collected from the accelerometer #1 & #2, which was gathered on the four different operating health conditions (BFF, ORF, IRF, and BBF), and the reference one that was located on the top of the BLDC motor under the variable speed environment that was varying between 1600 rpm and 2400 rpm for 60 s under 17.06 kHz sampling rate with its ProbPlot for each, are shown in Figures 14-15. Figure 15 again clarifies the conclusion drawn from Figure 9 and shows clearly that the considered experimental bearing faults are early bearing fault scenarios. However, the ProbPlot in Figure 15 shows more deviation in the normal order statistic medians than the one in Figure 9. These results come from the effect of the variable speed condition considered in the second environment.  
Figure 16: Raw vibration data signal from accelerometer #2 and its ProbPlot in bearing fault free (BFF) case under variable speed environment.  
Figure 17: Raw vibration data signal from accelerometer #2 and its ProbPlot in outer-race fault (ORF) case under variable speed environment.  
Figure 18: Raw vibration data signal from accelerometer #2 and its ProbPlot in inner-race fault (IRF) case under variable speed environment.  
After projection to the AVPCA subspace, the dimension was reduced to one dimension, as can be seen in Figure 20, in which the SCREE [40] plot indicates that the number of retained principal components (PCs) can be chosen as one (\(p=1\)). Using the cumulative percent of variance (CPV) criteria defined as, respectively,  
\[CPV(p)=100\cdot\sum_{i=1}^{p}\frac{\lambda_{i}}{\sum_{i=1}^{n}\lambda_{i}} \tag{10}\]  
indicates that for \(p=1\), the CPV = 99.76%, which means that by only considering the one-space (the 1 st eigenvector that corresponds to the largest eigenvalue), we could reach only 99.76% of precision. Thus, to reach at least a 99.95% precision, the number of retained PCs is fixed at six (\(p=6\)), as shown in Table 2.  
A comparison between the features (images) that are obtained: (1) directly in the time domain from the original raw data of the vibration signals; (2) by capturing the Fast Fourier Transformation (FFT) of the vibration signals as in [23]; and (3) by generating the ProbPlot of the vibration signals as proposed in this paper, is considered to demonstrate the performance and effectiveness of the proposed ProbPlot via IR-AVPCA method at online bearing fault detection and diagnosis. Furthermore, both classical PCA and AVPCA are used and compared in terms of bearing faults classification success rate. It should be noted that for the second working conditions (i.e., nonstationary conditions under variable speed environment), the proposed algorithm is compared only to the first case (i.e., when directly generating the features (images) from the raw vibration data), since the FFT is not convenient under those operating conditions in which the needed bearing fault information that changes over time will be lost.  
Figure 19: Raw vibration data signal from accelerometer #2 and its ProbPlot in ball bearing fault (BBF) case under variable speed environment.  
Figure 20: SCREE plot of the first 6 th eigenvalues.