## 2 The ProbPlot via IR-AVPCA Technique for Bearing Fault Detection and Diagnosis (BFDD)
### ProbPlot of the Vibration Signals and Features Creation Tool  
By definition, the probability plot [35] is a \"graphical technique for assessing whether or not a data set follows a given distribution, such as the normal or Weibull\"; this paper considers the normal probability plot due to its importance in many statistical applications [36]. The ProbPlot strategy plots the vibration raw data according to a theoretical distribution (i.e., normal or Weibull) so that the resulted points should be bordered and form a straight line. Any deviation from this bordered straight line indicates a deviation from the current specified distribution, since the correlation coefficient (of the ProbPlot) associated with the linear fit of the considered raw data is a measure of the fit itself. Thus, using this merit, the different bearing fault can be detected and then diagnosed, in which the further the ProbPlot results vary from the straight line, the greater the detection and the diagnosis of the bearing faults that is achieved.  
For generating the best features (images), the highest correlation coefficient must be chosen, since it will generate the straightest probability plot. The features (images) created through the ProbPlot will be formed by a _vertical axis_, which comprises ordered response values, and _horizontal axis_, which comprises order statistic medians for the given distribution. These order statistic medians, \(Me_{i}\), can be approximated by [37]:  
\[Me_{i}=f(Mu_{i}) \tag{1}\]in which \(f\) is the percent point function for the desired distribution, which is the inverse of the cumulative distribution function, and \(Mu_{i}\) are the uniform order statistic medians defined as [37]:  
\[\left\{\begin{array}{cc}Mu_{i}=1-Mu_{i}&\quad\quad\quad\quad\text{for $i=1$}\\ Mu_{i}=\frac{(i-0.3175)}{(n+0.365)}&\quad\text{for $i=2,3,\cdots,n-1$}\\ Mu_{i}=0.5^{(\frac{1}{n})}&\quad\quad\quad\text{for $i=n$}\end{array}\right. \tag{2}\]  
One advantage of the proposed ProbPlot-based features generation in the form of images (as proposed in this paper) is that it is very easy; further, it is a straightforward method, and the results can be directly interpreted.  
In order to generate the features (images), both \(x\)-axis and \(y\)-axis are auto-scaled when capturing the image so the only parameter of the image (feature) is the size in pixels. Then, all irrelevant information, such the axis, the titles, labels, and so on, were eliminated, and the only pure ProbPlots of the vibration signals were captured to form the features, as can be seen in Figure 1.  
Figure 1: Features creation using the probability plot (ProbPlot): (**a**) the raw data of the vibration signal in the BFF case, (**b**) the ProbPlot of the vibration signal, and (**c**) the created feature (image of 100 dpi (dots per inch).  
The flowchart of the ProbPlot-based features generation in Matlab is shown in Figure 2. Note that \(m\) is the number of all possible faulty cases and the fault-free case, which are known as clusters. In our application,  
* \(m=1\) for the BFF cluster,
* \(m=2\) for the ORF cluster,
* \(m=3\) for the IRF cluster, and
* \(m=4\) for the BBF cluster.