## 4 Validation of CNN-NAP  
In this section, CNN-NAP is applied to the bearing measured signal under various rotating speeds to verify the validity of CNN-NAP. The bearing fault simulation bench is shown in Figure 3, and the test bearing is cylindrical roller bearing of NU250 EM whose parameters are shown in Table 1.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Inside Diameter** & **Outside Diameter** & **Thickness** & **Pitch Diameter** & **No. of Balls** \\ \hline
25 mm & 52 mm & 15 mm & 44.2 mm & 13 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bearing parameters.  
The bearing faults are artificially created by electron discharge machining technology and the crack depth is 0.5 mm on the outer-race, inner-race, and roller (OF, IF, and RF, respectively). The signals of five bearings health conditions (normal, OF, IF, RF and concurrent faults in the outer race and roller (ORF)) run under different rotating speeds are collected. The number of different bearing and roller (ORF)) run under different rotating speeds are collected. The number of different bearing signal sets under various rotating speeds and labels are shown in Table 2. Each sample contains 1200 data points, and the sampling frequency is 25.6 kHz.  
The testing data sets are time-varying signals under high-speed oscillation, and the rotation speed is irregularly changed in the interval of 0-1500 rpm, as shown in Figure 4. One hundred samples of each fault are randomly selected at three rotational speeds as training samples, respectively. Thus, there are 500 training samples, i.e., \(n=500\) in Figure 2. The input is a matrix of \(500\times 1200\); each row represents a sample with 1200 data points. In the designed CNN model, there are the three'same' convolutional layers, three'max' pooling layers and four fully connection layers. The size of the convolution kernel is \(1\times 10\), and the pooling window is \(1\times 10\). The activation function of convolution and connection layers is ReLU while C9 layer without it. It should be noticed that the features obtained  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
\begin{tabular}{l} Sets \\ \end{tabular} & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Bearing signal sets.  
Figure 3: Bearing fault simulation experiment platform.  
in the C9 layer need to go through the NAP projection. Other specific parameters are shown in Figure 2. In the process of back propagation, loss function is defined as Equation (11) to find the optimal weight and projection matrix \(P\). Training epochs is 200 and training error is 0.001.  
In addition, 10,000 samples are randomly selected as testing data sets. The average training accuracy of 15 experiments is 100% and average testing accuracy of CNN-NAP is 99.57%. The specific results are shown in Figure 5. Taking inner-race fault as an example, Figure 5(b) shows the results of CNN-NAP learning in C9 layer, and it can be seen from the comparison Figure 5(a) that the features tend to be consistent. Therefore, this method can be used for rolling bearing fault diagnosis under various speeds conditions.  
Figure 4: Testing datasets rotating speed.  
Figure 5: Results of 15 trials.  
Figure 6: (**a**) samples of inner-race fault at three different speeds: (1) 1000 rpm; (2) 1300 rpm; (3) 1500 rpm; (**b**) the learned features of C9 layer.