# Intelligent Fault Diagnosis of Rolling Element Bearings Based on Modified AlexNet +
## 2 Proposed Methods and Materials
### Feature Selection
#### 2.2.2 Modified AlexNet Model  
At present, AlexNet is the most widely used convolutional neural network and has become one of the hotspots in bearing fault diagnosis [17]. Due to some limitations of AlexNet, it is very challenging to achieve satisfactory results in bearing fault diagnosis. Figure 7 shows the traditional AlexNet model and the limitations of AlexNet are described below:  
Bearing fault signals are complex due to the high variance, nonlinear, and nonstationary characteristics of vibration signals. As a result, the input distributions of the layers of AlexNet differ from each other and internal covariate shifting occurs [17]. This can make achieving accuracy in parameter training very challenging and time-consuming, which requires proper setup [30]. In a traditional AlexNet, the FC layer is located in the final three layers, namely fc6, fc7, and fc8. An FC consists of many layers that are all connected [31]. The FC layer of AlexNet has the disadvantage of too many training parameters. The procedure for calculating the training parameters of FC layers is given below. In AlexNet, there are two types of FC layers. The first FC layer (fc6) is connected to the final conv layer, whereas subsequent FC layers (fc7 and fc8) are connected to additional FC layers. Each situ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Pre-Trained Model**} & \multicolumn{5}{c}{**Accuracy (\%)**} & \multirow{2}{*}{**Average**} \\ \cline{2-2} \cline{4-7}  & **Trial 1** & & **Trial 2** & & **Trial 3** & **Trial 4** & **Trial 5** \\ \hline GoogleNet & 95.30 & 94.98 & 94.65 & 95.60 & 95.80 & 95.27 \\ \hline VGG16 & 87.50 & 87.98 & 88.50 & 87.30 & 88.50 & 87.96 \\ \hline
**AlexNet** & **97.50** & **97.30** & **98.10** & **96.88** & **97.78** & **97.51** \\ \hline ResNet50 & 96.98 & 97.35 & 96.38 & 95.88 & 96.78 & 96.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classification of the results of the pre-trained model. Figure 6: Best pre-trained model selection process. Figure 7: AlexNet model. Case 1: An FC (fc6) layer's number of parameters connected to a conv layer can be calculated [30] by the following equations  
\[P_{cf}=W_{cf}+B_{cf} \tag{1}\]  
\[B_{cf}=F \tag{2}\]  
\[W_{cf}=F\times N\times O^{2} \tag{3}\]  
where:  
\(P_{cf}\) = number of parameters;  
\(W_{cf}\) = The number of weights in an FC layer that is linked to a conv layer;  
\(B_{cf}\) = The number of biases in an FC layer that is linked to a conv layer;  
\(O\) = The size of the previous conv layer's output image;  
\(N\) = The number of kernels in the last conv layer;  
\(F\) = The number of neurons in the FC layer. In the first FC layer (fc6) of AlexNet, \(F\) is 4096, \(N\) is 256, and \(O\) is 6. Therefore,  
\[ \begin{array}{ll}W_{cf}&=4096\times 256\times 6^{2}\\ &=37,748,736\\ \\ B_{cf}&=4096\\ \\ P_{cf}&=W_{cf}+B_{cf}\\ &=37,748,736+4096\\ \\ &=37,752,832\\ \end{array} \]  
Case 2: An FC layer's number of parameters connected to an FC layer can be calculated [30] by the following equations  
\[P_{ff}=B_{ff}+W_{ff} \tag{4}\]  
\[B_{ff}=F \tag{5}\]  
\[W_{ff}=F_{-1}\times F \tag{6}\]  
where:  
\(P_{cf}\) = number of parameters;  
\(W_{cf}\) = The number of weights in an FC layer that is linked to an FC layer;  
\(B_{ff}\) = The number of biases in an FC layer that is linked to an FC layer;  
F = The number of neurons in the FC layer;  
\(F_{-1}\) = The number of neurons in the just before FC layer. In the second FC layer (fc7) of AlexNet, \(F\) is 4096, and \(F_{-1}\) = 4096. Therefore,  
\[B_{ff}=\text{F }=4096\]  
\[W_{ff1}=F_{-1}\times F=4096\times 4096=16,777,216\]  
\[ \begin{array}{ll}P_{ff1}&=B_{ff}+W_{ff}=4096+16,777,216\\ &=16,781,312\\ \end{array} \]  
In the last FC layer (fc8) of AlexNet, \(F\) is 1000, and \(F_{-1}\) = 4096. Therefore,  
\[B_{ff}=\text{F }=1000\\ \\ \end{array}\]\[W_{ff}=F_{-1}\times F=4096\times 1000=4,096,000\]  
\[ \begin{array}{ll}P_{ff2}&=B_{ff}+W_{ff}=1000+4,096,000\\ &=4,097,000\end{array}