# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Fault Diagnosis and Prediction
#### 2.2.2 Method Based on Deep Learning
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**References** & **Application** &
\begin{tabular}{c} **Type of** \\ **Equipment** \\ \end{tabular} & **Signal** & **Fault Type** \\ \hline
[47] & Fault & Rolling & Vibration & \\
[47] & diagnosis & bearing & signal & \\
[54] & Fault & Rolling & Vibration & \\
[54] & diagnosis & bearing & signal & \\
[63] & Fault & Rolling & Vibration & \\
[63] & diagnosis & bearing & signal & \\
[64] & Fault & Spindle & Vibration & \\
[64] & diagnosis & bearing & signal & \\
[145] & Fault & Rolling & Vibration & \\
[146] & diagnosis & bearing & signal & \\
[147] & Fault & Rolling & Vibration & \\
[148] & diagnosis & bearing & signal & \\
[149] & diagnosis & bearing & signal & \\
[154] & diagnosis & bearing & signal & \\
[155] & Fault & Rolling & Vibration & \\
[155] & diagnosis & bearing & signal & \\
[155] & & Rolling & Vibration & \\
[161] & diagnosis & bearing & signal & \\ \hline \hline \end{tabular}
\end{table} 
Table 20: Bearing condition monitoring using convolutional neural network. Figure 6: Structure of a typical recurrent neural network. In Figure 7, x represents the inputs, h represents the hidden layer output, and y represents the output layer output. The right-hand side of Figure 7 is the form of an RNN expanded on the time axis, where the input is a time series \(\{\cdots,x_{t-1},x_{t},x_{t+1},\cdots\}\), where \(x_{t}\in R^{n}\) and n is the number of neurons in the input layer. Correspondingly, the hidden layer is \(\{\cdots,h_{t-1},h,h_{t+1},\cdots\}\), where \(h_{t}\in R^{m}\) and m is the number of neurons in the hidden layer. The output value y at the moment t is not only influenced by the input x at the same moment, but also by the state of the hidden layer s at the previous moment. Such feedback is called'memory' or'state' s in RNN and is at the heart of it. This leads to the following computational process for the RNN model. The hidden layer calculation formula can be expressed as:  
\[h_{t}{=}\ f_{h}(s_{t}) \tag{37}\]  
where,  
\[s_{t}{=}\ Wx_{t}{+}\ Uh_{t-1}{+}b_{h} \tag{38}\]  
\(f_{h}\) represents the activation function of the hidden layer, \(b_{h}\) represents the bias of the hidden layer, \(W\) is the weight matrix from the input layer to the hidden layer, U is the weight matrix from the hidden layer outputs at time t - 1 to the hidden layer outputs at time t, \(s_{t}\) represent the state of the neuron at moments t, and \(h_{t-1}\) represents the output of the hidden layer at the previous moment. The RNN outputs are \(\{\cdots,y_{t-1},y,y_{t+1},\cdots\}\), where \(y_{t}\in R^{P}\) with p being the number of neurons in the output layer. The output layer is a fully connected layer, which means that each neuron in the hidden layer is connected to all the neurons in the output layer, and the output layer expression is:  
\[y_{t}{=}\ f_{o}(Vh_{t}{+}b_{o}) \tag{39}\]  
where V is the weight matrix from the hidden layer to the output layer, \(f_{o}\) and \(b_{o}\) represent the output layer activation function and bias respectively. As with traditional neural networks, the training of the network pattern parameters is based on an error back-propagation algorithm. The difference is that the parameters of the recurrent neurons in the RNN are shared from moment to moment, and thus the calculation of the gradient relies on the gradient of all past moments. For this reason, the error back propagation algorithm in RNNs is also known as the back propagation through time (BPTT). In the following, the cross-entropy is considered as the loss function, the \(\tan h\) function is the hidden layer activation function and the final output of the RNN is y, the state and output at time t are:  
\[h_{t}{=}\ \tan h(s_{t}){=}\ \tan h(W) \tag{40}\]  
\[\hat{y}_{t}=\ softmax(Vs_{t}) \tag{41}\]  
Figure 7: The expanded form of a recurrent neural network.