# RUL Prediction of Rolling Bearings Based on a DCAE and CNN
## 3 Degradation Modeling of Rolling Bearings
#### Experimental Data
Step 3: Calculate the distance \(d_{t}\) between the vector representation of all \(d\) neurons in the topology layer and the input layer sample \(x_{t}\). Choose the neuron with the smallest distance from \(x_{t}\) as the best matching neuron \(l\), namely \(\|x_{t}-w_{l}\|=\min\{d_{t}\}\), where \(w_{l}\) is the vector representation of the best matching neuron. The vector of the connection weights between each neuron in the topological layer and the input layer neuron connected to it is represented by \(w_{\prime}=\{w_{\prime 1},w_{i^{\prime}2},\ldots,w_{i^{\prime}80}\}\), where \(i^{\prime}\) represents the \(i^{\prime}\)th neuron of the topological layer and \(i^{\prime}=1,2,\ldots,d\). Step 4: Update the best matching neuron and the connection weights of the neighborhood neuron and the input layer neuron:  
\[w_{i^{\prime\prime}j}(n+1)=w_{i^{\prime\prime}j}(n)+\eta(n)\cdot T_{i^{\prime \prime}j}\cdot\big{(}x_{jt}-w_{i^{\prime\prime}}(n)\big{)} \tag{6}\]  
where \(i^{\prime\prime}\) is the \(i^{\prime\prime}\)th neighborhood neuron in the topological layer around the best matching neuron \(l\), \(n\) is the training step, \(w_{i^{\prime\prime}j}(n+1)\) is the connection weight between the input layer  
Figure 8: DCAE structure diagram. neuron \(x_{jl}\) and the \(i^{\prime\prime}\)th neighborhood neuron in the topology layer during \(n+1\) training, \(\eta(n)\) is the gain function, \(0<\eta(n)<1\), \(\eta(n)\) gradually decreases as the number of training increases and \(T_{i^{\prime\prime},j}\) is the weight:  
\[T_{i^{\prime\prime},l}=\exp\left(-S_{i^{\prime\prime},l}^{2}/2\sigma^{2}\right) \tag{7}\]  
where \(S_{i^{\prime\prime},l}\) is the Euclidean distance between the \(i^{\prime\prime}\)th neighborhood neuron of the best matching neuron \(l\) of the topological layer and the best matching neuron \(l\). Here, \(\sigma\) is the standard deviation of the distance between each neighborhood neuron of the best matching neuron \(l\) of the topological layer and the best matching neuron \(l\). Step 5: Choose an 80-dimension sample at another point in time to provide to the input layer of the SOM network and return to step 3 until all the training set samples are provided to the SOM network. Step 6: Perform training step \(n=n+1\) and return to step 2 until the training step reaches the maximum number of training times \(N\). Step 7: After training, calculate the bearing health factor at each time point:  
\[HI=MQE=\|x_{t}-w_{l}\| \tag{8}\]  
where \(MQE\) is the minimum quantization error. Taking the features extracted by Bearing1_3 as an example, the number of nodes in the input layer of the SOM network is 80, and the corresponding HI can be calculated for each sampling point so that the HI curve of the training set can be constructed. Step 4: Algorithm evaluation and verification. In order to make the HI curve smoother, the obtained HI curve is subjected to sliding filter processing with a window size of 8. The normalized results of the seven bearings are shown in Figure 9. The degraded state modeling method of rolling bearings proposed in this paper can better extract the degraded curve of the original vibration signal. The whole method is carried out without supervision, does not rely on expert experience and has good practicability and versatility. In order to verify the advantages of the method in this paper, the HI curves constructed by four other methods are selected for comparison. The first method, the traditional time-frequency domain feature extraction of the signal, is a method to extract the following features: time domain characteristics (root mean square, peak value, peak-to-peak value, variance, kurtosis, skew, waveform index, peak index and margin index) and frequency  
Figure 9: HI curves of the 7 bearings. domain characteristics (root mean square frequency, center of gravity frequency, frequency standard deviation and frequency variance).