# RUL Prediction of Rolling Bearings Based on a DCAE and CNN
## 3 Degradation Modeling of Rolling Bearings  
Based on the full life cycle data of the rolling bearings, this article constructs a health indicator (HI) that can indicate the degradation state of the bearings. In the process of constructing the HI, performing effective feature extraction on the original data to mine its deeper characterization, followed by performing feature selection to remove redundant features, is crucial. On this basis, feature fusion is also required, which involves combining multiple effective features to establish a degradation curve that reflects the health of the bearing. Among them, feature extraction, feature selection and dimensionality reduction methods are the most essential parts.  
Since the original vibration data are one-dimensional time series data, this paper uses one-dimensional convolution for improving the autoencoder; that is, the encoding and decoding part makes use of the one-dimensional convolution operation so as to construct a feature extraction model based on DCAE. In time series data, there is a strong correlation among the data. One-dimensional convolution is employed to directly process the original data without destroying the correlation between the data. The deep structure in DCAE can obtain a higher-level representation of the input data layer by layer through feature extraction, with the output of the coding network being chosen as the extracted feature. The output of the hidden layer can be regarded as a better feature expression of the input data and has stronger robustness. Because of the high dimension of the original bearing vibration signal, the single feature, when extracted directly, cannot represent its degradation status well. An SOM is an effective unsupervised feature reduction method, and the HI value can be obtained by computing the distance among the input data and the vector representation of the low-dimensional network nodes. Thus, this paper makes use of the SOM network for feature reduction based on multiple high-level features.  
The process of the rolling bearing degradation modeling method is shown in Figure 5. After the original vibration data are processed in segments, they are directly input into the DCAE model. DCAE is composed of two parts: an encoding network and a decoding network. In DCAE, the coding part consists of convolutional layers and pooling layers. The decoding part includes convolutional layers and upsampling layers. The data dimension in the upsampling layers is expanded by  
\[U_{n,m}^{i}=upsampling\left({I_{n,m}^{i}}\right) \tag{4}\]  
where \(I_{n,m}^{i}\) denotes the input of the \(i\)th point at the \(n\)th block data of the \(m\)th layer and \(U_{n,m}^{i}\) denotes the output of the upsampling layer of the \(i\)th point at the \(n\)th block data of the \(m\)th layer.  
Figure 4: SOM network structure diagram.  
Suppose that \(\mathbf{X}=\left[\begin{array}{cccc}X_{1}&X_{2}&\cdots&X_{n}\end{array}\right]^{T}\) and \(\mathbf{X}^{\prime}=\left[\begin{array}{cccc}X_{1}^{\prime}&X_{2}^{\prime}& \cdots&X_{n}^{\prime}\end{array}\right]^{T}\) denote the input and output of DCAE, respectively. The loss function of DCAE is defined as  
\[J=\frac{1}{n}\sum_{j=1}^{n}\|X_{j}-X_{j}^{\prime}\|^{2} \tag{5}\]  
The processed low-dimensional features are selected as the input of the SOM network. The output value of the SOM network is the bearing degradation HI.