## 3 Proposed Smart Diagnosis Method  
The SAFC model is composed of SAE and SF, where SAE has five layers of network and SF has two layers of network, as shown in Figure 3. The specific steps of the extracting features are as follows:  
1. All the collected time domain signals are converted into frequency domain signal through FFT. According to different health conditions, the spectrum samples of vibration signals at different speeds are composed into a training set \(\left\{\boldsymbol{X}^{i},l^{i}\right\}_{i=1}^{M}\), where \(M\) is the number of samples, \(\boldsymbol{X}^{i}\in\Re^{N\times 1}\) means that the ith sample contains \(N\) Fourier coefficients, and \(l^{i}\) is the health label of the ith sample.  
2. The no label training set \(\left\{\mathbf{X}^{i}\right\}_{i=1}^{M}\) was used to pre-train the SAE layer by layer. The pre-training process was to superimpose \(N\) AE into \(N\) hidden layers, and the characteristics learned in the previous layer were used as the input of the next layer.
3. BP algorithm combined with label data \(\left\{\mathbf{X}^{i},l^{i}\right\}_{i=1}^{M}\) was used to realize weight updating and parameter fine-tuning of SAE, and back propagation training was carried out by minimizing the error between extracting features and health labels.
4. BN was applied to each activation layer of SAE, and the characteristic difference at different speeds was reduced by its translation and scaling.
5. The features initially extracted after dimension reduction through two-layer SAE were taken as new training samples and a new training sample set \(\left\{x^{i},l^{i}\right\}_{i=1}^{m}\) was formed. \(m\) is the number of samples [21], \(x^{j}\in R^{N_{in}\times 1}\) means the ith sample contains \(N_{in}\) data points and \(l_{i}\) is the label of the sample.
6. The sample set was formed into a matrix form \(T\in R^{N_{in}\times M}\), and then input into the sparse filtering model for training. The weight matrix \(W\) is obtained through the minimization formula (24).
7. The learning feature \(f^{i}\in R^{N_{out}\times 1}\) can be calculated by \(W\) and \(x_{i}\). ReLU was used as the activation function here to extend the sparse filtering to nonlinear mapping. Therefore, Equation (20) is extended to: \[f^{i}_{j}=\text{ReLU}(W^{T}_{j}x^{i})\] (25) When the learning feature set \(\left\{f^{i}\right\}_{i=1}^{M}\) is obtained, it is combined with the tag set \(\left\{l^{i}\right\}_{i=1}^{M}\) and input into the softmax regression classifier for training. Then, the remaining samples are used as test samples to test the accuracy of the proposed method [22].  
The testing process is as follows. First, each test sample is converted into spectrum after FFT transformation, and all spectrum samples are combined into matrix form as input; second dimension reduction and preliminary feature extraction are carried out in superposition self-coding to cluster the same type of signals with different speeds. Then, the trained sparse filter model is used to study the characteristics of the test samples. Finally, the learned features are combined with the real fault type label, and then input to the softmax regression model that has been trained for classification, thus obtaining the test accuracy rate.  
Figure 3: The overall architecture of the sparse filter rotating component comprehensive diagnosis model (SAFC) model.