## 3 Feature Extraction and Recognition Based on IPSO-VMD
### Multi-Dimension Sensitive Feature Optimization  
For the time-domain parameters, they are closely related to the mechanical system parameters, operating conditions, and signal distribution. For the frequency-domain parameters, they can well reflect the spectral changes of the vibration signals of the mechanical components and determine the operating status of the mechanical components. Because the VMD algorithm will have a greater degree of punishment for abrupt signals, some abrupt information in the original signal will be lost in the VMD decomposition process. Considering the time domain or the frequency domain alone has its own limitations, because they can capture different information and characteristics of the signals, and only by considering them together can the sensitive feature vectors be better constructed. 1. Time-Domain Parameters  
Time-domain parameters are closely related to mechanical system parameters, working conditions, and signal distribution. For time series signal \(x_{i}\), the time-domain parameters used in this paper include 12 indicators: maximum \(x_{\max}\), absolute mean \(\overline{x}\), effective value \(x_{rms}\), peak to peak value \(x_{p-p}\), average amplitude \(MA\), variance \(\sigma_{x}^{2}\), deviation \(S_{c}\), waveform index \(S_{f}\), peak index \(C_{f}\), pulse index \(I_{f}\), margin index \(CL_{f}\), kurtosis \(\beta\). 2. Frequency-Domain Parameters  
Based on the frequency-domain parameters, the frequency spectrum changes of the vibration signals of mechanical components can be well reflected, and the running state of the mechanical components can be judged. The frequency-domain parameter indexes adopted in this paper include: mean frequency \(Mf\), center frequency \(Fc\), root mean square frequency \(Rmsf\), standard deviation frequency \(Sdf\), frequency-domain amplitude skewness index \(Scf\), and frequency kurtosis \(\beta f\). 3. Other Parameters  
Energy operator [34] is as follows:  
\[ \begin{split} Teo=\frac{N\sum\limits_{n=1}^{N}\left(r(n)- \overline{r}(n)\right)^{4}}{\left(\sum\limits_{n=1}^{N}\left(r(n)-\overline{r} (n)\right)^{2}\right)^{2}}\end{split}  \tag{15}\]  
where, \(\overline{r}(n)\) represents the mean value of \(r(n)\), and \(r(n)\) represents the Teager energy differential signal, \(r(n)=x^{2}(n)-x(n-1)x(n+1)\). A total of 19 fault feature parameters are extracted based on time domain and frequency domain. Although they can reflect different information of vibration signals and system faults, the sensitivity of different fault feature parameters is different, and there is partial irrelevance or redundancy. The Laplacian score (LS) method is used to extract the inherent information architecture of the feature set, map the complex and high-dimensional feature space to the low dimensional space, and achieve the optimization of multi-dimensional feature [35]. The specific steps of the LS method are as follows:  
Step 1: Input the training sample feature matrix \(F\in\Re^{m\times n}\), where \(m\) is the number of samples, \(n\) is the feature dimension, and \(f_{ri}\) is the \(r\)-th dimension feature of the \(i\)-th sample. Step 2: Build a neighbor graph \(G\), where \(G\) has \(m\) sample points; \(x_{i}\) corresponds to the \(i\)-th node. If \(x_{i}\) and \(x_{j}\) are \"neighbors\", they are connected at two points. \(x_{i}\) and \(x_{j}\) are defined as neighbors. On the contrary, the two points are not connected. Step 3: Define the weight matrix \(S_{ij}\) as below:  
\[S_{ij}= \begin{cases}e^{\frac{|x_{i}-x_{j}|^{2}}{t}}&\text{ if }x_{i}\text{ {close to }}x_{j}\\ 0&\text{ otherwise}\end{cases}  \tag{16}\]  
where \(t\) is a suitable constant. Step 4: \(L_{r}\) is defined as the Laplace score of the \(r\)-th feature \(f_{r}\).