## 3 Experiment and Analysis Results
### Analysis Results
According to the _PCA-WLE_[1] feature selection method, the root mean square value \(RMS\), wavelet packet total energy \(E_{T}\) and node 1 wavelet packet energy entropy \(S_{1}\) are selected as sensitive features. Meanwhile, the obtained features are normalized. The RMS of the signal can describe the vibration energy and reflect the wear degree of the bearings, while the wavelet packet energy and its entropy have a high noise resistance, and the higher the energy in the sub-band is, the more notable the fault information is. The running state of the bearings is divided into four types: normal operation, mild anomaly, moderate anomaly, and severe anomaly, namely, \(K=4\). With the selected sensitive features as clustering parameters, according to the historical data, the _K-means_ clustering method is applied to obtain the clustering centre of the operating conditions, as summarized in Table 6. The results of K-means clustering are shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Wavelet Base** & **Threshold Method** & **SNR** & **RMSE** & **Correlation Coefficient** \\ \hline \multirow{4}{*}{dmey} & hard threshold & 6.799 & 0.969 & 0.889 \\  & soft threshold & 5.730 & 1.096 & 0.857 \\  & fixed threshold & 6.697 & 0.980 & 0.887 \\  & hard threshold & 6.677 & 0.983 & 0.886 \\ db10 & soft threshold & 5.616 & 1.110 & 0.852 \\  & fixed threshold & 6.569 & 0.995 & 0.883 \\  & hard threshold & 6.664 & 0.984 & 0.886 \\ sym8 & soft threshold & 5.588 & 1.114 & 0.851 \\  & fixed threshold & 6.554 & 0.997 & 0.883 \\  & hard threshold & 6.718 & 0.978 & 0.887 \\ coif5 & soft threshold & 5.653 & 1.106 & 0.853 \\  & fixed threshold & 6.718 & 0.978 & 0.887 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: SNR and RMSE of the four wavelet bases after noise reduction with three layers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
\begin{tabular}{c} I = 1, 2... 1 6 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Relevant features of bearings. The kernel density of the fault data in the four states is estimated, and the corresponding probability distribution diagram is obtained. When the value of the probability distribution function reaches 99%, it is used as the initial alarm threshold to realize early warning. The results are shown in Figure 7, and the initial threshold is summarized in Table 6. The part above the threshold of the severe anomaly is considered to be in the state of complete failure. Then, the vibration signal of the online running bearings is collected. The experiment is performed at a speed of 2100 \(\mathrm{\SIUnitSymbolMicro\meter}\)/min and a radial force of 12 kN. The time series data of the online running bearings in the normal running state are \(Y=[Y_{1},Y_{2},\ldots,Y_{n}]\), and \(n\) is the number of data points in the sampling period. The normal data of the historical equipment and online equipment in the same time period are \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Clustering Centre (Normalization)**} \\ \hline _RMS_ & _ET_ & _S_**1** & **Threshold(RMS)** & **Classification** \\ \hline
0.0143 & 0.0009 & 0.9998 & 0.7281 & Normal operation \\
0.2743 & 0.0616 & 0.9563 & 2.9079 & Mild anomaly \\
0.4046 & 0.1259 & 0.8903 & 3.8243 & Moderate anomaly \\
0.9263 & 0.6173 & 0.1734 & 7.7653 & Severe anomaly \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Historical data clustering centres and thresholds. Figure 6: Results of _k-means_ clustering analysis: (**a**) spatial representation of clustering results; (**b**) temporal variation of different categories. Figure 7: Probability distribution diagram. selected for similar proportion analysis, and the Euclidean distance and cluster centre of the two sets are calculated. Through correlation calculation, the similarity proportions are \(\eta_{1}=0.8491\) and \(\eta_{1}=0.8491\). The distance weight \(w_{1}\) and center weight \(w_{2}\) are both set to 0.5. Finally, the similarity ratio \(\eta=0.8014\) is obtained.