## 4 Case Study
### Improved MSPCA Fault Data Recognition Results  
After de-noising by wavelet threshold, the reconstructed signal is decomposed into three-scale wavelet packets. The energy of two nodes is obtained in the first scale, four in the second scale and eight in the third scale. The energy difference between the wavelet packet energy and its corresponding normal data at each scale and the same node at the same scale is calculated, respectively. The result of each scale is shown in Figures 10-12.  
Figure 8: Control limit chart of the PCA (Principal Component Analysis) fault recognition model: (a) PCA-\(T^{2}\); (b) PCA-SPE.  
Figure 9: Control limit chart of MSPCA fault diagnosis model: (a) MSPCA-\(T^{2}\); (b) MSPCA-SPE.  
Figure 11 shows the energy difference of eight vectors decomposed by layer 3 wavelet packet of the detector data. The dashed line represents the energy difference threshold of each component.  
In comparison, it can be found that, in Figure 10, no fault information is found in the energy difference of the first layer wavelet packet decomposition signal node [1, 0], and fault information is found in the node [1, 1] near the data point 175-225, 300-350. In Figure 11, the energy of wavelet packet of node [2, 1] detects data faults at about 225-275 data points, and the range of data faults detected by node [2, 2] is about 200-275, 300-350 data points. In the nodes [2, 0] and [2, 3], there is no fault information. At the same time, it can be found that, in the nodes [2, 0], the energy difference is almost zero, which is the same as in the first layer of nodes [1, 0]. Although the energy difference of nodes [2, 3] fluctuates, it does not exceed the set threshold, so it is considered that the change is mainly caused by the noise contained in the data. In Figure 12, node [3, 0] and node [1, 0] have no fault data as  
Figure 11: Energy difference result of second layer decomposition.  
Figure 10: Energy difference result of first layer decomposition.  
in [2, 0]. However, data faults occurred in the vicinity of 150-200, 300-350 in nodes [3, 1], nodes [3, 2], nodes [3, 3], and nodes [3, 4].  
Through three-level decomposition of the wavelet packet, the suspected fault location of the data to be checked can be obtained between 150-225 and 275-350 data points. Parameters and principal component modeling analysis are carried out for nodes beyond the threshold at each level, principal component space and residual space are established, and \(T^{2}\) control limit is set. The data with fault information is projected into the principal component space, and the result is shown in Figure 13.  
In contrast to the results of the three models, we can find that the improved MSPCA model of this paper can be more effective at making noise reduction in the data, and the error reporting in the malfunction diagnosis is obviously reduced, and it can be obvious from the control limits and the SPE  
Figure 12: Energy difference result of third layer decomposition.  
Figure 13: Control chart of improved MSPCA fault diagnosis model: (**a**) improved MSPCA-\(T^{2}\); (**b**) improved MSPCA-SPE.  
control chart to find the location of the two points of failure. Evaluate the accuracy of each model, as shown in Table 1.