## 4 Results and Analysis
### Numerical Example
However, there are still about 30% of the fault windows below the fault detection threshold. The result of detecting the fault \(f_{2}\) by using the proposed method is shown in Figure 7. It can be seen from Figure 7 that only about 12% of the fault windows were below the fault detection threshold, which is better than the results of PVs obtained by PCA. Figure 4: The results of detecting the incipient fault \(f_{1}\) by using the principal component analysis (PCA) + Kullback-Leibler divergence (KLD)1 method and the PCA + KLD2 method: (**a**) the result of first projection vector (PV) for the fault \(f_{1}\); (**b**) the result of second PV for the fault \(f_{1}\); (**c**) the result of third PV for the fault \(f_{1}\); (**d**) the result of fourth PV for the fault \(f_{1}\). Figure 3: Results of detecting fault \(f_{1}\) using the \(T^{2}\) and squared prediction error (SPE) statistics: (**a**) \(T^{2}\) statistic result for all on-line samples; (**b**) the result for the \(T^{2}\) statistic at 1000 samples before and after insertion of the fault; (**c**) SPE statistic result for all on-line samples; (**d**) the result for the SPE statistic at 1000 samples before and after insertion of the fault. Figure 5: The detection result of the proposed method for the incipient fault \(f_{1}\). \[FAR =\mathit{prob}\{h(w_{k})>\epsilon_{k}|H_{0}\} \tag{23}\]0 The results of detecting the incipient fault \(f_{2}\) by using the PCA + KLD1 method and the PCA + KLD2 method: (**a**) the result of first PV for the fault \(f_{2}\); (**b**) the result of second PV for the fault \(f_{2}\); (**c**) the result of third PV for the fault \(f_{2}\); (**d**) the result of fourth PV for the fault \(f_{2}\). Figure 8 shows the results of detecting the incipient fault \(f_{3}\) by using the PCA + KLD1 method and the PCA + KLD2 method. Both the PCA + KLD1 method and the PCA + KLD2 method have poor detection results for \(f_{3}\). There are still about 60% of the fault windows below the fault detection threshold. Although we could increase the detection rate by decreasing the detection threshold, because the fault windows and the normal windows are not clearly separated, the false alarm rate will also increase. It can be seen from Figure 9 that the proposed method has a very good detection effect on \(f_{3}\), and the fault detection rate is 98% with similar false alarm rate as the PCA + KLD2 method. \[FAR =\mathit{prob}\{h(w_{k})>\epsilon_{k}|H_{0}\} \tag{23}\]1 The results of detecting the incipient fault \(f_{3}\) by using the PCA + KLD1 method and the PCA + KLD2 method: (**a**) the result of first PV for the fault \(f_{3^{*}}\)(**b**) the result of second PV for the fault \(f_{3^{*}}\)(**c**) the result of third PV for the fault \(f_{3^{*}}\)(**d**) the result of fourth PV for the fault \(f_{3}\). \[FAR =\mathit{prob}\{h(w_{k})>\epsilon_{k}|H_{0}\} \tag{23}\]2 The detection result of the proposed method for the incipient fault \(f_{2}\). The three incipient faults were simulated randomly for a total of 100 times and then the average values of the fault detection results were calculated and are summarized in \[FAR =\mathit{prob}\{h(w_{k})>\epsilon_{k}|H_{0}\} \tag{23}\]3 In Table 1, the traditional fault detection methods using PCA and \(T^{2}\) or SPE statistics were very poor at detecting the three incipient faults. The PCA + KLD1, and PCA + KLD2 methods using fixed PVs and KL divergence greatly improved the detection of the three incipient faults compared to the \(T^{2}\) and SPE statistics. However, under the condition of similar false alarm rate, the average fault detection rate of the proposed method in this paper for the three faults was 60% and 30% higher than the PCA + KLD1, and PCA + KLD2 methods with fixed PVs, respectively. In Table 1, the proposed method and the PCA + KLD2 method had similar effects for the incipient fault \(f_{1}\), while the detection rate of the incipient fault \(f_{3}\) was improved by up to 60%.