## 4 Results and Analysis  
### Numerical Example  
The effectiveness of the proposed method is verified through a numerical example which includes three incipient faults. The system is modeled by  
\[ \begin{array}{c}x_{1}=s_{1}+s_{2}+e_{1}\\ x_{2}=s_{2}-s_{3}+f_{1}+e_{2}\\ x_{3}=s_{1}-(1+f_{2})s_{4}+e_{3}\\ x_{4}=(1+f_{3})x_{1}+x_{3}+e_{4}\end{array}  \tag{20}\]  
In Equation (20), \([s_{1},s_{2},s_{3},s_{4}]^{T}\) and \([e_{1},e_{2},e_{3},e_{4}]^{T}\) are Gaussian-distributed source signals and noises. All the source signals obey the standard normal distribution \(\mathrm{N}(0,1)\), and \(f_{1}\),\(f_{2}\) and \(f_{3}\) are three incipient faults, respectively, and \([x_{1},x_{2},x_{3},x_{4}]^{T}\) are the four variables that need be monitored [22]. Three incipient faults were simulated in this numericalexperiment. Fault \(f_{1}\) is the offset fault, and fault \(f_{2}\) and fault \(f_{3}\) are the gain faults. The three faults were inserted as follows:  
\[f_{1} = \begin{cases}0&1 st-30000 th\\ 0.35&30001 st-60000 th\\ \end{cases}  \tag{21}\] \[f_{2} = \begin{cases}0&1 st-30000 th\\ 0.25&30001 st-60000 th\\ \end{cases} \] \[f_{3} = \begin{cases}0&1 st-30000 th\\ 0.05&30001 st-60000 th\\ \end{cases} \]  
The fault detection rate (\(FDR\)) and false alarm rate (\(FAR\)) are calculated as follows:  
\[FDR =\mathit{prob}\{h(w_{k})>\epsilon_{k}|H_{1}\} \tag{22}\] \[FAR =\mathit{prob}\{h(w_{k})>\epsilon_{k}|H_{0}\} \tag{23}\]  
The other parameters of the numerical simulation experiment were set as follows. The total number of samples were 120,000, of which 60,000 were historical normal samples and 60,000 were online samples for testing. The sliding window length for both the historical normal data and the on-line test data in the experiment was 300. After using the sliding windows, a total of 200 windows of on-line data were obtained, of which the first 100 windows are normal windows, and the last 100 windows are fault windows. The default Signal-to-Noise Ratio (SNR) was 20 dB. For the purpose of comparison with other PCA-based fault detection methods, the proposed method was compared with the conventional \(T^{2}\) statistic [33] (PCA + \(T^{2}\)), the SPE statistic [33] (PCA + SPE), the method of using KL divergence to monitor the PVs in the principal subspace [21] (PCA + KLD1), and the method of using KL divergence to monitor the PVs both in the principal subspace and the residual subspace [22] (PCA + KLD2). The principal subspace was selected with a cumulative variance contribution of more than 90%. The confidence level for the \(T^{2}\) statistic and the SPE statistic were both set at 0.99. The significance level for the PCA + KLD1 method and the PCA + KLD2 method were both set at 0.05. The significance level for proposed method was set at 0.01. The results of the \(T^{2}\) statistic and the SPE statistic for the incipient fault \(f_{1}\) are shown in Figure 3. It can be seen from Figure 3 that the \(T^{2}\) statistic and SPE statistic did not change significantly before and after insertion of the fault \(f_{1}\). After \(f_{1}\) occurred, most of the online fault samples were still within the detection threshold. The \(T^{2}\) statistic and the SPE statistic were equally ineffective in detecting the incipient faults \(f_{2}\) and \(f_{3}\). The results of detecting the incipient fault \(f_{1}\) by using the PCA + KLD1 method and the PCA + KLD2 method are shown in Figure 4. A comparison of Figure 4a-d shows that the second and third PV were sensitive to the incipient fault \(f_{1}\). The remaining two PVs were less effective at detecting \(f_{1}\). The detection result of the proposed method for the incipient fault \(f_{1}\) is shown in Figure 5. It can be seen from Figure 5 that the proposed method also has a good detection effect for the fault \(f_{1}\). The results of detecting the incipient fault \(f_{2}\) by using the PCA + KLD1 method and the PCA + KLD2 method are shown in Figure 6. Comparing Figure 6a-d shows that only the third PV of the four PVs obtained by PCA was sensitive to \(f_{2}\).