## 5 Internal Leakage Fault Diagnosis
### Accuracy Verification of the Proposed Fault Features  
In the existing work, the PCA-based fault diagnosis method transforms the three energy features into principal component and residual subspaces [31]. The T\({}^{2}\) and SPE statistics were used to measure the deviation of test data from the internal leakage threshold, indicating the severity of leakage. As shown in Figure 11, the T\({}^{2}\) statistics and SPE statistics based on the three energy features and the proposed five features are presented, respectively. The dashed line represents the control limit at a confidence level of 95%. The fault diagnosis based on five features exhibited a more significant difference in distribution between the internal leakage and healthy datasets in both subspaces.  
Figure 11: The T\({}^{2}\) and SPE statistics of three features and five features: **(a)** T\({}^{2}\) statistics of internal leakage in different severity; **(b)** SPE statistics of internal leakage in different severity.  
Figure 10: The redless chamber pressure of leakage cylinder with different working conditions.  
PCA was used to reduce the dimension of health and minor leakage data to obtain its corresponding feature matrix (Figure 12). The deviation between minor leakage and healthy data for the five features is greater than for the three energy features. Furthermore, the Euclidean distance between data cluster centers was used to calculate the deviation between each internal leakage condition and the healthy condition, as shown in Figure 13. It is clear that the newly proposed features are more accurate in diagnosing the internal leakage by their deviation.  
In practical engineering, it is difficult to obtain sufficient historical data, and it is more common to use the initial data with certain leakage to train the fault diagnosis model, and new data are constantly added to dynamically update the fault diagnosis model. Therefore, in order to further verify the effectiveness of the proposed features, the data set was further divided, as shown in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases** & **Training Set** & **Testing Set** \\ \hline Data 1 & Healthy & Minor leakage \& Medium leakage \& Serious leakage \\ Data 2 & Healthy \& Minor leakage & Medium leakage \& Serious leakage \\ Data 3 & Healthy \& Minor leakage \& Medium leakage \& Serious leakage \\ \hline \hline \end{tabular}
\end{table}
Table 2: Partitioned dataset.  
Figure 12: The feature matrix obtained by PCA dimensionality reduction.  
Figure 13: The Euclidean distance between healthy data and internal leakage data.  
Figure 14 shows the detection accuracy of T\({}^{2}\) statistics and SPE statistics based on three features and five features under different training sets. The newly proposed features method always maintains a high detection success rate as new internal leakage data is added to the training set. However, in T\({}^{2}\) statistics, the existing three energy features method suffers a decline in detection success rate with the addition of new internal leakage data. As a result, the newly proposed five features has stronger generalization ability for the internal leakage diagnosis.