*4.2. Discussions*

In order to prove the reliability of the method in this paper, several points will be discussed: (1) the problem solved by this method; (2) the comparison analysis based on the FAR and FDR; (3) the feasibility of the proposed algorithms is testified.

CCA-JITL FD model was applied to detect fault signals of the running gears in two groups in which the results were compared to each other to improve detection accuracy. The method uses CCA to group data and a JITL algorithm to optimize selection of sample data points, so as to achieve better FD performance based on the data shown in Figure 4. Figure 4a depicts the CCA-JITL FD output based on data set *Px*, and Figure 4b is the result of CCA-JITL FD based on data set *Py*. The number of singular vectors, h values, decide the proximity of dimension reduction and affect FAR and FDR very much. We tuned parameters and concluded that when *h* = 2, CCA-JITL models achieved the best performance.

Figure 5 shows FD experiment results using only CCA. The system infrastructure of CCA-JITL was generalized to be utilized to other FD models using PCA and PLS. FD experiment results using SVD-based PLS and JITL are shown in Figure 6. FD experiment results using PCA and JITL are shown in Figure 7. FD experiment results using only PCA are shown in Figure 7.

 **Figure 5.** Experiment results of the FD model using CCA.

**Figure 6.** Experiment results of the FD model using PLS. (**a**) Online testing of the FD model using PLS; (**b**) Online testing of the FD model using PLS and JITL.

**Figure 7.** Experiment results of the FD model using PCA. (**a**) Online testing of the FD model using PCA; (**b**) Online testing of the FD model using PCA and JITL.

Based on the detection results shown in Figure 4, the detection of data set *Px* after injection fault data is normal, and the detection results of data set *Py* show short-term and transient fluctuations after fault injection. Then, the statistics fall back above the threshold. The detection results of the datasets *Px* and *Py* are compared with each other to verify the performance after injecting fault data, Figure 4a detects a fault, and Figure 4b shows short-term fluctuations. Then, the statistics fall back above the threshold. According to the comparison and verification of the FD results, it was proved that the fault detection at the 500th sample was accurate. Based on the detection results shown in Figure 5, it was observed that statistics were above the threshold before the injection failure time, so false positives have occurred. Moreover, after the injection fault, there is a fluctuation of the statistical value lower than the threshold value, and there is a situation of false negatives. Based on the detection results shown in Figure 6, in Figure 6a it was observed that statistics were above the threshold before the injection failure time, so false positives have occurred. Additionally, after the injection fault, there is a fluctuation of the statistical value lower than the threshold value, and there is a situation of false negatives. Based on the detection results shown in Figure 6b, detection after fault injection is normal, but the fluctuation of statistical value before injecting fault data was partly above the threshold, so a false positive situation had occurred.

Based on the detection results shown in Figure 7. Based on the detection results shown in Figure 7a. *T*<sup>2</sup> statistics showed a few statistical fluctuations higher than the threshold before the fault injection, so false positive situations have occurred. SPE statistics fluctuated a few times above the threshold before the injection of failure signals and kept below the threshold many times after the fault injection. There are serious false positives and omissions. In Figure 7b it was observed that the short-term or instantaneous fluctuations of *T*<sup>2</sup> scores were above the threshold before the fault injection time, so false positive situations had occurred. In the detection results of this method, SPE statistics fluctuation indicates that the SPE scores fluctuate higher than the thresholds before a fault signal is added and then if the SPE scores remain below the threshold, thus, false negative situations have occurred.

As shown in Figure 8, the receiver operating characteristic (ROC) curves of CCA-JITL, CCA, PLS-JITL, PLS, PCA-JITL and PCA are compared. The ROC curves of each method from top to bottom represent the ROC curves of the model when *T*<sup>2</sup> statistics and SPE statistics are used, respectively. Combined with the area under the curve (AUC) the score of each method shown in Table 1. It proved that the performance of CCA-JITL is better than other methods. The AUC values of CCA-JITL, PLS-JITL and PCA-JITL were mostly higher compared with those of PCA, CCA and PLS. The AUC scores of the models increases after adding JITL.

**Figure 8.** The ROC curves of FD models. (**a**) A ROC curve of CCA and JITL (*Px*); (**b**) A ROC curve of CCA and JITL (*Py*); (**c**) A ROC curve of CCA; (**d**) A ROC curve of PCA and JITL; (**e**) A ROC curve of PCA; (**f**) A ROC curve of PLS and JITL; (**g**) A ROC curve of PLS.

Comparisons of FAR and FDR measures on FD models using CCA-JITL, CCA, PLS-JITL, PLS, PCA-JITL and PCA are shown in Table 1. By comparing FAR and FDR among all algorithms, CCA-JITL worked best for online testing. FAR and FDR scores were calculated regarding *T*<sup>2</sup> and SPE statistics, respectively. The average scores of FAR and FDR were considered to be used in the result comparison. Compared with the CCA method, the av-

erage FAR score of CCA-JITL was reduced by 54.44% across both *T*<sup>2</sup> and SPE measures, and the average FDR score was increased by 34.05%. Compared with algorithms based on PCA-JITL or PLS-JITL, the FAR score of CCA-JITL is lower. The JITL component improved all the FDR scores of all the FD models. PLS, PCA, and CCA showed that the FDR score increased by 32.20%, 29.2%, and 34.05%, respectively, after using JITL. Since the variables of the real data used in this paper are not independent, CCA-JITL method are more favorable for FD of the data in the running gears. The feasibility of the proposed algorithms were testified by the above comparative experiments. JITL is also useful to shape the visual representation of data fitting so that the fault signals were displayed more distinguished.


**Table 1.** Online Testing Results of FD models.

The model was tested using a new 1000×8 data set as the independent testing set. According to Table 2, compared with the CCA model, the results of independent testing of the CCA-JITL FD model showed that the AUC score increased, FAR decreased by 33.2%, and FDR increased by 60.65%. It proved that this approach is generalizable and still had good performance when random new data was applied.

**Table 2.** Independent testing of FD models.


### **5. Conclusions and Future Studies**

In this study, the proposed algorithms have demonstrated significant advantages on the fault detectability in the running gear systems. This paper presents an FD algorithm based on CCA and JITL. After data preprocessing and normalization, CCA transforms highdimension historical input data matrices from the database into low-dimension subspaces to maximize correlations between the most important latent dimensions. Then, online input sample data is mapped to these subspaces with coordinates. Finally, JITL components measure Euclidean similarity between query samples and historical samples in subspaces and search subsets of query sample data points with largest distance to training data to

build local fault detection models. The evaluation results of the case study showed CCA-JITL outperformed traditional CCA very much in terms of FAR and FDR. This approach was also applied to the FD models based on PCA and PLS and achieved better outcomes, which suggested our system infrastructure was transferable to PCA and PLS FD models.

In future, there are still many research directions that are worth further study. The evaluation results in Tables 1 and 2 suggested that PCA, PLS and CCA FD models have their unique strengths using different evaluation methods, and thus, the study of model fusion strategies will be promising. Moreover, only FD was investigated in this paper, without classifying and diagnosing positions and categories of faults. Different types of FDD machine learning models will be meaningful to detect specific failure points. Another possible direction for optimization is to change the fitting methods of JITL, such as clustering, and the derivation method of CCA, such as Kernel-based CCA, to enhance the performance of the systems. The third possibility is to improve the scope of the model, that is, how to apply the models to dynamic systems. Furthermore, the research investigation on how to support multi-sensor data acquisition will be very useful, for instance, the data acquisition system using FUSED deposition modeling [39]. Moreover, the method of using prior prediction to detect the remaining useful life is also an important research direction. These research topics will be considered in order to successfully implement and deliver real-world FDD applications for high-speed train running gear systems of high-speed trains.

**Author Contributions:** Conceptualization, H.Z.; supervision, C.C.; writing—original draft preparation, K.Z.; visualization, Z.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
