## 4 Experiments
### Discussion  
Considering all the above-mentioned experimental results, the five main conclusions are listed as follows:  
1. TfEn can extract more intrinsic fault features of non-stationary series from both time and frequency domains.  
As shown in Figures 11 and 12, TfEn achieves the higher classification accuracy than all the other existing entropy methods. Through jointly taking into account nonlinear behaviors from both dimensions of time and frequency, TfEn can extract more intrinsic fault features, even if the decomposed components are non-stationary or approximately stationary (shown in Figures 8 and 9), which avoids the risk of losing useful information caused by existing entropy methods aimed at processing the 1D sequence.  
2. TfEn achieves the good balance between the performance and calculation efficiency.  
As shown in Table 3, compared with the existing SampleEn (2862.31 s) and ApEn (2740.10 s), t TfEn presents the significantly shorter running time (1.61 s) but the higher classification, since it does not rely on the state space reconstruction and lots of loop comparisons that need considerable computation resources. Additionally, compared with another two entropy methods, SvdEn (0.24 s) and SpectEn (0.14 s), the running time of TfEn is slightly longer (1.61 s), while it brings about the considerable improvement on classification accuracy with variable training sizes (shown in Table 5). The relatively better balance between performance and calculation efficiency makes TfEn more suitable for practices.  
3. RFE exhibits superior performance and interpretability than the mainstream PCA and KPCA.  
As shown in Figure 10 and Table 2, using the RFE, the difference of TfEn features among each fault mode is further highlighted, and its average classification accuracy is improved by 11.20% and 17.60%, compared with mainstream PCA and KPCA. For the other four entropy methods, the RFE also brings about the better results than the PCA and KPCA-based ones (shown in Figure 11 and Table 5), which reveals its superiority on the feature selection. Besides that, the optimal features selected by the RFE retain their original physical meanings and thus having the better interpretability (shown in Table 1).  
4. Effective feature extraction designed manually is still necessary under the scenery of few failure samples.  
As described in Section 4.3, various sizes of training samples (4/6/8/10/12/14) are defined to test the stability and adaptiveness for methods. According to Figure 12 and Table 5, we can see that, with the help of strong capability on the feature self-learning, the DNN-based methods do achieve superior performance (accuracy of 100%) when training sizes are relatively large (size of 12/14 per fault mode). However, their classification accuracy is overtaken by the entropy-based methods designed manually when the training samples are gradually decreased, which means that the effective manual feature extraction is still necessary for the practices where the failure data is usually hard to collect.  
5. The proposed method provides a novel concept for fault diagnosis subject to the limitation of few failure samples.  
With the help of TfEn and RFE, robust fault representations are constructed for centrifugal pumps, which presents the superior stability and adaptiveness for the variable sizes of training samples. When the training samples are extremely scarce (size of four per fault mode), the average classification accuracy is significantly improved by 12.95% and 33.27%, compared with the mainstream entropy-based methods and the DNN-based ones. Thanks to the remarkable fault representation, the proposed method has no requirements for massive data, and is thus more suitable for practical engineering.