## 4 Experiments
### Comparisons with Variable Taining Sample Sizes  
In this subsection, two sets of representative methods are applied for comparisons, namely the entropy-based methods and the deep neural network (DNN)-based methods. In addition, in order to further test the adaptability of these methods to extremely few failure samples, a variety of training sample sizes are designed and applied for fault diagnosis  
* Category I entropy-based methods  
In this category, still applying CEEMD for the signal processing, 4 existing mainstream entropy-based feature extraction methods are utilized to implement the comparisons, which consist of the singular value decomposition entropy (SvdEn) [31], sample entropy (SampleEn) [32], approximately entropy (ApEn) [31] and spectral entropy (SpecEn) [33]. Additionally, in the feature selection stage, the PCA and KPCA are still utilized to compare with the RFE, which means that there are 3 classes of representations for each feature extraction method applied to achieve the fault diagnosis, namely, features selected by PCA, features selected by KPCA and features selected by RFE. In the fault classification stage, as in Section 4.2, there are still 10 samples under each mode applied to train an SVM classifier model. The corresponding results are shown in Figure 11.  
As shown in Figure 11, under the condition of 10 training samples per fault mode, the best testing accuracy (100%) is achieved by the proposed method mainly based on TfEn and RFE. For each entropy  
Figure 11: Comparisons with 4 mainstream entropy-based feature extraction methods. method, the RFE-based feature selection outperforms another two based on PCA and KPCA, and the advantages are quite obvious, especially for SvdEn and SpectEn, which illustrates the superiority of the RFE-based feature selection. In addition, using the same feature selection based on RFE, the proposed TfEn obtains a better classification accuracy that the other 4 existing entropy methods, where the accuracy is respectively improved by 12.80%, 5.20%, 5.60% and 6.40%. It further proves that our proposed TfEn can extract the more intrinsic fault representations than the other 4 mainstream entropy methods aimed at processing the 1D sequence, which is thus more suitable for the fault classification especially with few failure samples. In addition to the performance on feature extraction, we compare the calculation efficiency among these entropy methods by the quantified running time, which is illustrated in Table 3.  
As shown in Table 3, due to the state space reconstruction and lots of loops, SampleEn and ApEn take 2862.31 s and 2740.10 s, respectively, to complete the feature extraction for each 50 samples. Although these two entropy methods obtain a slightly higher classification accuracy than SvdEn and SpectEn, the overly low calculation efficiency still makes them almost intolerable in practices. To sum up, the proposed TfEn achieves the best balance between the performance and computation efficiency. Although its running time (1.61 s per 50 samples) is slightly longer than SvdEn (0.24 s per 50 samples) and SpectEn (0.14 s per 50 samples), it is still acceptable, taking into account its considerable performance on fault classification. \(\bullet\) Category II DNN-based methods  
In this category, 3 representative DNN models, namely, multilayer perceptron (MLP) [34], stacked autoencoder (SAE) [35] and convolutional neural network (CNN) [36], are applied to compare with our proposed method. Referring to the related literatures [37; 38], the preprocessing and model architectures are respectively set as follows. Besides that, in order to further test the adaptability of methods to limiting few failure samples, the training sample size under each fault mode is respectively set as 4, 6, 8, 10, 12 and 14 in this subsection. In addition to the above 3 deep models, the 5 entropy-based methods in Category **I** are also applied to the comparisons, which are respectively SvdEn-RFE, SampleEn-RFE, ApEn-RFE, SpecEn-RFE and our proposed TfEn-RFE.