## 1 Introduction
In the feature selection stage, the redundant components will be eliminated from the fault features and the optimal feature subset with lower dimensions will be obtained. In order to overcome the curse of dimension [17] and further improve the performance of fault diagnosis, there are large amounts of feature selection methods applied to select optimal feature subset from the high-dimensional features. As an unsupervised dimensionality reduction method, principal component analysis (PCA) obtains the most widely applications on the feature selection, because of its simple calculation and stable effects [18; 19]. However, the PCA still has two shortcomings that need to be solved. The first one is that the feature subset selected by PCA is usually difficult to achieve the satisfied diagnostic accuracy, due to its unsupervised mechanism. Another one is that the feature subset is fused from the raw feature set, and it thus damages the original physical meaning. In scenarios where the diagnostic accuracy and feature interpretability are well emphasized, the PCA is limited to further applications, due to the above-mentioned shortcomings [20]. To solve the above-mentioned issues, a novel fault representation method is designed and proposed in this paper, which is mainly composed of the time-frequency entropy (TfEn)-based feature extraction and recursive feature elimination (RFE)-based feature selection. Compared with the existing mainstream entropy-based methods aimed at processing the 1D sequence, our proposed method can jointly highlight and measure the intrinsic characteristics of nonlinear and non-stationary signals from both dimensions of time and frequency, and it can achieve more accurate fault diagnosis, even with limited failure samples. In the feature extraction stage, TfEn is proposed to comprehensively quantify the complexity and dynamic change by taking into account nonlinear behaviors of signals in the form of a 2D time-frequency matrix. In the feature selection stage, RFE is applied to select the optimal feature subset and eliminate the redundant components. Abandoning the feature fusion of PCA, RFE retains the original physical meanings of selected features, and thus has the better interpretability [21]. In addition, it is worth noting that our proposed method has no rigorous requirements for calculation resources and failure samples, which means that it more suitable for practical scenes with few failure samples. To sum up, the main contributions of this paper can be listed as follows:  
1. A novel feature extraction called TfEn is proposed to jointly quantify the robust nonlinear characteristics from both time and frequency dimensions, which overcomes the limitations of existing entropy based on 1D sequence and thus being more suitable for non-stationary signals;
2. RFE-based feature selection is applied to select the optimal feature subset form the high-dimensional feature vector, which achieves better performance on improving the classification accuracy and retaining the feature interpretability compared with mainstream methods;
3. The proposed two-stage fault representation has the capability of highlighting and depicting the intrinsic differences among vibration signals, which presents the stability against the variable sizes of training samples, and obtains the satisfied results even with extremely few failure data;
4. Compared with the existing entropy variants and mainstream deep models, the experiments with the real vibration signals of centrifugal pumps illustrate the validity and superiority of our proposed fault diagnosis method. The remaining of the paper is organized as follows. Section 2 introduces the necessary background. Section 3 describes the proposed fault diagnosis. Section 4 implements the experiments, then presents and discusses the results. Finally, the conclusion is shown in Section 5.