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In practical engineering, the vibration-based fault diagnosis with few failure samples is gaining more and more attention from researchers, since it is generally hard to collect sufficient failure records of centrifugal pumps. In such circumstances, effective feature extraction becomes quite vital, since there may not be enough failure data to train an end-to-end classifier, like the deep neural network (DNN). Among the feature extraction, the entropy combined with signal decomposition algorithms is a powerful choice for fault diagnosis of rotating machinery, where the latter decomposes the non-stationary signal into multiple sequences and the former further measures their nonlinear characteristics. However, the existing entropy generally aims at processing the 1D sequence, which means that it cannot simultaneously extract the fault-related information from both the time and frequency domains. Once the sequence is not strictly stationary (hard to achieve in practices), the useful information will be inevitably lost due to the ignored domain, thus limiting its performance. To solve the above issue, a novel entropy method called time-frequency entropy (TfEn) is proposed to jointly measure the complexity and dynamic changes, by taking into account nonlinear behaviors of sequences from both dimensions of time and frequency, which can still fully extract the intrinsic fault features even if the sequence is not strictly stationary. Successively, in order to eliminate the redundant components and further improve the diagnostic accuracy, recursive feature elimination (RFE) is applied to select the optimal features, which has better interpretability and performance, with the help of the supervised embedding mechanism. To sum up, we propose a novel two-stage method to construct the fault representation for centrifugal pumps, which develops from the TfEn-based feature extraction and RFE-based feature selection. The experimental results using the real vibration data of centrifugal pumps show that, with extremely few failure samples, the proposed method respectively improves the average classification accuracy by 12.95% and 33.27%, compared with the mainstream entropy-based methods and the DNN-based ones, which reveals the advantage of our methodology.

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