The prognosis of bearings is vital for condition-based maintenance of rotating machinery. This article proposes a systematic prognostic scheme for rolling element bearings. The proposed scheme infers the degradation progression by developing a novel health indicator (HI). This novel HI, derived from the spectral correlation, Wasserstein distance, and linear rectification, can reflect the changes in the probability distribution of all cyclic power-spectra over time. In other words, any form of variation in modulation characteristics can be revealed through the proposed novel indicator, even for the weak information buried by the internal or external noise. Furthermore, the developed HI can eliminate random fluctuations that often impair the remaining useful life (RUL) prediction accuracy. Then, a 3 ${\boldsymbol{\sigma }}$ criterion-based technique is introduced to divide health stages. After that, the gated recurrent unit network is employed to predict the RUL of the bearing system, integrated with the Bayesian optimization algorithm to tune the optimal hyperparameters adaptively. This renders the establishment of an intelligent prognosis model with high prediction accuracy and generalization ability. Finally, experimental validations are conducted using the run-to-failure datasets of bearings. The obtained results demonstrate that the proposed HI has better monotonicity, and the proposed prognostic scheme can predict the RUL with high accuracy.

The theoretical developments of data-driven fault diagnosis methods have yielded fruitful achievements and significantly benefited industry practices. However, most methods are developed based on the assumption of data balance, which is incompatible with engineering scenarios. First, the normal state accounts for the majority of the equipment’s lifespan; second, the probability of various faults varies, both of which result in an imbalance in the data. The consequence of data imbalance in intelligent fault diagnosis methods has attracted extensive attention from the research community, and a significant number of papers have been published. Nevertheless, a comprehensive review of achievements in this field is still missing, and the research perspectives have not been thoroughly investigated. To end this, we review and discuss all the research achievements in fault diagnosis under data imbalance in this paper, based on our best knowledge. First, the existing imbalanced learning methods are classified into three categories: data processing methods, model construction methods, and training optimization methods. Then, the three methodologies are introduced and discussed in detail: the data processing method is to optimize the inputs of the intelligent fault diagnosis model so that the imbalance rate of the sample set involved in training is reduced; the model construction method is to design the structure and the features of the intelligent fault diagnosis model so that the model itself is resistant to the effects of imbalance; the training optimization method is an optimisation of the training process for intelligent fault diagnosis models, raising the importance of the minority class in the training. Finally, this paper summarizes the prospects of the imbalanced learning problem in intelligent fault diagnosis, discusses the possible solutions, and provides some recommendations.

With significant advantages in feature learning, the deep learning based compound fault diagnosis method has brought many successful applications for industrial equipment. However, few studies focus on the interpretability of intelligent compound fault diagnosis methods, and the diagnosis results are hard to interpret which prevents the wide application of these methods in practical industrial scenarios. To solve the above challenging problems, an intelligent and interpretable compound fault diagnosis framework, called wavelet capsule network (WavCapsNet), is proposed for machinery by leveraging the backward tracking technique. First, the WavCapsNet is constructed with a wavelet kernel convolutional layer which is employed to learn the features with interpretable meaning from vibration signals, and two capsule layers which endow the diagnosis model with the ability to decouple the compound fault intelligently. Second, the WavCapsNet is trained and optimized with the normal and single fault samples (without compound fault samples). Finally, the interpretable analysis is launched by backward tracking the coupling matrices in capsule layers, which is focused on the relationship between the learned features and different health conditions. The experimental results on a five-speed transmission dataset show that the proposed method, compared to other methods, not only achieves higher compound fault decoupling accuracy under the scenario of incomplete fault data but also improves the transparency and interpretability in the decision-making process of fault diagnosis.