# Bearing Health Monitoring Using Relief-F-Based Feature Relevance Analysis and HMM
## 1 Introduction
Nonlinear dynamic methods emerged as alternatives to the process of feature estimation in machinery fault diagnosis, such as permutation entropy (PE) [20], multiscale entropy (MSE) [21], and multiscale permutation entropy (MPE) [22]. PE measures the randomness of the time series [23]; however, it analyzes the signal using one scale; so, fault information contained in other ranges is not coded. MSE uses a coarse-grained procedure to obtain a multiple-scale time series [24]. In MPE analysis, the entropy of the coarse-grained time series at each scale is calculated by the neighborhood preserving embedding algorithm. Still, MPE can not reveal relevant fault information because of its averaging algorithm. In contrast, authors in [25] proposes a generalized composite multiscale permutation entropy (GCMPE). [PERSON] et al. [26] introduce a fault diagnosis method by combining extreme-point symmetric mode decomposition and composite multiscale weighted permutation entropy. Though entropy-based features can obtain reliable discrimination performances, the parameters estimated lack of suitable interpretability concerning the bearing fault diagnosis. For the learning stage applied to the fault assessment, knowledge-based frameworks have been carried out by different unsupervised and supervised strategies, such as support vector machines [27]; metaheuristic-based algorithms [28]; Bayesian algorithms [29,30]; hidden Markov models (HMM) [31]; and deep learning [32,33]. Notably, authors in [34] introduce a method for diagnosing bearing faults based on deep learning and convolutional neural network layers. They use the raw vibration signals as input data, and the approach does not require any feature estimation. Moreover, authors in [35] propose a capsule network for fault diagnosis, which achieves high diagnostic accuracy in different working conditions. [PERSON] et al. [36] proposed a hierarchical adaptive deep belief network, holding as inputs the frequency domain information and achieving high performance in the diagnosis of bearing fault types and damage levels. Further, authors in [37,38] proposed unsupervised diagnosis models based on frequency-domain features and deep learning. Yet, interpreting the relevance of the learned features through deep learning is not clear, and the overfitting is latent. HMM is a practical stochastic approach to code time-series dependencies that have been demonstrated promising vibration-basedbearing fault diagnosis results [39]. In particular, as a dual random process, HMM has two main parameters, hidden states, and observation vector, which allow revealing the signal patterns from a stochastic perspective. Nevertheless, the learning stage often consists of a vast number of features, e.g., time, frequency, and time-frequency measures, meanwhile, the number of samples is limited due to the workload of data acquisition [40]. Hence, feature embedding and selection techniques are often used for dimensionality reduction to enhance the classification performance. The principal aim of this kind of methods is to reduce the number of features while increasing the classification accuracy [41]. Some feature-embedding techniques have been explored, including principal component analysis (PCA), which is a dimension compression method based on the minimum mean square error [42], and kernel principal component analysis (KPCA); as a nonlinear PCA extension by the introduction of a kernel-based mapping [43]. The former lack suitable representation when dealing with nonlinear dependencies, and the latter computes a feature space that loses the original engineering meaning (data interpretability). Accordingly, other dimensionality reduction methods known as feature selection are recommended [44; 45].