# Weak Multiple Fault Detection Based on Weighted
## 1 Introduction
[23] used basis pursuit to diagnose rolling bearing faults. The result shows that the basis pursuit can represent features with fine resolution in the time-frequency domain, which makes explaining the fault features easy. [PERSON] and [PERSON] [24] applied some typical atomic decomposition methods, such as the method of frames, best orthogonal basis, and matching and basis pursuits, to analyze the vibration signals of damaged gearboxes. Moreover, [PERSON] et al. [25] used the matching pursuit and time-frequency atom to analyze the bearing vibration signals and extract the vibration signatures. In recent years, structural group sparse methods have received extensive attention in the fields of statistics, machine learning, signal processing, computer vision, and biological information [26; 27; 28; 29]. Structural group sparse indicates not only that the signal is sparse, but also that the signal has a simple form of structural sparsity [30]. Group sparsity can be divided into non-overlapping group sparse and overlapping group sparse models. If no coupling exists between adjacent groups in the signal, then constructing a non-overlapping group sparse model can simplify the optimization problem [31]. Non-overlapping group sparse is often not satisfactory to obtain good noise reduction results; therefore, the overlapping group sparse (OGS) model is introduced. For instance, OGS has been implemented for estimating sparse signals in noise [32]. OGS is a special structure in the structural group sparse model, indicating that an overlap exists between adjacent groups. In addition to the sparsity of the wavelet coefficients of the multiple fault features, interrelated structures also exist between the coefficients. OGS has been introduced to the field of fault diagnosis in recent years. For instance, on the basis of the overlapping group shrinkage and majorization-minimization (MM), [PERSON] et al. [33; 34] extracted the periodic group sparse signal from the vibration signal and realized the compound fault diagnosis of rolling bearings. The proposed algorithm requires a priori knowledge of the location of the fault to further separate the different faults. OGS is an effective method for extracting compound faults. However, the following problems also exist. (1) Determining the location of the fault before disassembling is impossible. (2) The literature [33; 34] is based on the prior knowledge of the sparsity of the fault signal itself, but the fault signal is sparse under wavelet transform, and the signal itself is not sparse. In this research, OGS is improved to overcome these shortcomings. In industry, during the degradation of a bearing, multiple types of faults may coexist, and the signal is weak due to the long signal transmission path. Therefore, in this research, a multiple fault feature extraction method based on the WOGS is proposed. Overlaps emerge in the time domain due to the different types of faults; thus, the wavelet transform coefficients must also overlap. The OGS model is constructed on the basis of the overlapping wavelet transform coefficients, and the sparse model is solved by MM. The weight coefficients in the OGS model are then estimated by analyzing the salient features of the vibration signal. Consequently, the weak impulse features in the vibration signal can be enhanced, which are then evaluated for multiple fault diagnosis. The rest of this paper is organized as follows. In Section 2, the OGS is reviewed and the majorization-minimization algorithm for the OGS problem is presented. Section 3 details the proposed algorithm to extract multiple fault features. Section 4 covers the verification of the proposed algorithm by using a simulated signal model. Section 5 presents the experimental study to further validate the proposed algorithm. Finally, Section 6 concludes the paper.