# Weak Fault Detection for Gearboxes Using Majorization–Minimization and Asymmetric Convex Penalty Regularization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Unique dictionaries’ atoms and optimal wavelet basis cannot simultaneously match the natural structure of every real vibration signal well;
- (2)
- A large number of observed signals should be collected to form a training dictionary before diagnosis, which is always infeasible in practical applications;
- (3)
- Computational complexity and time-consuming problems occur simultaneously in dictionary training, such as with the K-SVD training and SI-K-SVD dictionaries training [31].

- (1)
- Generally, the penalty functions that are established in a low-rank matrix approximation (LRMA) model are symmetric functions, e.g., the absolute value function (AVF); the common drawback is that this penalty function is non-differentiable at the zero point, which can lead to some numerical issues, such as a local optimum and early termination of algorithm.
- (2)
- In a conventional low-rank matrix approximation (LRMA) method, the convex regularizer, such as the L1-norm, usually underestimates the sparse signal when the absolute value function (AVF) is used as a sparsity regularizer; the nonconvex regularizer suffers from several issues, such as a strict convexity problem of objective cost function (OCF), a non-convergence problem, etc. Additionally, both the convex and nonconvex regularizers shrink all the coefficients equally and remove too much energy from the useful signal, resulting in the estimation of the fault signal becoming more challenging.
- (3)
- When the useful fault characteristics signals are very weak but additive noise extremely strong, the conventional LRMA method cannot estimate low-dimensional feature distribution accurately.

## 2. Majorization–Minimization Algorithm

^{k}), G(u, u

^{k}), which is called the majorization function of F(u) at the k-th iteration point u

^{k}. More specifically, consider the minimization problem

^{k}) is a continuously differentiable function, satisfying

- (1)
- Initialize u
^{0}and k = 0; - (2)
- Construct a majorization function G(u, u
^{k}); - (3)
- Operate the iteration ${u}^{(k+1)}=\mathrm{arg}\underset{u}{\mathrm{min}}G(u,{u}^{(k)})$;
- (4)
- If the stopping criterion is satisfied, then output ${u}^{\mathrm{opt}}$; otherwise, k = k + 1, and go to step (2);
- (5)
- Output ${u}^{\mathrm{opt}}$.

## 3. Asymmetric Convex Penalty Regularization Algorithm

#### 3.1. Sparse Representation and Filter Banks

**LPF**is a specified low-pass filter. Substituting Equation (7) into $\stackrel{\wedge}{f}\approx f$, we have

**H**is a specified high-pass filter (i.e., the

**HPF**in Equation (11)), λ is a regularization parameter, and

**D**is a matrix defined as $\mathit{D}=\left[\begin{array}{cccc}-1,\text{}1& & & \\ & \cdots & & \\ & & \cdots & \\ & & & -1,\text{}1\end{array}\right]$, which controls the sparsity of the approximating value of x. If x is a sparse signal, i.e., most of the amplitude values in x tend to zero, then the problem in Equation (12) can also be solved by the L1-norm fused lasso optimization (LFLO) algorithm, i.e.,

_{0}and λ

_{1}are regularization parameters. The solution of the LFLO algorithm can be given by a soft-threshold function [43]. In that case, we have

#### 3.2. Asymmetric Convex Penalty Regularization Model

**D**

_{1}is defined as ${\mathit{D}}_{1}=\left[\begin{array}{cccc}-1,\text{}1& & & \\ & \cdots & & \\ & & \cdots & \\ & & & -1,\text{}1\end{array}\right]$, and the matrix

**D**

_{2}is defined as ${\mathit{D}}_{2}=\left[\begin{array}{cccccc}-1,& 2,& -1& & & \\ & -1,& 2,& -1& & \\ & & & \cdots & & \\ & & & -1,& 2,& -1\end{array}\right]$. The innovations of the novel compound regularizer model are as follows:

- (I)
- The M-term compound regularizers estimate the fault transient impulses;
- (II)
- The compound regularizers model consists of symmetric and asymmetric penalty functions, wherein the symmetric penalty function is a differentiable function, compared with the nondifferentiable function $\Vert {x}_{i}\Vert $ at i = 0.
- (III)
- The MM algorithm is introduced for the solution of the proposed compound regularization method.

#### 3.3. The Solution of the Proposed Model Based on the Majorization–Minimization Algorithm

- (a)
- The majorizer of the symmetric and differentiable function $\varphi ({[{\mathit{D}}_{i}x]}_{n})$ based on the MM algorithm.
- (b)
- The majorizer of the asymmetric and differentiable function ${\theta}_{\epsilon}({x}_{n};r)$ based on the MM algorithm.
- (c)
- The majorizer of the objective cost function F(x) based on the MM algorithm.

^{−1}into Equation (39), we have

- (1)
**Input**: signal y, r ≥ 1, matrix**A**, matrix**B**, ${\lambda}_{i},i=0,1,\dots ,M$, k = 0;- (2)
- $\mathit{E}={\mathit{B}}^{T}\mathit{B}{\mathit{A}}^{-1}y-{\lambda}_{0}{\mathit{A}}^{T}b$
- (3)
- Initialize x = y;
- (4)
- Repeat the following iterations:$${[\mathbf{\Gamma}(v)]}_{n}=(1+r)/4\left|{v}_{n}\right|,\left|{v}_{n}\right|\ge \epsilon $$$${[\mathbf{\Gamma}(v)]}_{n}=(1+r)/4\epsilon ,\left|{v}_{n}\right|\le \epsilon ;$$$${[\mathbf{\Lambda}({\mathit{D}}_{i}v)]}_{n}=\frac{{\varphi}^{\prime}({[{\mathit{D}}_{i}v]}_{n})}{{[{\mathit{D}}_{i}v]}_{n}},\hspace{1em}i=0,1,2,\dots ,M;$$$${\mathit{M}}^{(k)}=2{\lambda}_{0}\mathbf{\Gamma}({x}^{(k)})+{\displaystyle \sum _{i=1}^{M}{\lambda}_{i}{\mathit{D}}_{i}^{T}[\mathbf{\Lambda}({\mathit{D}}_{i}{x}^{(k)})]{\mathit{D}}_{i}};$$$${\mathit{Q}}^{(k)}={\mathit{B}}^{T}\mathit{B}+{\mathit{A}}^{T}{\mathit{M}}^{(k)}\mathit{A};$$$${x}^{(k+1)}=\mathit{A}{[{\mathit{Q}}^{(k)}]}^{-1}\mathit{E};$$
- (5)
- If the stopping criterion is satisfied, then output signal x—otherwise, k = k + 1, and go to step (4).
- (6)
**Output**: signal x.

#### 3.4. Parameter Selection

_{0}and β

_{1}are the constants, so as to maximize the signal-to-noise (SNR); here, parameters β

_{0}and γ are typically set up to be constant value, i.e., β

_{0}= [0.5, 1], γ = [7.5, 8]. In practice, the SD of the background noise in Equation (44) can be computed using both the fault signal and healthy data under same operating environment. Moreover, when the healthy data is not available or is unknown, the standard deviation of the background noise can still be estimated by the following formula:

## 4. Numerical Simulation

_{0}= 1 m/s

^{2}is the intensity of the fault impulse impact, A

_{1}= 0.1 m/s

^{2}is the intensity of the systematic vibration signal, and the damping ratio is a = 0.1; in addition, f

_{n}= 2000 Hz represents the natural frequency of excited structure, the length of vibration signal is N = 8192, the rotating frequencies are f

_{1}= 280 Hz and f

_{2}= 400 Hz, and the sampling frequency f

_{s}= 20 KHz. Additionally, in this simulated case, the additive white noise x

_{3}(t) with SDs sigma = 0.5, sigma = 0.7, and sigma = 0.9 are respectively added to the simulated signal, in order to test the noise tolerance of the proposed ACPR method. Therefore, it can be calculated that the gear fault frequency is 100 Hz (because the repetition interval T = 0.01, the sampling point of a single period is NT = round(fs*T), and the sampling time series is t

_{0}= (0:NT-1)/fs; therefore, the resonant frequency is 100 Hz). Figure 1a,b depict the obtained periodical impulse of the gear fault and natural modulated signal, respectively.

_{0}= 0.7 and γ = 7.5. The parameters of the proposed ACPR can be obtained as follows: the standard deviation σ = 0.5, regularization term parameters λ

_{0}= 0.7 × 0.5 = 0.35, λ

_{1}= 7.5 × (1 − 0.7) × 0.5 = 1.125, and λ

_{2}= 0.7 × 0.5 = 0.35.

## 5. Experimental Validation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Jing, L.Y.; Zhao, M.; Li, P.; Xu, X.Q. A convolutional neural network based feature learning and fault diagnosis method for the condition monitoring of gearbox. Measurement
**2017**, 111, 1–10. [Google Scholar] [CrossRef] - Li, Y.B.; Li, G.Y.; Yang, Y.T.; Liang, X.H.; Xu, M.Q. A fault diagnosis scheme for planetary gearboxes using adaptive multi-scale morphology filter and modified hierarchical permutation entropy. Mech. Syst. Signal Process.
**2018**, 105, 319–337. [Google Scholar] [CrossRef] - Kia, S.H.; Henao, H.; Capolino, G.A. Fault index statistical study for gear fault detection using stator current space vector analysis. IEEE Trans. Ind. Appl.
**2016**, 52, 781–4788. [Google Scholar] [CrossRef] - Teng, W.; Ding, X.; Zhang, X.L.; Liu, Y.B.; Ma, Z.Y. Multi-fault detection and failure analysis of wind turbine gearbox using complex wavelet transform. Renew. Energy
**2016**, 93, 591–598. [Google Scholar] [CrossRef] - Hemmati, F.; Orfali, W.; Gadala, M.S. Roller bearing acoustic signature extraction by wavelet packet transform, applications in fault detection and size estimation. Appl. Acoust.
**2016**, 104, 101–118. [Google Scholar] [CrossRef] - Qu, J.X.; Zhang, Z.S.; Gong, T. A novel intelligent method for mechanical fault diagnosis based on dual-tree complex wavelet packet transform and multiple classifier fusion. Neurocomputing
**2016**, 171, 837–853. [Google Scholar] [CrossRef] - Wang, L.H.; Zhao, X.P.; Wu, J.X.; Xie, Y.Y.; Zhang, Y.H. Motor fault diagnosis based on short-time Fourier transform and convolutional neural network. Chin. J. Mech. Eng.
**2017**, 30, 1357–1368. [Google Scholar] [CrossRef] - Bouchikhi, E.H.E.; Choqueuse, V.; Benbouzid, M.E.H. Current frequency spectral subtraction and its contribution to induction machines bearings condition monitoring. IEEE Trans. Energy Convers.
**2013**, 28, 135–144. [Google Scholar] [CrossRef] [Green Version] - Wang, H.C.; Chen, J.; Dong, G.M. Feature extraction of rolling bearing’s early weak fault based on EEMD and tunable Q-factor wavelet transform. Mech. Syst. Signal Process.
**2014**, 48, 103–119. [Google Scholar] [CrossRef] - Li, Q.; Liang, S.Y. Bearing incipient fault diagnosis based upon maximal spectral kurtosis TQWT and group sparsity total variation de-noising approach. J. Vibroeng.
**2018**, 20, 1409–1425. [Google Scholar] [CrossRef] - Li, R.Y.; He, D. Rotational machine health monitoring and fault detection using EMD-based acoustic emission feature quantification. IEEE Trans. Instrum. Meas.
**2012**, 61, 990–1001. [Google Scholar] [CrossRef] - Yuan, J.; Ji, F.; Gao, Y.; Zhu, J.; Wei, C.J.; Zhou, Y. Integrated ensemble noise-reconstructed empirical mode decomposition for mechanical fault detection. Mech. Syst. Signal Process.
**2018**, 104, 323–346. [Google Scholar] [CrossRef] - Wang, L.; Liu, Z.W.; Miao, Q.; Zhang, X. Time-frequency analysis based on ensemble local mean decomposition and fast kurtogram for rotating machinery fault diagnosis. Mech. Syst. Signal Process.
**2018**, 103, 60–75. [Google Scholar] [CrossRef] - Li, Q.; Liang, S.Y.; Song, W.Q. Revision of bearing fault characteristic spectrum using LMD and interpolation correction algorithm. Procedia CIRP
**2016**, 56, 182–187. [Google Scholar] [CrossRef] - Li, Z.X.; Jiang, Y.; Guo, Q.; Hu, C.; Peng, Z.X. Multi-dimensional variational mode decomposition for bearing-crack detection in wind turbines with large driving-speed variations. Renew. Energy
**2018**, 116, 55–73. [Google Scholar] [CrossRef] - Li, Q.; Ji, X.; Liang, S.Y. Incipient fault feature extraction for rotating machinery based on improved AR-minimum entropy deconvolution combined with variational mode decomposition approach. Entropy
**2017**, 19, 7. [Google Scholar] [CrossRef] - Zhang, R.; Tao, H.Y.; Wu, L.F.; Guan, Y. Transfer learning with neural networks for bearing fault diagnosis in changing working conditions. IEEE Access
**2017**, 5, 14347–14357. [Google Scholar] [CrossRef] - Masud, A.A.; Albarracín, R.; Rey, J.A.A.; Sukki, F.M.; Illias, H.A.; Bani, N.A.; Munir, A.B. Artificial Neural Network Application for Partial Discharge Recognition: Survey and Future Directions. Energies
**2016**, 9, 574. [Google Scholar] [CrossRef] - He, M.; He, D. Deep learning based approach for bearing fault diagnosis. IEEE Trans. Ind. Appl.
**2017**, 53, 3057–3065. [Google Scholar] [CrossRef] - Jia, F.; Lei, Y.G.; Guo, L.; Lin, J.; Xing, S.B. A neural network constructed by deep learning technique and its application to intelligent fault diagnosis of machines. Neurocomputing
**2018**, 272, 619–628. [Google Scholar] [CrossRef] - Li, Q.; Liang, S.Y. Multiple faults detection for rotating machinery based on Bi-component sparse low-rank matrix separation approach. IEEE Access
**2018**, 6, 20242–20254. [Google Scholar] [CrossRef] - Tang, H.F.; Chen, J.; Dong, G.M. Sparse representation based latent components analysis for machinery weak fault detection. Mech. Syst. Signal Process.
**2014**, 46, 373–388. [Google Scholar] [CrossRef] - Zhou, H.T.; Chen, J.; Dong, G.M.; Wang, R. Detection and diagnosis of bearing faults using shift-invariant dictionary learning and hidden Markov model. Mech. Syst. Signal Process.
**2016**, 72–73, 65–79. [Google Scholar] [CrossRef] - Yang, B.Y.; Liu, R.N.; Chen, X.F. Fault diagnosis for a wind turbine generator bearing via sparse representation and shift-invariant K-SVD. IEEE Trans. Ind. Inform.
**2017**, 13, 1321–1331. [Google Scholar] [CrossRef] - Feng, Z.P.; Liang, M. Complex signal analysis for planetary gearbox fault diagnosis via shift invariant dictionary learning. Measurement
**2016**, 90, 382–395. [Google Scholar] [CrossRef] - Qin, Y. A new family of model-based impulsive wavelets and their sparse representation for rolling bearing fault diagnosis. IEEE Trans. Ind. Electron.
**2018**, 65, 2716–2726. [Google Scholar] [CrossRef] - Cui, L.L.; Wang, J.; Lee, S. Matching pursuit of an adaptive impulse dictionary for bearing fault diagnosis. J. Sound Vib.
**2014**, 333, 2840–2862. [Google Scholar] [CrossRef] - Cui, L.L.; Gong, X.Y.; Zhang, J.Y.; Wang, H.Q. Double-dictionary matching pursuit for fault extent evaluation of rolling bearing based on the Lempel-Ziv complexity. J. Sound Vib.
**2016**, 385, 372–388. [Google Scholar] [CrossRef] - Cui, L.L.; Wu, N.; Ma, C.Q.; Wang, H.Q. Quantitative fault analysis of roller bearings based on a novel matching pursuit method with a new step-impulse dictionary. Mech. Syst. Signal Process.
**2016**, 68–69, 34–43. [Google Scholar] [CrossRef] - Ding, X.X.; He, Q.B. Time-frequency manifold sparse reconstruction: A novel method for bearing fault feature extraction. Mech. Syst. Signal Process.
**2016**, 80, 392–413. [Google Scholar] [CrossRef] - Li, Q.; Liang, S.Y. Incipient fault diagnosis of rolling bearings based on impulse-step impact dictionary and re-weighted minimizing nonconvex penalty Lq regular technique. Entropy
**2017**, 19, 8. [Google Scholar] [CrossRef] - Ding, Y.; He, W.P.; Chen, B.Q.; Zi, Y.Y.; Selesnick, I.W. Detection of faults in rotating machinery using periodic time-frequency sparsity. J. Sound Vib.
**2016**, 382, 357–378. [Google Scholar] [CrossRef] [Green Version] - He, W.P.; Ding, Y.; Zi, Y.Y.; Selesnick, I.W. Sparsity-based algorithm for detecting faults in rotating machines. Mech. Syst. Signal Process.
**2016**, 72–73, 46–64. [Google Scholar] [CrossRef] - He, W.P.; Ding, Y.; Zi, Y.Y.; Selesnick, I.W. Repetitive transients extraction algorithm for detecting bearing faults. Mech. Syst. Signal Process.
**2017**, 84, 227–244. [Google Scholar] [CrossRef] [Green Version] - Zhang, H.; Chen, X.F.; Du, Z.H.; Li, X.; Yan, R.Q. Nonlocal sparse model with adaptive structural clustering for feature extraction of aero-engine bearings. J. Sound Vib.
**2016**, 368, 223–248. [Google Scholar] [CrossRef] - Du, Z.H.; Chen, X.F.; Zhang, H.; Yang, B.Y.; Zhai, Z.; Yan, R.Q. Weighted low-rank sparse model via nuclear norm minimization for bearing fault detection. J. Sound Vib.
**2017**, 400, 270–287. [Google Scholar] [CrossRef] - Mourad, N.; Reilly, J.P.; Kirubarajan, T. Majorization minimization for blind source separation of sparse sources. Signal Process.
**2017**, 131, 120–133. [Google Scholar] [CrossRef] - Qiu, T.Y.; Palomar, D.P. Undersampled sparse phase retrieval via majorization minimization. IEEE trans. Signal Process.
**2017**, 65, 5957–5969. [Google Scholar] [CrossRef] - Ndoye, M.; Anderson, J.M.M.; Greene, D.J. An MM-based algorithm for ℓ1-regularized least-squares estimation with an application to ground penetrating radar image reconstruction. IEEE Trans. Image Process.
**2016**, 25, 2206–2221. [Google Scholar] [CrossRef] [PubMed] - Donoho, D.L.; Elad, M.; Temlyakov, V.N. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory
**2006**, 52, 6–18. [Google Scholar] [CrossRef] [Green Version] - Donoho, D.L. Compressed sensing. IEEE Trans. Inform. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Donoho, D.L.; Tsaig, Y. Fast solution of ℓ1-norm minimization problems when the solution may be sparse. IEEE Trans. Inform. Theory
**2008**, 54, 4789–4812. [Google Scholar] [CrossRef] - Donoho, D.L. De-noising by soft-thresholding. IEEE Trans. Inf. Theory
**1995**, 41, 613–627. [Google Scholar] [CrossRef] [Green Version] - Selesnick, I.W. Total variation denoising via the Moreau envelope. IEEE Signal Process. Lett.
**2017**, 24, 216–220. [Google Scholar] [CrossRef] - Selesnick, I.W.; Parekh, A.; Bayram, I. Convex 1-D total variation denoising with non-convex regularization. IEEE Signal Process. Lett.
**2015**, 22, 141–144. [Google Scholar] [CrossRef] - Selesnick, I.W.; Graber, H.L.; Pfeil, D.S.; Barbour, R.L. Simultaneous low-pass filtering and total variation denoising. IEEE Trans. Signal Process.
**2014**, 62, 1109–1124. [Google Scholar] [CrossRef] - Ning, X.R.; Selesnick, I.W.; Duval, L. Chromatogram baseline estimation and denoising using sparsity (BEADS). Chemom. Intell. Lab.
**2014**, 139, 156–167. [Google Scholar] [CrossRef] - Donoho, D.L.; Johnstone, I.M. Ideal spatial adaptation by wavelet shrinkage. Biometrika
**1993**, 81, 425–455. [Google Scholar] [CrossRef] - Annual Conference of the Prognostics and Health Management Society 2009. Available online: https://www.phmsociety.org/events/conference/phm/09 (accessed on 5 June 2018).
- Atat, H.A.; Siegel, D.; Lee, J. A systematic methodology for gearbox health assessment and fault classification. Int. J. Progn. Health Manag.
**2011**, 2, 1. [Google Scholar]

**Figure 1.**Simulated synthetic signal. (

**a**) Faulty periodical transient impulses of a gearbox; and (

**b**) the systematic natural modulated signal.

**Figure 2.**A simulated synthetic signal, its detected impulses, and its envelope spectrum, using the proposed non-convex penalty regularization and L1-norm fused lasso optimization (LFLO) method, when noise standard deviation is 0.5. (

**a**) Simulated synthetic signal; (

**b**) envelope spectrum of the simulated synthetic signal; (

**c**) detected impulses using the proposed ACPR method; (

**d**) envelope spectrum of detected impulses using proposed ACPR method; (

**e**) detected impulses using the nonconvex penalty regularization (NCPR) method; (

**f**) envelope spectrum of detected impulses using the NCPR method; (

**g**) detected impulses using the LFLO method; (

**h**) envelope spectrum of detected impulses using the LFLO method.

**Figure 3.**A simulated synthetic signal, its detected impulses, and its envelope spectrum, using the proposed non-convex penalty regularization and LFLO methods, when noise standard deviation sigma is 0.7. (

**a**) Simulated synthetic signal; (

**b**) envelope spectrum of simulated synthetic signal; (

**c**) detected impulses using the proposed ACPR method; (

**d**) envelope spectrum of detected impulses using the proposed ACPR method; (

**e**) detected impulses using the NCPR method; (

**f**) envelope spectrum of detected impulses using the NCPR method; (

**g**) detected impulses using the LFLO method; and (

**h**) envelope spectrum of detected impulses using the LFLO method.

**Figure 4.**The simulated synthetic signal, its detected impulses, and its envelope spectrum, using the proposed non-convex penalty regularization and LFLO methods, when the noise standard deviation sigma is 0.9. (

**a**) Simulated synthetic signal; (

**b**) envelope spectrum of the simulated synthetic signal; (

**c**) detected impulses using the proposed ACPR method; (

**d**) envelope spectrum of detected impulses using the proposed ACPR method; (

**e**) detected impulses using the NCPR method; (

**f**) envelope spectrum of detected impulses using the NCPR method; (

**g**) detected impulses using the LFLO method; and (

**h**) envelope spectrum of detected impulses using the LFLO method.

**Figure 5.**Experimental setup for the gearbox multiple faults. (

**a**) Overview of the experimental apparatus; (

**b**) the internal structure the gearbox; (

**c**) the internal structure of the gear meshing; (

**d**) the installation location of input shaft accelerometer; (

**e**) the installation location of the output shaft accelerometer; and (

**f**) the gears with failure [49,50].

**Figure 7.**The raw vibration signal and its Hilbert envelope spectrum. (

**a**) The raw vibration signal and (

**b**) the Hilbert envelope spectrum of that raw vibration signal.

**Figure 8.**The fault information extracted through the proposed approach. (

**a**) The time-domain waveform of the extracted fault signal and (

**b**) the envelope spectrum of the extracted fault signal.

**Figure 9.**The fault information extracted through the NCPR approach. (

**a**) The time-domain waveform of extracted fault signal and (

**b**) the envelope spectrum of the extracted fault signal.

**Figure 10.**The fault information extracted through the LFLO approach. (

**a**) The time-domain waveform of the extracted fault signal and (

**b**) the envelope spectrum of the extracted fault signal.

Functions | $\mathit{\varphi}\mathbf{(}\mathit{x}\mathbf{)}$ | ${\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathit{x}\mathbf{)}$ |
---|---|---|

${\varphi}_{A}(x)$ | $\Vert x\Vert $ | Signal(x) |

${\varphi}_{B}(x)$ | $\sqrt{{\left|x\right|}^{2}+\epsilon}$ | $x/\sqrt{{\left|x\right|}^{2}+\epsilon}$ |

${\varphi}_{c}(x)$ | $\left|x\right|-\epsilon \mathrm{log}(\left|x\right|+\epsilon )$ | $x/(\left|x\right|+\epsilon )$ |

**Table 2.**Parameter settings of the proposed asymmetric convex penalty regularization (ACPR) method for gear fault detection.

Regularization Parameter λ_{0} | Regularization Parameter λ_{1} | Regularization Parameter λ_{2} | M-Term | Iteration Times |
---|---|---|---|---|

λ_{0} = 0.35 | λ_{1} =1.125 | λ_{2} = 0.35 | 2 | 50 |

Noise Standard Deviation | ACPR Algorithm | NCPR Algorithm | LFLO Algorithm |
---|---|---|---|

sigma = 0.5 | 0.288113 s | 0.05391 s | 0.000486 s |

sigma = 0.7 | 0.283184 s | 0.073986 | 0.000743 s |

sigma = 0.9 | 0.308288 s | 0.052860 s | 0.000524 s |

Regularization Parameter λ_{0} | Regularization Parameter λ_{1} | Regularization Parameter λ_{2} | M-Term | Iteration Times |
---|---|---|---|---|

λ_{0} = 0.02863 | λ_{1} = 0.09203 | λ_{2} = 0.02863 | 2 | 50 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Q.; Liang, S.Y.
Weak Fault Detection for Gearboxes Using Majorization–Minimization and Asymmetric Convex Penalty Regularization. *Symmetry* **2018**, *10*, 243.
https://doi.org/10.3390/sym10070243

**AMA Style**

Li Q, Liang SY.
Weak Fault Detection for Gearboxes Using Majorization–Minimization and Asymmetric Convex Penalty Regularization. *Symmetry*. 2018; 10(7):243.
https://doi.org/10.3390/sym10070243

**Chicago/Turabian Style**

Li, Qing, and Steven Y. Liang.
2018. "Weak Fault Detection for Gearboxes Using Majorization–Minimization and Asymmetric Convex Penalty Regularization" *Symmetry* 10, no. 7: 243.
https://doi.org/10.3390/sym10070243