# Group-Sparse Feature Extraction via Ensemble Generalized Minimax-Concave Penalty for Wind-Turbine-Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

#### 2.1. GMC Penalty

#### 2.2. Overlapping Group Shrinking Algorithm

## 3. The Proposed Group-Sparse Feature Extraction Method Based on Ensemble GMC

#### 3.1. Optimization Problem Formulation

#### 3.2. Convexity Condition

#### 3.3. Algorithm Implementation

^{−4}.

Algorithm 1 FBGFE: Forward–backward group feature extraction algorithm |

Input:${y}_{ik}\in {\mathbb{R}}^{N},i\in I,k\in {R}^{+},\lambda >0$,$\lambda $,$\gamma $,$k$ |

Initialization:$\rho =\mathrm{max}\left\{1,\gamma /(1-\gamma )\right\}$,$0<\mu <2/\rho $ |

For $i=0,1,2,\dots $${w}_{ik}^{(i)}=\frac{1}{k}conv({x}_{ik}^{(i)}-\mu ({x}_{ik}^{(i)}+\gamma ({v}_{ik}^{(i)}-{x}_{ik}^{(i)})-{y}_{ik}),h)$ ${v}_{ik}^{(i)}=\frac{1}{k}conv({x}_{ik}^{(i)}-\mu ({x}_{ik}^{(i)}+\gamma ({v}_{ik}^{(i)}-{x}_{ik}^{(i)})-{y}_{ik}),h)$ ${x}_{ik}^{(i+1)}=soft({w}_{ik}^{(i)},\mu \lambda )$ ${v}_{ik}^{(i+1)}=soft({u}_{ik}^{(i)},\mu \lambda )$ endwhere $i$ is the iteration counter. Return: $x$ |

#### 3.4. Remark of the Proposed Algorithm

## 4. Simulation Study

#### 4.1. Simulation Validation

#### 4.2. Selection of Regularization Parameter

## 5. Experimental Validation

#### 5.1. Case 1: High-Speed Bearing Outer-Race Fault

#### 5.2. Case 2: Fault Diagnosis of a Wind Turbine Pinion Gear

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The non−convex degree of two functions varies with the parameter. (

**a**) MC function. (

**b**) GMC function.

**Figure 3.**Noise−free signal and noise signal. (

**a**) Noise-free signal, (

**b**) Noise signal with Gaussian noise of $\sigma =0.1$.

**Figure 4.**Denoising results obtained using the comparison methods and proposed method. (

**a**) Denoising using ${\U0001d4c1}_{1}$ norm regularization with $\lambda =0.14$. (

**b**) Denoising using OGS method with $\lambda =0.07$. (

**c**) Denoising using GMC penalty with $\lambda =0.26$ and $\gamma =0.8$. (

**d**) Denoising using the proposed method with $\lambda =0.15$, $\gamma =0.8$ and $K=3$.

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## Share and Cite

**MDPI and ACS Style**

He, W.; Zhang, P.; Liu, X.; Chen, B.; Guo, B.
Group-Sparse Feature Extraction via Ensemble Generalized Minimax-Concave Penalty for Wind-Turbine-Fault Diagnosis. *Sustainability* **2022**, *14*, 16793.
https://doi.org/10.3390/su142416793

**AMA Style**

He W, Zhang P, Liu X, Chen B, Guo B.
Group-Sparse Feature Extraction via Ensemble Generalized Minimax-Concave Penalty for Wind-Turbine-Fault Diagnosis. *Sustainability*. 2022; 14(24):16793.
https://doi.org/10.3390/su142416793

**Chicago/Turabian Style**

He, Wangpeng, Peipei Zhang, Xuan Liu, Binqiang Chen, and Baolong Guo.
2022. "Group-Sparse Feature Extraction via Ensemble Generalized Minimax-Concave Penalty for Wind-Turbine-Fault Diagnosis" *Sustainability* 14, no. 24: 16793.
https://doi.org/10.3390/su142416793