# Health Indicator Similarity Analysis-Based Adaptive Degradation Trend Detection for Bearing Time-to-Failure Prediction

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## Abstract

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## 1. Introduction

- (1)
- A bearing TTF prognostic approach is proposed to provide a continuously updated prediction of TTF by adaptive DT detection based on HI similarity analysis.
- (2)
- The specific FTH for each tested bearing is dynamically estimated with online monitored data and a configured SP to improve the accuracy of TTF prediction.
- (3)
- The DT of bearings is adaptively detected with the fitted degradation curve and truncated HI trajectory to address the issue of DT shifts.

## 2. System Scheme of Time-to-Failure Prediction of Bearings

#### 2.1. Health Indicator Construction by Fusing Compound Features

_{i}denotes the ith feature value, and $\delta (\xb7)$ indicates the simple unit step function. The F represents one of the extracted compound features.

_{i}is the HI value of ith sampling, and S

_{*j}denotes the first PCs of the current sample. μ and C are the learned mean and covariance of PCs decomposed from compound features of training bearings, respectively.

#### 2.2. Failure Threshold Estimation Based on the Online HI Values

_{l}samples $({x}_{i}^{l},\text{}{y}_{i}^{l})$, a linear SVM classifier ${h}_{l}(x)$ is built, and $1\le l\le C$. The ${h}_{l}(x)$ is defined as

_{dp}. Subsequently, the FTH of the unit can be obtained by

#### 2.3. Degradation Trend Detection Using Similarity Analysis for TTF Prediction

_{t}represents the Ts at time t, and Fc

_{i}indicate q predicted values of the Fc from t + I − q to t + I, where $i\in {N}^{+}$. Then, the similarity between them can be computed as

_{i}and Ts

_{t}, which should receive more attention.

_{t}is the TI at t, HIs represent all HI values obtained at t, and $H(mad+r)$ denotes the predictive HI at mad + r. In addition, the predictive HI values follow a t-distribution, and the confidence interval can be computed as follows:

_{t}to t, which is expressed as

_{t}composed of the latest serval HI points slides along the fitted curve to calculate their similarity, which is shown as S1, S2, and S3 in the figure. Supposing S2 is the best, the corresponding point of the fitted curve is selected as the DT. Finally, the TI

_{t}, as well as the TTF, can be predicted using Equation (11).

#### 2.4. Performance Evaluation Metrics

_{i}and Er

_{i}denote the RA and error of predicted TTF at the time i, respectively. Thus, the best CRA of models is 1.

## 3. Case Study 1: XJUT-SY Bearing Dataset

#### 3.1. Data Description

#### 3.2. Data Preprocessing and HI Construction

#### 3.3. TTF Prediction

_{t}for DT detection with the Fc, which is presented by a dotted orange line in Figure 6a. The Ts

_{t}circled by a yellow rectangle is used to calculate the similarity with the Fc point-wise for proper DT detection. After the mad with the best similarity, 0.8525, is selected, the TI

_{t}can be acquired with Equation (11). $\Delta H{I}_{t}$ in the equation is set as the difference value between the HI value at time t and the estimated FTH, and the future HI values can be predicted by extrapolating the Fc. Consequently, the TI

_{t}is calculated as the time interval from mad to the time point when the predicted HI value increases by $\Delta H{I}_{t}$. Finally, the TTF of bearing 1_5 at time t is deduced with Equation (15). Moreover, the 95% confidence interval of the TTF prediction result, which is depicted by solid orange lines, is also obtained with Equation (13) by calculating the upper and lower confidence limits of the predicted HI values.

#### 3.4. Ablation Experiments

#### 3.5. Comparisons with Other Approaches

## 4. Case Study 2: FEMTO-ST Bearing Dataset

#### 4.1. Data Description

#### 4.2. Data Preprocessing and HI Construction

#### 4.3. TTF Prediction

_{t}is acquired to detect proper DT. It can be found that the DT of the fitted degradation model at 150.5 min shows the best similarity with Ts

_{t}after exploration, which is 0.8735. This indicates that ordinary TTF prediction methods with only degradation model fitting can be a particular example of our proposed approach. Subsequently, future HI values can be predicted by extrapolating the fitted model from the detected DT, and the time interval TI

_{t}to the TTF can be calculated once the predictive HI values hit the estimated FTH from previous steps. Moreover, the 95% confidence interval of the predictive HI values, which is depicted by an orange area, is also obtained via Equation (15).

#### 4.4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\left\{{v}_{i}\right\}}_{i=1}^{N}$ | Raw vibration signal. |

f_{i} | Extracted compound features. |

HI_{i} | Constructed HI value. |

Sm | Score matrix computed in the DPCA algorithm. |

Ds | Input dataset for CSVM algorithm. |

C_{l} | The number of samples within the ith cluster. |

dp | Degradation time point. |

Sp | Scaling parameter. |

Ts | Tested segment. |

Fc | Fitted degradation curve. |

TI_{t} | Time interval at time t. |

mad | Detected the most appropriate degradation trend. |

## References

- Xia, M.; Li, T.; Shu, T.X.; Wan, J.F.; de Silva, C.W.; Wang, Z.R. A Two-Stage Approach for the Remaining Useful Life Prediction of Bearings Using Deep Neural Networks. IEEE Trans. Ind. Inform.
**2019**, 15, 3703–3711. [Google Scholar] [CrossRef] - Pan, D.; Liu, J.-B.; Cao, J. Remaining useful life estimation using an inverse Gaussian degradation model. Neurocomputing
**2016**, 185, 64–72. [Google Scholar] [CrossRef][Green Version] - Lei, Y.; Li, N.; Gontarz, S.; Lin, J.; Radkowski, S.; Dybala, J. A Model-Based Method for Remaining Useful Life Prediction of Machinery. IEEE Trans. Reliab.
**2016**, 65, 1314–1326. [Google Scholar] [CrossRef] - Xia, J.; Feng, Y.W.; Lu, C.; Fei, C.W.; Xue, X.F. LSTM-based multi-layer self-attention method for remaining useful life estimation of mechanical systems. Eng. Fail. Anal.
**2021**, 125, 105385. [Google Scholar] [CrossRef] - Ragab, M.; Chen, Z.H.; Wu, M.; Foo, C.S.; Kwoh, C.K.; Yan, R.Q.; Li, X.L. Contrastive Adversarial Domain Adaptation for Machine Remaining Useful Life Prediction. IEEE Trans. Ind. Inform.
**2021**, 17, 5239–5249. [Google Scholar] [CrossRef] - Cheng, Y.; Wang, C.; Wu, J.; Zhu, H.; Lee, C.K.M. Multi-dimensional recurrent neural network for remaining useful life prediction under variable operating conditions and multiple fault modes. Appl. Soft Comput.
**2022**, 118, 108507. [Google Scholar] [CrossRef] - Song, Y.; Gao, S.Y.; Li, Y.B.; Jia, L.; Li, Q.Q.; Pang, F.Z. Distributed Attention-Based Temporal Convolutional Network for Remaining Useful Life Prediction. IEEE Internet Things J.
**2021**, 8, 9594–9602. [Google Scholar] [CrossRef] - Wang, X.; Wang, T.Y.; Ming, A.B.; Zhang, W.; Li, A.H.; Chu, F.L. Spatiotemporal non-negative projected convolutional network with bidirectional NMF and 3DCNN for remaining useful life estimation of bearings. Neurocomputing
**2021**, 450, 294–310. [Google Scholar] [CrossRef] - Ding, Y.F.; Jia, M.P.; Miao, Q.H.; Huang, P. Remaining useful life estimation using deep metric transfer learning for kernel regression. Reliab. Eng. Syst. Saf.
**2021**, 212, 107583. [Google Scholar] [CrossRef] - Li, N.; Lei, Y.; Lin, J.; Ding, S.X. An Improved Exponential Model for Predicting Remaining Useful Life of Rolling Element Bearings. IEEE Trans. Ind. Electron.
**2015**, 62, 7762–7773. [Google Scholar] [CrossRef] - Wu, J.; Wu, C.; Cao, S.; Or, S.W.; Deng, C.; Shao, X. Degradation Data-Driven Time-To-Failure Prognostics Approach for Rolling Element Bearings in Electrical Machines. IEEE Trans. Ind. Electron.
**2019**, 66, 529–539. [Google Scholar] [CrossRef] - Duan, J.; Shi, T.; Zhou, H.; Xuan, J.; Wang, S. A novel ResNet-based model structure and its applications in machine health monitoring. J. Vib. Control
**2020**, 27, 1036–1050. [Google Scholar] [CrossRef] - Wang, Y.; Peng, Y.; Zi, Y.; Jin, X.; Tsui, K.L. A two-stage data-driven-based prognostic approach for bearing degradation problem. IEEE Trans. Ind. Inform.
**2016**, 12, 924–932. [Google Scholar] [CrossRef] - Yang, C.S.; Lou, Q.F.; Liu, J.; Yang, Y.B.; Cheng, Q.Q. Particle filtering-based methods for time to failure estimation with a real-world prognostic application. Appl. Intell.
**2018**, 48, 2516–2526. [Google Scholar] [CrossRef] - Liao, L. Discovering Prognostic Features Using Genetic Programming in Remaining Useful Life Prediction. IEEE Trans. Ind. Electron.
**2014**, 61, 2464–2472. [Google Scholar] [CrossRef] - Cheng, Y.; Zhu, H.; Hu, K.; Wu, J.; Shao, X.; Wang, Y. Reliability prediction of machinery with multiple degradation characteristics using double-Wiener process and Monte Carlo algorithm. Mech. Syst. Signal Process.
**2019**, 134, 106333. [Google Scholar] [CrossRef] - Kundu, P.; Chopra, S.; Lad, B.K. Multiple failure behaviors identification and remaining useful life prediction of ball bearings. J. Intell. Manuf.
**2019**, 30, 1795–1807. [Google Scholar] [CrossRef] - Witczak, M.; Mrugalski, M.; Lipiec, B. Remaining Useful Life Prediction of MOSFETs via the Takagi–Sugeno Framework. Energies
**2021**, 14, 2135. [Google Scholar] [CrossRef] - Chen, Z.; Zhu, H.; Wu, J.; Fan, L. Health indicator construction for degradation assessment by embedded LSTM–CNN autoencoder and growing self-organized map. Knowl.-Based Syst.
**2022**, 252, 109399. [Google Scholar] [CrossRef] - Liu, K.; Huang, S. Engineering. Integration of data fusion methodology and degradation modeling process to improve prognostics. IEEE Trans. Autom. Sci. Eng.
**2014**, 13, 344–354. [Google Scholar] [CrossRef] - Chehade, A.; Bonk, S.; Liu, K. Sensory-Based Failure Threshold Estimation for Remaining Useful Life Prediction. IEEE Trans. Reliab.
**2017**, 66, 939–949. [Google Scholar] [CrossRef] - Liu, K.; Gebraeel, N.; Shi, J. Engineering. A data-level fusion model for developing composite health indices for degradation modeling and prognostic analysis. IEEE Trans. Autom. Sci. Eng.
**2013**, 10, 652–664. [Google Scholar] [CrossRef] - Cheng, Y.; Wang, J.; Wu, J.; Zhu, H.; Wang, Y. Abnormal symptom-triggered remaining useful life prediction for rolling element bearings. J. Vib. Control
**2022**. [Google Scholar] [CrossRef] - Hou, M.; Pi, D.; Li, B. Similarity-based deep learning approach for remaining useful life prediction. Measurement
**2020**, 159, 107788. [Google Scholar] [CrossRef] - Cheng, Y.; Zhu, H.; Hu, K.; Wu, J.; Shao, X.; Wang, Y. Health Degradation Monitoring of Rolling Element Bearing by Growing Self- Organizing Mapping and Clustered Support Vector Machine. IEEE Access
**2019**, 7, 135322–135331. [Google Scholar] [CrossRef] - Wang, B.; Lei, Y.; Li, N.; Li, N. A Hybrid Prognostics Approach for Estimating Remaining Useful Life of Rolling Element Bearings. IEEE Trans. Reliab.
**2018**, 69, 401–412. [Google Scholar] [CrossRef] - Wang, Y.; Deng, C.; Wu, J.; Xiong, Y. Failure time prediction for mechanical device based on the degradation sequence. J. Intell. Manuf.
**2013**, 26, 1181–1199. [Google Scholar] [CrossRef] - Har-Peled, S.; Raichel, B. The Frechet distance revisited and extended. ACM Trans. Algorithms
**2014**, 10, 1–22. [Google Scholar] [CrossRef][Green Version] - Suh, S.; Lukowicz, P.; Lee, Y.O. Generalized multiscale feature extraction for remaining useful life prediction of bearings with generative adversarial networks. Knowl.-Based Syst.
**2022**, 237, 107866. [Google Scholar] [CrossRef] - Chang, Y.; Chen, J.; Liu, Y.; Xu, E.; He, S. Temporal convolution-based sorting feature repeat-explore network combining with multi-band information for remaining useful life estimation of equipment. Knowl.-Based Syst.
**2022**, 249, 108958. [Google Scholar] [CrossRef] - Lin, T.; Wang, H.; Guo, X.; Wang, P.; Song, L. A novel prediction network for remaining useful life of rotating machinery. Int. J. Adv. Manuf. Technol.
**2022**, 124, 4009–4018. [Google Scholar] [CrossRef] - Nectoux, P.; Gouriveau, R.; Medjaher, K.; Ramasso, E.; Chebel-Morello, B.; Zerhouni, N.; Varnier, C. PRONOSTIA: An experimental platform for bearings accelerated degradation tests. In Proceedings of the IEEE International Conference on Prognostics and Health Management, PHM’12, Beijing, China, 23–25 May 2012; pp. 1–8. [Google Scholar]
- Cartella, F.; Lemeire, J.; Dimiccoli, L.; Sahli, H. Hidden Semi-Markov Models for Predictive Maintenance. Math. Probl. Eng.
**2015**, 2015, 278120. [Google Scholar] [CrossRef][Green Version] - Zhao, M.; Tang, B.; Tan, Q. Bearing remaining useful life estimation based on time–frequency representation and supervised dimensionality reduction. Measurement
**2016**, 86, 41–55. [Google Scholar] [CrossRef]

**Figure 5.**HIs for TTF prognostics of (

**a**) bearing 1_3, (

**b**) bearing 1_5, (

**c**) bearing 2_1, and (

**d**) bearing 3_1.

**Figure 6.**Degradation predictions of (

**a**) bearing 1_5, (

**b**) bearing 1_3, (

**c**) bearing 2_1, and (

**d**) bearing 3_1.

**Figure 7.**TTF prognostic results and its 95% confidence interval of (

**a**) bearing 1_3, (

**b**) bearing 1_5, (

**c**) bearing 2_1, and (

**d**) bearing 3_1.

**Figure 8.**TTF prognostic results of the tested bearings (

**a**) bearing 1_3, (

**b**) bearing 1_5, (

**c**) bearing 2_1, and (

**d**) bearing 3_1 using an original method (without FTH estimation and DT detection), the method with FTH estimation, and the proposed method.

**Figure 9.**TTF prediction results of different approaches for (

**a**) bearing 1_3, (

**b**) bearing 1_5, (

**c**) bearing 2_1, and (

**d**) bearing 3_1.

**Figure 11.**HIs for TTF prognostics of (

**a**) bearing 1_1, (

**b**) bearing 1_2, (

**c**) bearing 2_1, and (

**d**) bearing 2_2.

**Figure 12.**Degradation prediction of (

**a**) bearing 1_1, (

**b**) bearing 1_2, (

**c**) bearing 2_1, and (

**d**) bearing 2_2.

**Figure 13.**TTF prognostic results and the 95% confidence interval of (

**a**) bearing 1_1, (

**b**) bearing 1_2, (

**c**) bearing 2_1, and (

**d**) bearing 2_2.

**Figure 14.**TTF prediction results of different approaches for (

**a**) bearing 1_1, (

**b**) bearing 1_2, (

**c**) bearing 2_1, and (

**d**) bearing 2_2.

Method | Errors (%) | CRA | ||||||
---|---|---|---|---|---|---|---|---|

Bearing 1_3 | Bearing 1_5 | Bearing 2_1 | Bearing 3_1 | Bearing 1_3 | Bearing 1_5 | Bearing 2_1 | Bearing 3_1 | |

Proposed approach | 3.27 | 0.11 | 0.61 | 0 | 0.9091 | 0.9827 | 0.9803 | 0.9987 |

LSTM | 25.88 | 13 | 2.59 | 0.79 | 0.7162 | 0.8604 | 0.9651 | 0.9771 |

SVR | 33.63 | 7.27 | 2.14 | 1.02 | 0.6148 | 0.9258 | 0.9736 | 0.9806 |

GR | 37.25 | 14.05 | 3.09 | 0.49 | 0.6444 | 0.8567 | 0.9623 | 0.9768 |

Wang et al. [26] | N/A | N/A | N/A | N/A | 0.8482 | 0.7878 | 0.8621 | 0.8942 |

Sun et al. [29] | 7.01 | N/A | 2.59 | N/A | 0.53.6 | N/A | 0.5721 | N/A |

Chang et al. [30] | N/A | 2.74 | N/A | N/A | N/A | 0.955 | N/A | N/A |

Lin et al. [31] | N/A | N/A | 3.48 | 0 | N/A | N/A | 0.9727 | 0.9706 |

Method | Errors (%) | CRA | ||||||
---|---|---|---|---|---|---|---|---|

Bearing 1_1 | Bearing 1_2 | Bearing 2_1 | Bearing 2_2 | Bearing 1_1 | Bearing 1_2 | Bearing 2_1 | Bearing 2_2 | |

Proposed approach | 0.04 | 0.0 | 0.0 | 0.13 | 0.9995 | 0.9954 | 1.0 | 0.9868 |

LSTM | 0.6 | 0.34 | 0.3 | 2.44 | 0.9929 | 0.9767 | 0.9911 | 0.7643 |

SVR | 2.4 | 1.18 | 0.54 | 16.4 | 0.974 | 0.9608 | 0.9885 | 0.7788 |

Original model | 0.04 | 8.15 | 0.88 | 34.88 | 0.972 | 0.8912 | 0.9923 | 0.5458 |

Wang et al. [26] | N/A | N/A | N/A | N/A | 0.9047 | 0.8546 | 0.8621 | 0.6521 |

Wu et al. [11] | 0.02 | N/A | 0.22 | 0.37 | 0.98 | N/A | 0.78 | 0.63 |

Cartella et al. [33] | 37.72 | 49.73 | 27.18 | 18.1 | N/A | N/A | N/A | N/A |

Zhao et al. [34] | 13.9 | 65.55 | 47.2 | 17.3 | N/A | N/A | N/A | N/A |

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## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Zhu, H.; Fan, L.; Lu, Z. Health Indicator Similarity Analysis-Based Adaptive Degradation Trend Detection for Bearing Time-to-Failure Prediction. *Electronics* **2023**, *12*, 1569.
https://doi.org/10.3390/electronics12071569

**AMA Style**

Chen Z, Zhu H, Fan L, Lu Z. Health Indicator Similarity Analysis-Based Adaptive Degradation Trend Detection for Bearing Time-to-Failure Prediction. *Electronics*. 2023; 12(7):1569.
https://doi.org/10.3390/electronics12071569

**Chicago/Turabian Style**

Chen, Zhipeng, Haiping Zhu, Liangzhi Fan, and Zhiqiang Lu. 2023. "Health Indicator Similarity Analysis-Based Adaptive Degradation Trend Detection for Bearing Time-to-Failure Prediction" *Electronics* 12, no. 7: 1569.
https://doi.org/10.3390/electronics12071569