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A fault diagnosis method of rotating machinery based on improved multiscale attention entropy and random forests

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Abstract

In order to precisely diagnose the fault type of rotating machinery, a fault diagnosis method for rotating machinery based on improved multiscale attention entropy and random forests is proposed in this study. Firstly, a nonlinear dynamics technique without hyperparameters namely multiscale attention entropy is proposed for measuring signal complexity by extending attention entropy to multiple time scales. Secondly, aiming at the insufficient coarse-graining of multiscale attention entropy, composite multiscale attention entropy is exploited to extract the features of rotating machinery faults. Then, t-distributed stochastic neighbor embedding is used to overcome the feature redundancy problem by reducing the dimension of the extracted features. Finally, the reduced-dimensional features are inputted into the random forests model to complete fault pattern recognition of rotating machinery. The results of the experiment indicate that the proposed method achieves the optimal diagnostic performance on two different fault datasets respectively, showing an extremely competitive advantage in comparison with conventional diagnosis models. Meanwhile, the proposed method is adopted to the actual hydropower unit without misjudgment, which verifies its strong adaptability. The research proposes a novel method for detecting faults in rotating machinery such as hydropower units.

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Data will be made available on reasonable request.

References

  1. Hu, J.H., Yang, J.B., He, X.H., Zeng, W., Zhao, Z.G., Yang, J.D.: Transition of amplitude–frequency characteristic in rotor–stator interaction of a pump-turbine with splitter blades. Renew. Energy 205, 663–677 (2023). https://doi.org/10.1016/j.renene.2023.02.008

    Article  Google Scholar 

  2. Huang, Y.F., Tao, J., Sun, G., Wu, T.Y., Yu, L.L., Zhao, X.B.: A novel digital twin approach based on deep multimodal information fusion for aero-engine fault diagnosis. Energy 270, 126894 (2023). https://doi.org/10.1016/j.energy.2023.126894

    Article  Google Scholar 

  3. Karatu, C., Arslanoglu, Y.: Development of condition-based maintenance strategy for fault diagnosis for ship engine systems. Ocean Eng. 256, 111515 (2022). https://doi.org/10.1016/j.oceaneng.2022.111515

    Article  Google Scholar 

  4. Xu, Y.D., Feng, K., Yan, X.A., Yan, R.Q., Ni, Q., Sun, B.B., Lei, Z., Zhang, Y.C., Liu, Z.: CFCNN: A novel convolutional fusion framework for collaborative fault identification of rotating machinery. Inf. Fusion. 95, 1–16 (2023). https://doi.org/10.1016/j.inffus.2023.02.012

    Article  Google Scholar 

  5. He, D.Q., Lao, Z.P., Jin, Z.Z., He, C.F., Shan, S., Miao, J.: Train bearing fault diagnosis based on multi-sensor data fusion and dual-scale residual network. Nonlinear Dyn. 111, 14901–14924 (2023). https://doi.org/10.1007/s11071-023-08638-w

    Article  Google Scholar 

  6. Yang, C., Li, H.K., Cao, S.X.: Unknown fault diagnosis of planetary gearbox based on optimal rank nonnegative matrix factorization and improved stochastic resonance of bistable system. Nonlinear Dyn. 111, 217–242 (2023). https://doi.org/10.1007/s11071-022-07846-0

    Article  Google Scholar 

  7. Mehta, M., Chen, S.Y., Tang, H.C., Shao, C.H.: A federated learning approach to mixed fault diagnosis in rotating machinery. J. Manuf. Syst. 68, 687–694 (2023). https://doi.org/10.1016/j.jmsy.2023.05.012

    Article  Google Scholar 

  8. Gawde, S., Patil, S., Kumar, S., Kamat, P., Kotecha, K., Abraham, A.: Multi-fault diagnosis of Industrial rotating machines using data-driven approach: a review of two decades of research. Eng. Appl. Artif. Intell. 123, 106139 (2023). https://doi.org/10.1016/j.engappai.2023.106139

    Article  Google Scholar 

  9. Brito, L.C., Susto, G.A., Brito, J.N., Duarte, M.A.V.: An explainable artificial intelligence approach for unsupervised fault detection and diagnosis in rotating machinery. Mech. Syst. Sign. Process. 163, 108105 (2022). https://doi.org/10.1016/j.ymssp.2021.108105

    Article  Google Scholar 

  10. Zhang, Z.W., Shao, M.Y., Ma, C.C., Lv, Z., Zhou, J.L.: An enhanced domain-adversarial neural networks for intelligent cross-domain fault diagnosis of rotating machinery. Nonlinear Dyn. 108, 2385–2404 (2022). https://doi.org/10.1007/s11071-022-07341-6

    Article  Google Scholar 

  11. Ruan, H.L., Wang, Y., Li, X.M., Qin, Y., Tang, B.P.: An enhanced non-local weakly supervised fault diagnosis method for rotating machinery. Measurement 189, 110433 (2022). https://doi.org/10.1016/j.measurement.2021.110433

    Article  Google Scholar 

  12. Jin, Z.Z., He, D.Q., Lao, Z.P., Wei, X.H., Yang, W.F.: Early intelligent fault diagnosis of rotating machinery based on IWOA-VMD and DMKELM. Nonlinear Dyn. 111, 5287–5306 (2023). https://doi.org/10.1007/s11071-022-08109-8

    Article  Google Scholar 

  13. Lin, J.S., Dou, C.H., Liu, Y.J.: Feature extraction based on generalized permutation entropy for condition monitoring of rotating machinery. Nonlinear Dyn. 107, 855–870 (2022). https://doi.org/10.1007/s11071-021-07054-2

    Article  Google Scholar 

  14. Kianat, J., Khan, M.A., Sharif, M., Akram, T., Rehman, A., Saba, T.: A joint framework of feature reduction and robust feature selection for cucumber leaf diseases recognition. Optik 240, 166566 (2021). https://doi.org/10.1016/j.ijleo.2021.166566

    Article  Google Scholar 

  15. Miao, Y.H., Wang, J.J., Zhang, B.Y., Li, H.: Practical framework of Gini index in the application of machinery fault feature extraction. Mech. Syst. Sign. Process. 165, 108333 (2022). https://doi.org/10.1016/j.ymssp.2021.108333

    Article  Google Scholar 

  16. Han, M.H., Pan, J.L.: A fault diagnosis method combined with LMD, sample entropy and energy ratio for roller bearings. Measurement 76, 7–19 (2015). https://doi.org/10.1016/j.measurement.2015.08.019

    Article  Google Scholar 

  17. Sharma, S., Tiwari, S.K., Singh, S.: Integrated approach based on flexible analytical wavelet transform and permutation entropy for fault detection in rotary machines. Measurement 169, 108389 (2021). https://doi.org/10.1016/j.measurement.2020.108389

    Article  Google Scholar 

  18. Zhang, X., Zhang, M., Wan, S.T., He, Y.L., Wang, X.L.: A bearing fault diagnosis method based on multiscale dispersion entropy and GG clustering. Measurement 185, 110023 (2021). https://doi.org/10.1016/j.measurement.2021.110023

    Article  Google Scholar 

  19. Deng, W., Zhang, S.J., Zhao, H.M., Yang, X.H.: A novel fault diagnosis method based on integrating empirical wavelet transform and fuzzy entropy for motor bearing. IEEE Access 6, 35042–35056 (2018). https://doi.org/10.1109/ACCESS.2018.2834540

    Article  Google Scholar 

  20. Dang, P.F., Yang, Z.X., Wen, B.G., Wang, M.G., Han, Q.K.: Fault severity assessment of rolling bearing based on optimized multi-dictionaries matching pursuit and Lempel-Ziv complexity. ISA Trans. 116, 191–202 (2021). https://doi.org/10.1016/j.isatra.2021.01.042

    Article  Google Scholar 

  21. Zhang, Y.Q., Ren, G.Q., Wu, D.H., Wang, H.G.: Rolling bearing fault diagnosis utilizing variational mode decomposition based fractal dimension estimation method. Measurement 181, 109614 (2021). https://doi.org/10.1016/j.measurement.2021.109614

    Article  Google Scholar 

  22. Costa, M., Goldberger, A.L., Peng, C.K.: Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 89, 068102 (2002). https://doi.org/10.1103/PhysRevLett.89.068102

    Article  Google Scholar 

  23. Aziz, W., Arif, M.: Multiscale permutation entropy of physiological time series. Proceedings of the INMIC 2005: 9TH International Multitopic Conference Proceedings 12, 368–373 (2005) DOI: https://doi.org/10.1109/INMIC.2005.334494

  24. Shang, H.K., Li, F., Wu, Y.J.: Partial discharge fault diagnosis based on multi-scale dispersion entropy and a hypersphere multiclass support vector machine. Entropy 21, 81 (2019). https://doi.org/10.3390/e21010081

    Article  Google Scholar 

  25. Tan, H.C., Xie, S.C., Zhou, H., Ma, W., Yang, C.X., Zhang, J.: Sensible multiscale symbol dynamic entropy for fault diagnosis of bearing. Int. J. Mech. Sci. 256, 108509 (2023). https://doi.org/10.1016/j.ijmecsci.2023.108509

    Article  Google Scholar 

  26. Zheng, J.D., Ying, W.M., Tong, J.Y., Li, Y.B.: Multiscale three-dimensional Holo-Hilbert spectral entropy: a novel complexity-based early fault feature representation method for rotating machinery. Nonlinear Dyn. 111, 10309–10330 (2023). https://doi.org/10.1007/s11071-023-08392-z

    Article  Google Scholar 

  27. Wang, Z.Y., Li, G.S., Yao, L.G., Cai, Y.X., Lin, T.X., Zhang, J., Dong, H.: Intelligent fault detection scheme for constant-speed wind turbines based on improved multiscale fuzzy entropy and adaptive chaotic Aquila optimization-based support vector machine. ISA Trans. 138, 582–602 (2023). https://doi.org/10.1016/j.isatra.2023.03.022

    Article  Google Scholar 

  28. Li, Y.B., Wang, S., Yang, Y., Deng, Z.C.: Multiscale symbolic fuzzy entropy: An entropy denoising method for weak feature extraction of rotating machinery. Mech. Syst. Sign. Process. 162, 108052 (2022). https://doi.org/10.1016/j.ymssp.2021.108052

    Article  Google Scholar 

  29. Shi, J.J., Su, Z., Qin, H.Y., Shen, C.Q., Huang, W.G., Zhu, Z.K.: Generalized variable-step multiscale Lempel-Ziv complexity: a feature extraction tool for bearing fault diagnosis. IEEE Sens. J. 22(15), 15296–15305 (2022). https://doi.org/10.1109/JSEN.2022.3187763

    Article  Google Scholar 

  30. Li, Y.B., Wang, S., Deng, Z.C.: Intelligent fault identification of rotary machinery using refined composite multi-scale Lempel-Ziv complexity. J. Manuf. Syst. 61, 725–735 (2021). https://doi.org/10.1016/j.jmsy.2020.05.004

    Article  Google Scholar 

  31. Yan, X.A., She, D.M., Xu, Y.D., Jia, M.P.: Application of generalized composite multiscale Lempel-Ziv complexity in identifying wind turbine gearbox faults. Entropy 23(11), 1372 (2021). https://doi.org/10.3390/e23111372

    Article  Google Scholar 

  32. Yan, X.A., Xu, Y.D., Jia, M.P.: Intelligent fault diagnosis of rolling-element bearings using a self-adaptive hierarchical multiscale fuzzy entropy. Entropy 23, 1128 (2021). https://doi.org/10.3390/e23091128

    Article  MathSciNet  Google Scholar 

  33. Li, Y.B., Yang, Y.T., Wang, X.Z., Liu, B.B., Liang, X.H.: Early fault diagnosis of rolling bearings based on hierarchical symbol dynamic entropy and binary tree support vector machine. J. Sound Vib. 428, 72–86 (2018). https://doi.org/10.1016/j.jsv.2018.04.036

    Article  Google Scholar 

  34. Li, Y.X., Jiang, X.R., Tang, B.Z., Ning, F.Y., Lou, Y.L.: Feature extraction methods of ship-radiated noise: From single feature of multi-scale dispersion Lempel-Ziv complexity to mixed double features. Appl. Acoust. 199, 109032 (2022). https://doi.org/10.1016/j.apacoust.2022.109032

    Article  Google Scholar 

  35. Li, Y.X., Geng, B., Jiao, S.B.: Dispersion entropy-based Lempel-Ziv complexity: A new metric for signal analysis. Chaos Solitons Fractals 161, 112400 (2022). https://doi.org/10.1016/j.chaos.2022.112400

    Article  MathSciNet  Google Scholar 

  36. Tan, H.C., Xie, S.C., Liu, R.D., Cheng, J.Q., Jing, K.K.: Fatigue condition diagnosis of rolling bearing based on normalized balanced multiscale sample entropy. Int. J. Fatigue 172, 107642 (2023). https://doi.org/10.1016/j.ijfatigue.2023.107642

    Article  Google Scholar 

  37. Yang, J., Bai, Y.P., Cheng, Y.Y., Cheng, R., Zhang, W.D., Zhang, G.J.: A new model for bearing fault diagnosis based on optimized variational mode decomposition correlation coefficient weight threshold denoising and entropy feature fusion. Nonlinear Dyn. 111, 17337–17367 (2023). https://doi.org/10.1007/s11071-023-08728-9

    Article  Google Scholar 

  38. Gong, J.C., Yang, X.Q., Qian, K., Chen, Z.Y., Han, T.: Application of improved bubble entropy and machine learning in the adaptive diagnosis of rotating machinery faults. Alex. Eng. J. 80, 22–40 (2023). https://doi.org/10.1016/j.aej.2023.08.006

    Article  Google Scholar 

  39. Zhu, Z.Y., Cheng, J.S., Wang, P., Wang, J., Kang, X.: A novel fault diagnosis framework for rotating machinery with hierarchical multiscale symbolic diversity entropy and robust twin hyperdisk-based tensor machine. Reliab. Eng. Syst. Saf. 231, 109037 (2023). https://doi.org/10.1016/j.ress.2022.109037

    Article  Google Scholar 

  40. He, C., Wu, T., Gu, R.W., Jin, Z.Y., Ma, R.J., Qu, H.Y.: Rolling bearing fault diagnosis based on composite multiscale permutation entropy and reverse cognitive fruit fly optimization algorithm-extreme learning machine. Measurement 173, 108636 (2021). https://doi.org/10.1016/j.measurement.2020.108636

    Article  Google Scholar 

  41. Yang, J.W., Choudhary, G.I., Rahardja, S., Franti, P.: Classification of interbeat interval time-series using attention entropy. IEEE Trans. Affect. Comput. 14(1), 321–330 (2020). https://doi.org/10.1109/TAFFC.2020.3031004

    Article  Google Scholar 

  42. Cho, P., Kim, K.: Global collective dynamics of financial market efficiency using attention entropy with hierarchical clustering. Fractal fract. 6, 562 (2022). https://doi.org/10.3390/fractalfract6100562

    Article  Google Scholar 

  43. Zhang, B.Y., Shang, P.J., Mao, X.G., Liu, J.Z.: Dispersion heterogeneous recurrence analysis and its use on fault detection. Commun. Nonlinear Sci. Numer. Simul. 117, 106902 (2023). https://doi.org/10.1016/j.cnsns.2022.106902

    Article  MathSciNet  Google Scholar 

  44. van der Maaten, L., Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9, 2579–2605 (2008)

    Google Scholar 

  45. He, Y.L., Hua, Q., Zhu, Q.X., Lu, S.: Enhanced virtual sample generation based on manifold features: Applications to developing soft sensor using small data. ISA Trans. 126, 398–406 (2022). https://doi.org/10.1016/j.isatra.2021.07.033

    Article  Google Scholar 

  46. Wang, Q., Nguyen, T.T., Huang, J.Z., Nguyen, T.T.: An efficient random forests algorithm for high dimensional data classification. Adv. Data Anal. Classif. 12, 953–972 (2018). https://doi.org/10.1007/s11634-018-0318-1

    Article  MathSciNet  Google Scholar 

  47. Wang, Z.Y., Zuo, R.G., Jing, L.H.: Fusion of geochemical and remote-sensing data for lithological mapping using random forest metric learning. Math. Geosci. 53, 1125–1145 (2021). https://doi.org/10.1007/s11004-020-09897-8

    Article  MathSciNet  Google Scholar 

  48. Alam, M.Z., Rahman, M.S., Rahman, M.S.: A random forest based predictor for medical data classification using feature ranking. Inform. Med. Unlocked 15, 100180 (2019). https://doi.org/10.1016/j.imu.2019.100180

    Article  Google Scholar 

  49. He, S., Wu, J.H., Wang, D., He, X.D.: Predictive modeling of groundwater nitrate pollution and evaluating its main impact factors using random forest. Chemosphere 290, 133388 (2022). https://doi.org/10.1016/j.chemosphere.2021.133388

    Article  Google Scholar 

  50. Gan, L., Wu, H., Zhong, Z.: Fatigue life prediction considering mean stress effect based on random forests and kernel extreme learning machine. Int. J. Fatigue 158, 106761 (2022). https://doi.org/10.1016/j.ijfatigue.2022.106761

    Article  Google Scholar 

  51. Imane, M., Rahmoune, C., Zair, M., Benazzouz, D.: Bearing fault detection under time-varying speed based on empirical wavelet transform, cultural clan-based optimization algorithm, and random forest classifier. J. Vib. Control 29(1–2), 286–297 (2023). https://doi.org/10.1177/10775463211047034

    Article  MathSciNet  Google Scholar 

  52. Guo, Z., Pu, Z.Q., Du, W.L., Wang, H.C., Wang, H.C., Li, C.: Improved adversarial learning for fault feature generation of wind turbine gearbox. Renew. Energ. 185, 255–266 (2022). https://doi.org/10.1016/j.renene.2021.12.054

    Article  Google Scholar 

  53. Chegini, S.N., Amini, P., Ahmadi, B., Bagheri, A., Amirmostofian, I.: Intelligent bearing fault diagnosis using swarm decomposition method and new hybrid particle swarm optimization algorithm. Soft. Comput. 26, 1475–1497 (2022). https://doi.org/10.1007/s00500-021-06307-x

    Article  Google Scholar 

  54. Dou, L.J., Wan, S.T., Zhan, C.G.: Application of multiscale entropy in mechanical fault diagnosis of high voltage circuit breaker. Entropy 20(5), 325 (2018). https://doi.org/10.3390/e20050325

    Article  Google Scholar 

  55. Huang, P.J., Huang, Z.W., Lu, X.D., Cao, Y.Q., Yu, J., Hou, D.B., Zhang, G.X.: Study on glycoprotein terahertz time-domain spectroscopy based on composite multiscale entropy feature extraction method. Spectrochim. Acta A Mol. Biomol. Spectrosc. 229, 117948 (2020). https://doi.org/10.1016/j.saa.2019.117948

    Article  Google Scholar 

  56. Azami, H., Rostaghi, M., Abasolo, D., Escudero, J.: Refined composite multiscale dispersion entropy and its application to biomedical signals. IEEE Trans. Biomed. Eng. 64, 2872–2879 (2017). https://doi.org/10.1109/TBME.2017.2679136

    Article  Google Scholar 

  57. Gan, X., Lu, H., Yang, G.Y.: Fault diagnosis method for rolling bearings based on composite multiscale fluctuation dispersion entropy. Entropy 21(3), 290 (2019). https://doi.org/10.3390/e21030290

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to the editor and anonymous referees for their careful reading and valuable comments.

Funding

This research was supported by the scientific research foundation of the Young Scholar Project of Cyrus Tang Foundation, the Shaanxi Province Key Research and Development Plan (Number 2021NY-181), the scientific research foundation of China Postdoctoral Science Foundation (2021M702527), the coordinates scientific research projects of State Power Investment Corporation Limited (Number TC2020SD01), China Postdoctoral Innovation Talents Support Program (BX20220243) and the National Natural Science Foundation of China (Numbers 51909222 and 51509210).

Author information

Authors and Affiliations

  1. College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, 712100, Shaanxi, China

    Fei Chen, Liyao Zhang, Wenshen Liu, Tingting Zhang, Diyi Chen & Bin Wang

  2. State Key Laboratory of Water Resources Engineering and Management, Wuhan University, Wuhan, 430072, Hubei, China

    Zhigao Zhao

  3. Wuling Power Corporation LTD, Changsha, 41004, China

    Weiyu Wang

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  1. Fei Chen
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  2. Liyao Zhang
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  3. Wenshen Liu
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Contributions

F.C.: Conceptualization, investigation, validation, visualization, writing—original draft. L.Z.: Investigation, validation. W.L.: Methodology, software. T.Z.: Investigation, formal analysis. Z.Z.: Supervision, funding acquisition. W.W.: Funding acquisition, data support. D.C.: Funding acquisition, supervision. B.W.: Funding acquisition, supervision, writing—review & editing.

Corresponding authors

Correspondence to Zhigao Zhao or Bin Wang.

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Appendices

Appendix A: A full list of technical terms.

AE

Attention entropy

MHHSE3D

Multiscale three-dimensional Holo–Hilbert spectral entropy

AED

Average Euclidean distance

min

minimum

AITHE

Ahrar Institute of Technology and Higher Education

MSDE

Multiscale symbolic dynamics entropy

CC

Correlation coefficient

MSE

Multiscale sample entropy

CMAE

Composite multiscale attention entropy

MSFE

Multiscale symbolic fuzzy entropy

CMDE

Composite multiscale dispersion entropy

MWPE

Multiscale weighted permutation entropy

CMFDE

Composite multiscale fluctuation dispersion entropy

PE

Permutation entropy

CMSE

Composite multiscale sample entropy

RCMDE

Refined composite multiscale dispersion entropy

DE

Dispersion entropy

RCMLZC

Refined composite multiscale Lempel–Ziv complexity

FE

Fuzzy entropy

RF

Random forests

KL

Kullback-Leiber

RTSMFE

Refined time-shifted multiscale fuzzy entropy

LZC

Lempel–Ziv complexity

SD

Standard deviation

MAE

Multiscale attention entropy

SE

Sample entropy

max

Maximum

SMSDE

Sensible multiscale symbol dynamic entropy

MDE

Multiscale dispersion entropy

SNE

Stochastic neighbor embedding

MEP

Maximum entropy partitioning

TSNE

t-distributed stochastic neighbor embedding

Appendix B A full list of mathematical symbols.

\(b\)

The number of interval point types

\(C\)

Cost function

\(p\left( y \right)\)

the probability of occurrence of \(y\)

\(S\)

The Shannon entropy of the interval between adjacent key points

\(h_{i}\)

The \(i - {\text{th}}\) high-dimensional data sample

\(H\)

The high-dimensional original data

\(w_{i}\)

The \(i - {\text{th}}\) low-dimensional data sample

\(W\)

The low-dimensional data

\(d\)

The dimension of low-dimensional data

\(D\)

The dimension of high-dimensional data

\(KL\left( {P_{i} \left\| {Q_{i} } \right.} \right)\)

The similarity between high and low dimensional data

\(p_{ij}\)

The probability density function of high-dimensional space sample

\(\sigma_{i}\)

The standard deviation of Gaussian distribution

\(n\)

The number of data points

\(q_{ij}\)

The probability density function

\(\eta\)

The learning rate

\(t\)

The number of iterations

\(\mu \left( t \right)\)

The momentum factor

\(T\)

Maximum number of iterations

\(P\)

Number of decision trees

\(M\)

Dimension of features

\(TP\)

True positive

\(TN\)

True negative

\(FP\)

False positive

\(FN\)

False negative

\(X\)

The time series signal

\(x_{i}\)

Sample points of timing signals

\(Y_{i}^{\left( \tau \right)}\)

Coarse-grained time series

\(N\)

The length of the time series

\(\tau\)

Scale factor

\(\tau_{\max }\)

Maximum scale factor value

\({\text{AE}}\left( {Y_{i}^{\left( \tau \right)} } \right)\)

The AE value of coarse-grained time series

\(\xi_{k}^{\left( \tau \right)}\)

The set of coarse-grained time series

\(n(t)\)

Gaussian white noise

\(m\)

Embedding dimension

\(\chi\)

Delay time

\(c\)

Integer indices

\(r\)

Similar tolerance

  

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Chen, F., Zhang, L., Liu, W. et al. A fault diagnosis method of rotating machinery based on improved multiscale attention entropy and random forests. Nonlinear Dyn 112, 1191–1220 (2024). https://doi.org/10.1007/s11071-023-09126-x

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