{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T16:14:33Z","timestamp":1761581673087,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,3]],"date-time":"2018-03-03T00:00:00Z","timestamp":1520035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Gears are key components in rotation machinery and its fault vibration signals usually show strong nonlinear and non-stationary characteristics. It is not easy for classical time\u2013frequency domain analysis methods to recognize different gear working conditions. Therefore, this paper presents a joint fault diagnosis scheme for gear fault classification via tensor nuclear norm canonical polyadic decomposition (TNNCPD) and multi-scale permutation entropy (MSPE). Firstly, the one-dimensional vibration data of different gear fault conditions is converted into a three-dimensional tensor data, and a new tensor canonical polyadic decomposition method based on nuclear norm and convex optimization called TNNCPD is proposed to extract the low rank component of the data, which represents the feature information of the measured signal. Then, the MSPE of the extracted feature information about different gear faults can be calculated as the feature vector in order to recognize fault conditions. Finally, this researched scheme is validated by practical gear vibration data of different fault conditions. The result demonstrates that the proposed scheme can effectively recognize different gear fault conditions.<\/jats:p>","DOI":"10.3390\/e20030161","type":"journal-article","created":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T07:37:25Z","timestamp":1520321845000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Joint Fault Diagnosis Scheme Based on Tensor Nuclear Norm Canonical Polyadic Decomposition and Multi-Scale Permutation Entropy for Gears"],"prefix":"10.3390","volume":"20","author":[{"given":"Mao","family":"Ge","sequence":"first","affiliation":[{"name":"Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, Wuhan 430081, China"},{"name":"Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Lv","sequence":"additional","affiliation":[{"name":"Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, Wuhan 430081, China"},{"name":"Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cancan","family":"Yi","sequence":"additional","affiliation":[{"name":"Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, Wuhan 430081, China"},{"name":"Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, Wuhan 430081, China"},{"name":"Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiangjun","family":"Chen","sequence":"additional","affiliation":[{"name":"Zhejiang Provincial Special Equipment Inspection Research Institute, Hangzhou 314415, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.mechmachtheory.2015.03.013","article-title":"Gear fault diagnosis based on support vector machine optimized by artificial bee colony algorithm","volume":"90","author":"Yang","year":"2015","journal-title":"Mech. Mach. Theory"},{"key":"ref_2","unstructured":"Forrester, B.D. (1996). Advanced Vibration Analysis Techniques for Fault Detection and Diagnosis in Geared Transmission Systems. [PhD Thesis, Swinburne University of Technology]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.ymssp.2005.09.015","article-title":"A comparative experimental study on the diagnostic and prognostic capabilities of acoustics emission, vibration and spectrometric oil analysis for spur gears","volume":"21","author":"Tan","year":"2007","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Heydarzadeh, M., Nourani, M., Hansen, J., and Kia, S.H. (2017, January 5\u20139). Non-invasive gearbox fault diagnosis using scattering transform of acoustic emission. Proceedings of the 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), New Orleans, LA, USA.","DOI":"10.1109\/ICASSP.2017.7952180"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1006\/mssp.1999.1294","article-title":"Effectiveness and sensitivity of vibration processing techniques for local fault detection in gears","volume":"14","author":"Dalpiaz","year":"2000","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.renene.2012.04.019","article-title":"Fault diagnosis for wind turbine planetary gearboxes via demodulation analysis based on ensemble empirical mode decomposition and energy separation","volume":"47","author":"Feng","year":"2012","journal-title":"Renew. Energy"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/S0888-3270(03)00080-3","article-title":"Classification of gear faults using cumulants and the radial basis function network","volume":"18","author":"Wu","year":"2004","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yi, C., Lv, Y., Xiao, H., Huang, T., and You, G. (2018). Multisensor signal denoising based on matching synchrosqueezing wavelet transform for mechanical fault condition assessment. Meas. Sci. Technol.","DOI":"10.1088\/1361-6501\/aaa50a"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.ymssp.2009.07.003","article-title":"The envelope order spectrum based on generalized demodulation time\u2014Frequency analysis and its application to gear fault diagnosis","volume":"24","author":"Cheng","year":"2010","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.mechmachtheory.2012.04.008","article-title":"An order tracking technique for the gear fault diagnosis using local mean decomposition method","volume":"55","author":"Cheng","year":"2012","journal-title":"Mech. Mach. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4168","DOI":"10.1016\/j.eswa.2009.11.006","article-title":"Incipient gear box fault diagnosis using discrete wavelet transform (DWT) for feature extraction and classification using artificial neural network (ANN)","volume":"37","author":"Saravanan","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3403","DOI":"10.1103\/PhysRevA.45.3403","article-title":"Determining embedding dimension for phase-space reconstruction using a geometrical construction","volume":"45","author":"Kennel","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s12206-010-1117-6","article-title":"Gear fault identification and classification of singular value decomposition based on Hilbert-Huang transform","volume":"25","author":"Su","year":"2011","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TNSRE.2014.2329557","article-title":"Tensor based singular spectrum analysis for automatic scoring of sleep EEG","volume":"23","author":"Kouchaki","year":"2015","journal-title":"IEEE Trans. Neural Syst. Rehabil. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yi, C., Lv, Y., Ge, M., Xiao, H., and Yu, X. (2017). Tensor Singular Spectrum Decomposition Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis. Entropy, 19.","DOI":"10.3390\/e19040139"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/MSP.2013.2297439","article-title":"Tensor decompositions for signal processing applications: From two-way to multiway component analysis","volume":"32","author":"Cichocki","year":"2015","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1137\/130916084","article-title":"Canonical polyadic decomposition of third-order tensors: Reduction to generalized eigenvalue decomposition","volume":"35","author":"Domanov","year":"2014","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3551","DOI":"10.1109\/TSP.2017.2690524","article-title":"Tensor decomposition for signal processing and machine learning","volume":"65","author":"Sidiropoulos","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1587\/nolta.1.37","article-title":"Tensor decompositions for feature extraction and classification of high dimensional datasets","volume":"1","author":"Phan","year":"2010","journal-title":"Nonlinear Theory Its Appl. IEICE"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1137\/070697835","article-title":"Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization","volume":"52","author":"Recht","year":"2010","journal-title":"SIAM Rev."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","article-title":"Exact matrix completion via convex optimization","volume":"9","author":"Recht","year":"2009","journal-title":"Found. Comput. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.sab.2017.10.014","article-title":"A novel baseline correction method using convex optimization framework in laser-induced breakdown spectroscopy quantitative analysis","volume":"138","author":"Yi","year":"2017","journal-title":"Spectrochim. Acta Part B At. Spectrosc."},{"key":"ref_23","unstructured":"De, S.C., Olukotun, K., and R\u00e9, C. (arXiv, 2014). Global convergence of stochastic gradient descent for some non-convex matrix problems, arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Gu, S., Zhang, L., Zuo, W., and Feng, X. (2014, January 23\u201328). Weighted nuclear norm minimization with application to image denoising. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, USA.","DOI":"10.1109\/CVPR.2014.366"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1909","DOI":"10.1109\/TSP.2015.2500179","article-title":"L\u00f6wner-Based Blind Signal Separation of Rational Functions with Applications","volume":"64","author":"Debals","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1214\/14-AOS1257","article-title":"Minimax risk of matrix denoising by singular value thresholding","volume":"42","author":"Donoho","year":"2014","journal-title":"Ann. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6077","DOI":"10.1016\/j.eswa.2010.02.118","article-title":"Bearing fault diagnosis using multi-scale entropy and adaptive neuro-fuzzy inference","volume":"37","author":"Zhang","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1186","DOI":"10.3390\/e14071186","article-title":"Multivariate multi-scale permutation entropy for complexity analysis of Alzheimer\u2019s disease EEG","volume":"14","author":"Morabito","year":"2012","journal-title":"Entropy"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6447","DOI":"10.3390\/e17096447","article-title":"Rolling bearing fault diagnosis based on wavelet packet decomposition and multi-scale permutation entropy","volume":"17","author":"Zhao","year":"2015","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wang, S., Liu, H., and Xu, X. (2016, January 27\u201328). Fan fault diagnosis based on symmetrized dot pattern and improved BP neural network. Proceedings of the 2016 3rd International Conference on Materials Engineering, Manufacturing Technology and Control, Taiyuan, China.","DOI":"10.2991\/icmemtc-16.2016.176"},{"key":"ref_31","unstructured":"Erichson, N.B., Voronin, S., Brunton, S.L., and Kutz, J.N. (arXiv, 2016). Randomized matrix decompositions using R, arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s10107-009-0306-5","article-title":"Fixed point and Bregman iterative methods for matrix rank minimization","volume":"128","author":"Ma","year":"2011","journal-title":"Math. Program."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ji, S., and Ye, J. (2009, January 14\u201318). An accelerated gradient method for trace norm minimization. Proceedings of the 26th Annual International Conference on Machine Learning, Montreal, QC, Canada.","DOI":"10.1145\/1553374.1553434"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0888-3270(87)90069-0","article-title":"Examination of a technique for the early detection of failure in gears by signal processing of the time domain average of the meshing vibration","volume":"1","author":"McFadden","year":"1987","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"S104","DOI":"10.1137\/120876459","article-title":"Normalized iterative hard thresholding for matrix completion","volume":"35","author":"Tanner","year":"2013","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/161\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:57:24Z","timestamp":1760194644000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/161"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,3]]},"references-count":35,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,3]]}},"alternative-id":["e20030161"],"URL":"https:\/\/doi.org\/10.3390\/e20030161","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,3,3]]}}}