{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T01:06:38Z","timestamp":1784682398701,"version":"3.55.0"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2015,9,21]],"date-time":"2015-09-21T00:00:00Z","timestamp":1442793600000},"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>This paper presents a rolling bearing fault diagnosis approach by integrating wavelet packet decomposition (WPD) with multi-scale permutation entropy (MPE). The approach uses MPE values of the sub-frequency band signals to identify faults appearing in rolling bearings. Specifically, vibration signals measured from a rolling bearing test system with different defect conditions are decomposed into a set of sub-frequency band signals by means of the WPD method. Then, each sub-frequency band signal is divided into a series of subsequences, and MPEs of all subsequences in corresponding sub-frequency band signal are calculated. After that, the average MPE value of all subsequences about each  sub-frequency band is calculated, and is considered as the fault feature of the corresponding sub-frequency band. Subsequently, MPE values of all sub-frequency bands are considered as input feature vectors, and the hidden Markov model (HMM) is used to identify the fault pattern of the rolling bearing. Experimental study on a data set from the Case Western Reserve University bearing data center has shown that the presented approach can accurately identify faults in rolling bearings.<\/jats:p>","DOI":"10.3390\/e17096447","type":"journal-article","created":{"date-parts":[[2015,9,21]],"date-time":"2015-09-21T10:17:32Z","timestamp":1442830652000},"page":"6447-6461","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":89,"title":["Rolling Bearing Fault Diagnosis Based on Wavelet Packet Decomposition and Multi-Scale Permutation Entropy"],"prefix":"10.3390","volume":"17","author":[{"given":"Li-Ye","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, No. 2, Sipailou, Nanjing 210096, China"},{"name":"Key Laboratory of Micro Inertial Instrument and Advanced Navigation Technology,  Ministry of Education, No. 2, Sipailou, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, No. 2, Sipailou, Nanjing 210096, China"},{"name":"Key Laboratory of Micro Inertial Instrument and Advanced Navigation Technology,  Ministry of Education, No. 2, Sipailou, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ru-Qiang","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Instrument Science and Engineering, Southeast University, No. 2, Sipailou, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.isatra.2011.06.003","article-title":"A weighted multi-scale morphological gradient filter for rolling element bearing fault detection","volume":"50","author":"Li","year":"2011","journal-title":"ISA Trans."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1177\/1475921710395806","article-title":"Wavelet domain principal feature analysis for spindle health diagnosis","volume":"10","author":"Yan","year":"2011","journal-title":"Struct. 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