{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T13:44:38Z","timestamp":1765547078360,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T00:00:00Z","timestamp":1686873600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007128","name":"Natural Science Foundation of Shaanxi Province","doi-asserted-by":"publisher","award":["2022JM-337","252082220"],"award-info":[{"award-number":["2022JM-337","252082220"]}],"id":[{"id":"10.13039\/501100007128","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Xi\u2019an University of Technology Excellent Seed Fund","award":["2022JM-337","252082220"],"award-info":[{"award-number":["2022JM-337","252082220"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Slope entropy (SlopEn) has been widely applied in fault diagnosis and has exhibited excellent performance, while SlopEn suffers from the problem of threshold selection. Aiming to further enhance the identifying capability of SlopEn in fault diagnosis, on the basis of SlopEn, the concept of hierarchy is introduced, and a new complexity feature, namely hierarchical slope entropy (HSlopEn), is proposed. Meanwhile, to address the problems of the threshold selection of HSlopEn and a support vector machine (SVM), the white shark optimizer (WSO) is applied to optimize both HSlopEn and an SVM, and WSO-HSlopEn and WSO-SVM are proposed, respectively. Then, a dual-optimization fault diagnosis method for rolling bearings based on WSO-HSlopEn and WSO-SVM is put forward. We conducted measured experiments on single- and multi-feature scenarios, and the experimental results demonstrated that whether single-feature or multi-feature, the WSO-HSlopEn and WSO-SVM fault diagnosis method has the highest recognition rate compared to other hierarchical entropies; moreover, under multi-features, the recognition rates are all higher than 97.5%, and the more features we select, the better the recognition effect. When five nodes are selected, the highest recognition rate reaches 100%.<\/jats:p>","DOI":"10.3390\/s23125630","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T02:54:33Z","timestamp":1686884073000},"page":"5630","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["A Dual-Optimization Fault Diagnosis Method for Rolling Bearings Based on Hierarchical Slope Entropy and SVM Synergized with Shark Optimization Algorithm"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5035-223X","authenticated-orcid":false,"given":"Yuxing","family":"Li","sequence":"first","affiliation":[{"name":"School of Automation and Information Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"},{"name":"Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi\u2019an University of Technology, Xi\u2019an 710048, China"}]},{"given":"Bingzhao","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Automation and Information Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6736-1156","authenticated-orcid":false,"given":"Bo","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Automation and Information Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"}]},{"given":"Xiaohui","family":"Xue","sequence":"additional","affiliation":[{"name":"School of Automation and Information Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.knosys.2018.09.004","article-title":"Intelligent fault diagnosis of rotating machinery using improved multiscale dispersion entropy and mRMR feature selection","volume":"163","author":"Yan","year":"2019","journal-title":"Knowl. 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