{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:20:46Z","timestamp":1760239246023,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,20]],"date-time":"2020-10-20T00:00:00Z","timestamp":1603152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this research, the fourth axiom to improve the well-defined examination of similarity measures is studied, where the measures have a symmetric structure. We first provide a theoretic enhancement of three correlation coefficient similarity measures that were proposed by a source paper. Second, we use the same numerical example proposed by the source paper for pattern recognition problems to illustrate that the weighted correlation coefficient similarity measure is dependent on the set of weights. Finally, we demonstrate that the correlation coefficient similarity measure in the intuitionistic fuzzy set environment can address the issue of practical fault diagnosis when solving the turbine engine problems using similarity measures with symmetric characteristics.<\/jats:p>","DOI":"10.3390\/sym12101735","type":"journal-article","created":{"date-parts":[[2020,10,20]],"date-time":"2020-10-20T20:50:07Z","timestamp":1603227007000},"page":"1735","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Fourth Axiom of Similarity Measures"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1779-130X","authenticated-orcid":false,"given":"Chun-Hsiao","family":"Chu","sequence":"first","affiliation":[{"name":"Department of Tourism, Aletheia University, New Taipei City 25103, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1189-4922","authenticated-orcid":false,"given":"Chih-Ping","family":"Yen","sequence":"additional","affiliation":[{"name":"Department of Information Management, Central Police University, Taoyuan 33304, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3728-1013","authenticated-orcid":false,"given":"Yi-Fong","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Fashion Industry Management, Hsing Wu University, New Taipei City 25103, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Info. 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