{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T14:25:41Z","timestamp":1763389541677,"version":"3.45.0"},"reference-count":14,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,15]],"date-time":"2025-11-15T00:00:00Z","timestamp":1763164800000},"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>Music data exhibits numerous distinct symmetric and asymmetric patterns\u2014ranging from symmetric pitch sequences and rhythmic cycles to asymmetric phrase structures and dynamic shifts. These varied and often subjective patterns present notable challenges for data analysis, such as distinguishing meaningful structural features from noise and adapting analytical methods to accommodate both regularity and irregularity. To tackle this challenge, we present a novel uncertain hypothesis test, referred to as the conservative hypothesis test, which is designed to assess the validity of statistical hypotheses associated with the symmetric and asymmetric patterns exhibited by two multivariate normal uncertain populations. Specifically, we extend the uncertain hypothesis test for the mean difference between two single-characteristic normal uncertain populations to the multivariate case, filling a research gap in uncertainty theory. Building on this two-population multivariate hypothesis test, we propose the conservative hypothesis test\u2014a feasible uncertain hypothesis testing method for multivariable scenarios, developed based on multiple comparison procedures. To demonstrate the practical utility of these methods, we apply them to music-related statistical data, assessing whether two groups of evaluators use consistent criteria to score music. In essence, the hypothesis tests proposed in this paper hold significant value for social sciences, particularly music statistics, where data inherently contains ambiguity and uncertainty.<\/jats:p>","DOI":"10.3390\/sym17111973","type":"journal-article","created":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T14:04:07Z","timestamp":1763388247000},"page":"1973","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Conservative Hypothesis Test of Multivariate Data from an Uncertain Population with Symmetry Analysis in Music Statistics"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8259-9990","authenticated-orcid":false,"given":"Anshui","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics, Physics and Information, Shaoxing University, Shaoxing 312000, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-4475-5433","authenticated-orcid":false,"given":"Jiajia","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Physics and Information, Shaoxing University, Shaoxing 312000, China"}]},{"given":"Shiqi","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematics, Physics and Information, Shaoxing University, Shaoxing 312000, China"}]},{"given":"Wenxing","family":"Zeng","sequence":"additional","affiliation":[{"name":"Cai Yuanpei School of Art and Design, Shaoxing University, Shaoxing 312000, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,15]]},"reference":[{"key":"ref_1","unstructured":"Liu, B. (2007). Uncertainty Theory, Springer."},{"key":"ref_2","first-page":"363","article-title":"Prospect theory: An analysis of decision under risk","volume":"47","author":"KaiIneman","year":"1979","journal-title":"Econometrica"},{"key":"ref_3","unstructured":"Liu, B. (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5579","DOI":"10.1007\/s00500-017-2521-y","article-title":"Uncertain regression analysis: An approach for imprecise observations","volume":"22","author":"Yao","year":"2018","journal-title":"Soft Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2901","DOI":"10.1007\/s12652-024-04785-z","article-title":"Bayesian inference in the framework of uncertainty theory","volume":"15","author":"Li","year":"2024","journal-title":"J. Ambient. Intell. Humaniz. 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Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Dawson, M.R. (2020). Artificial neural networks solve musical problems with fourier phase spaces. Sci. Rep., 10.","DOI":"10.1038\/s41598-020-64229-4"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"226","DOI":"10.31811\/ojomus.1361656","article-title":"Four axioms for a theory of rhythmic sets and their implications","volume":"8","author":"Abarca","year":"2023","journal-title":"Online J. Music. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s10700-024-09436-8","article-title":"Music statistics: Uncertain logistic regression models with applications in analyzing music","volume":"23","author":"Lu","year":"2024","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_13","first-page":"3","article-title":"Some research problems in uncertainty theory","volume":"3","author":"Liu","year":"2009","journal-title":"J. Uncertain Syst."},{"key":"ref_14","unstructured":"Anderson, D.R., Williams, T.A., and Cochran, J.J. (2020). Statistics for Business & Economics, Cengage Learning."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1973\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T14:18:27Z","timestamp":1763389107000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1973"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,15]]},"references-count":14,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111973"],"URL":"https:\/\/doi.org\/10.3390\/sym17111973","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,15]]}}}