Conservative Hypothesis Test of Multivariate Data from an Uncertain Population with Symmetry Analysis in Music Statistics

Music data exhibits numerous distinct symmetric and asymmetric patterns—ranging 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—a 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.