Unimodality, pivotal in statistical analysis, offers insights into dataset structures and drives sophisticated analytical procedures. While unimodality's confirmation is straightforward for one-dimensional data using methods like Silverman's approach and Hartigans' dip statistic, its generalization to higher dimensions remains challenging. By extrapolating one-dimensional unimodality principles to multi-dimensional spaces through linear random projections and leveraging point-to-point distancing, our method, rooted in $\alpha$-unimodality assumptions, presents a novel multivariate unimodality test named mud-pod. Both theoretical and empirical studies confirm the efficacy of our method in unimodality assessment of multidimensional datasets as well as in estimating the number of clusters.

翻译：单峰性作为统计分析中的关键概念，能够揭示数据集的内在结构并推动复杂分析流程的发展。虽然对于一维数据，通过Silverman检验法和Hartigans峰度统计等方法可以便捷地验证单峰性，但该方法向高维空间的推广仍面临挑战。本文通过线性随机投影将一维单峰性原理拓展至多维空间，并利用点对点距离度量，在$\alpha$-单峰性假设基础上提出名为mud-pod的新型多元单峰性检验方法。理论与实证研究均证实，该方法在评估多维数据集单峰性及估计聚类数量方面具有显著效能。