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：单峰性是统计分析中的关键概念，有助于理解数据集结构并驱动复杂的分析流程。尽管对于一维数据，通过Silverman方法和Hartigans' dip统计量等方法可以简便地验证单峰性，但其在高维空间中的推广仍具挑战性。通过线性随机投影将一维单峰性原理推广至多维空间，并利用点对点距离，我们的方法基于$\alpha$-单峰性假设，提出了一种名为mud-pod的新型多变量单峰性检验。理论与实证研究均证实了该方法在多维数据集单峰性评估以及聚类数量估计中的有效性。