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：单峰性在统计分析中至关重要，既能揭示数据集结构，又能驱动复杂的分析方法。虽然对于一维数据，使用Silverman方法和Hartigans'双峰统计量等途径确认单峰性较为直接，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外推至多维空间，并利用点对点距离度量，我们提出了一种基于$\alpha$-单峰性假设的新型多变量单峰性检验方法，命名为mud-pod。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及聚类数量估计中的有效性。