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，它能揭示数据集的结构特征并推动复杂分析流程的发展。尽管在单维数据中，借助Silverman方法和Hartigans' dip统计量等工具可便捷地验证单峰性，但该方法向高维空间的推广仍面临挑战。通过将一维单峰性原理沿线性随机投影方向拓展至多维空间，并利用点对点距离度量，我们基于$\alpha$-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方法。理论与实证研究均证实，该方法在多维数据集的单峰性评估以及聚类数目估计中均具有显著有效性。