Spherical and hyperspherical data are commonly encountered in diverse applied research domains, underscoring the vital task of assessing independence within such data structures. In this context, we investigate the properties of test statistics relying on distance correlation measures originally introduced for the energy distance, and generalize the concept to strongly negative definite kernel-based distances. An important benefit of employing this method lies in its versatility across diverse forms of directional data, enabling the examination of independence among vectors of varying types. The applicability of tests is demonstrated on several real datasets.

翻译：球面及超球面数据在众多应用研究领域中普遍存在，突显了评估此类数据结构中独立性的关键任务。在此背景下，我们研究了基于距离相关度量的检验统计量的性质，该度量最初针对能量距离提出，并将其推广到基于强负定核的距离。采用这种方法的一个重要优势在于其对多种方向数据形式的灵活性，能够检验不同类型向量之间的独立性。通过在若干真实数据集上的应用，验证了这些检验的适用性。