Directional data consist of observations distributed on a (hyper)sphere, and appear in many applied fields, such as astronomy, ecology, and environmental science. This paper studies both statistical and computational problems of kernel smoothing for directional data. We generalize the classical mean shift algorithm to directional data, which allows us to identify local modes of the directional kernel density estimator (KDE). The statistical convergence rates of the directional KDE and its derivatives are derived, and the problem of mode estimation is examined. We also prove the ascending property of the directional mean shift algorithm and investigate a general problem of gradient ascent on the unit hypersphere. To demonstrate the applicability of the algorithm, we evaluate it as a mode clustering method on both simulated and real-world data sets.

翻译：方向数据包含分布在( 超) 范围上的观测数据, 并出现在天文学、生态学和环境科学等许多应用领域。 本文研究了为方向数据平滑内核的统计和计算问题。 我们将古典平均转换算法推广到方向数据, 从而使我们能够确定方向内核密度测深仪(KDE)的本地模式。 引出方向 KDE 及其衍生物的统计趋同率, 并研究模式估算问题。 我们还证明了方向平均转换算法的不断上升的属性, 并调查了单位双曲线梯度上升的一般问题。 为了证明算法的适用性, 我们将其评估为模拟和真实世界数据集的一种模式组合方法。