张一坤 （1）， 陈延志 （2） （1） 华盛顿大学， 西雅图， （2） 华盛顿大学统计系， 西雅图
定向数据由分布在（超）球面上的观测结果组成，并出现在许多应用领域，如天文学、生态学和环境科学。本文研究了定向数据内核平滑的统计和计算问题。我们将经典平均移位算法概括为定向数据，这使我们能够识别定向内核密度估计器 （KDE） 的本地模式。推导出定向KDE及其衍生物的统计收敛率，并研究模式估计问题。我们还证明了定向均移算法的提升属性，并调查了单位超球上的梯度上升的一般问题。为了演示我们提议的算法的适用性，我们把它评估为模拟数据集和真实数据集上的模式聚类方法。
Kernel Smoothing, Mean Shift, and Their Learning Theory with Directional Data
Yikun Zhang (1), Yen-Chi Chen (2) ((1) University of Washington, Seattle, (2) Department of Statistics, University of Washington, Seattle)
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 our directional mean shift algorithm and investigate a general problem of gradient ascent on the unit hypersphere. To demonstrate the applicability of our proposed algorithm, we evaluate it as a mode clustering method on both simulated and real-world datasets.
