Kernel Smoothing, Mean Shift, and Their Learning Theory with Directional Data

Abstract

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.

arXiv comment92 pages, 11 figures. Accepted to the Journal of Machine Learning Research

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• The Sloan Digital Sky Survey: Technical Summary
• The Dark Energy Survey: more than dark energy - an overview
• Uniform in bandwidth consistency of kernel-type function estimators
• A review of mean-shift algorithms for clustering
• Kernel density estimation for directional-linear data
• Toroidal diffusions and protein structure evolution
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• On the Convergence of the Mean Shift Algorithm in the One-Dimensional Space
• Mode-Seeking Clustering and Density Ridge Estimation via Direct Estimation of Density-Derivative-Ratios

Cited by in corpus (4)

• Nonparametric multimodal regression for circular data
• The EM Perspective of Directional Mean Shift Algorithm
• Linear Convergence of the Subspace Constrained Mean Shift Algorithm: From Euclidean to Directional Data
• Mode and Ridge Estimation in Euclidean and Directional Product Spaces: A Mean Shift Approach