Principal Nested Spheres for Time-Warped Functional Data Analysis
针对函数数据中的水平（相位）变异，提出基于主嵌套球面的分析方法，通过平方根速度函数表示将扭曲函数流形映射到希尔伯特球面，实现高效且可解释的变异分解。
There are often two important types of variation in functional data: the horizontal (or phase) variation and the vertical (or amplitude) variation. These two types of variation have been appropriately separated and modeled through a domain warping method (or curve registration) based on the Fisher–Rao metric. This article focuses on the analysis of the horizontal variation, captured by the domain warping functions. The square-root velocity function representation transforms the manifold of the warping functions to a Hilbert sphere. Motivated by recent results on manifold analogs of principal component analysis, we propose to analyze the horizontal variation via a principal nested spheres approach. Compared with earlier approaches, such as approximating tangent plane principal component analysis, this is seen to be an efficient and interpretable approach to decompose the horizontal variation in both simulated and real data examples.