This paper presents a new approach for dimension reduction of data observed on spherical surfaces. Several dimension reduction techniques have been developed in recent years for non-euclidean data analysis. As a pioneer work, (Hauberg 2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to process data on Riemannian manifolds, resulting in distorted results. This study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces. Our approach lies in the same line of (Hastie and Stuetzle et al. 1989) that proposed principal curves for data on euclidean space. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on spherical surfaces. The results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach.

中文翻译：

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球形主曲线

本文提出了一种新的方法来减少在球形表面上观察到的数据的尺寸。近年来，已经针对非欧几里得数据分析开发了几种降维技术。作为一项开创性的工作，（Hauberg，2016年）尝试在黎曼流形上实现主曲线。但是，这种方法使用近似值来处理黎曼流形上的数据，从而导致结果失真。这项研究提出了一种将数据投影到连续曲线上以在球面上构造主曲线的新方法。我们的方法与（Hastie and Stuetzle et al。1989）提出的有关欧氏空间数据的主曲线的主线相同。我们进一步研究满足球面自一致性的拟议主曲线的平稳性。