This paper presents a new approach for dimension reduction of data observed in a sphere. Several dimension reduction techniques have recently developed for the analysis of non-Euclidean data. As a pioneer work, Hauberg (2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to deal with data on Riemannian manifolds, which causes distorted results. In this study, we propose a new approach to construct principal curves on a sphere by a projection of the data onto a continuous curve. Our approach lies in the same line of Hastie and Stuetzle (1989) that proposed principal curves for Euclidean space data. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on a sphere. Results from real data analysis with earthquake data and simulation examples demonstrate the promising empirical properties of the proposed approach.

翻译：本文介绍了减少一个领域所观察到的数据的维度的新方法,最近为分析非欧洲域的数据开发了几种维度的减少技术。作为一项先驱工作,Hauberg(Hauberg)(2016年)试图在里曼尼方块上实施主要曲线,但这一方法使用近似值处理里曼尼方块的数据,造成扭曲的结果。在本研究中,我们提出一种新的方法,通过将数据投射到连续曲线上,在某一领域构建主要曲线。我们的方法与Hastie(1989年)和Stuetzle(1989年)的同一线相同,后者提出了欧洲域空间数据的主要曲线。我们进一步调查了满足一个领域的自我一致性的拟议主要曲线的静态性。从地震数据实际分析得出的结果和模拟实例显示了拟议方法有希望的经验特性。