  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.
