While change point detection in time series data has been extensively studied, little attention has been given to its generalisation to data observed on spheres or other manifolds, where changes may occur within spatially complex regions with irregular boundaries, posing significant challenges. We propose a new class of estimators, namely, Change Region Identification and SeParation (CRISP), to locate changes in the mean function of a signal-plus-noise model defined on $d$-dimensional spheres. The CRISP estimator applies to scenarios with a single change region, and is extended to multiple change regions via a newly developed generic scheme. The convergence rate of the CRISP estimator is shown to depend on the VC dimension of the hypothesis class that characterises the change regions in general. We also carefully study the case where change regions have the geometry of spherical caps. Simulations confirm the promising finite-sample performance of this approach. The CRISP estimator's practical applicability is further demonstrated through two real data sets on global temperature and ozone hole.

翻译：尽管时间序列数据中的变点检测已被广泛研究，但将其推广至球面或其他流形上观测数据的研究鲜有涉及——此类数据的变化可能发生在边界不规则的复杂空间区域内，从而带来显著挑战。我们提出一类新型估计器，即变化区域识别与分离（CRISP），用于定位定义在$d$维球面上的信号加噪声模型中均值函数的变化。CRISP估计器适用于单变化区域场景，并通过新开发的通用方案扩展至多变化区域情形。该估计器的收敛速度被证明依赖于特征变化区域的假设类VC维数。我们同时深入研究了变化区域具有球冠几何特征时的特例。仿真实验验证了该方法在有限样本下的优异性能，全球气温与臭氧空洞两个真实数据集进一步展示了CRISP估计器的实际应用价值。