In many temporal datasets, the parameters of the underlying distribution may change abruptly at unknown times. Detecting such changepoints is crucial for numerous applications. Although such a problem has been extensively studied for linear data, there has been notably less research on bivariate angular data. To the best of our knowledge, this paper presents the first attempt to address the changepoint detection problem for the mean direction of toroidal and spherical data. By defining the ``square of an angle'' through intrinsic geometry, we construct a curved dispersion matrix for bivariate angular data, analogous to the linear dispersion matrix in Euclidean space. Using the analogous measure of the ``Mahalanobis distance,'' we develop two new non-parametric tests to identify changes in the mean direction parameters for toroidal and spherical distributions. The pivotal distributions of the test statistics are shown to follow the Kolmogorov distribution under the null hypothesis. Under the alternative hypothesis, we establish the consistency of the proposed tests. We also apply the proposed methods to detect changes in mean direction for hourly wind-wave direction (toroidal) measurements and the path (spherical) of the cyclonic storm ``Biporjoy,'' which occurred between 6th and 19th June 2023 over the Arabian Sea, western coast of India.

翻译：在许多时间序列数据集中，底层分布的参数可能在未知时刻发生突变。检测此类变点对众多应用至关重要。尽管针对线性数据的此类问题已得到广泛研究，但对二元角度数据的研究却明显不足。据我们所知，本文首次尝试解决环面与球面数据平均方向的变点检测问题。通过内蕴几何定义“角度的平方”，我们为二元角度数据构建了一个曲率弥散矩阵，类似于欧几里得空间中的线性弥散矩阵。利用“马氏距离”的类比度量，我们开发了两种新的非参数检验方法，用于识别环面与球面分布中平均方向参数的变化。在原假设下，检验统计量的枢轴分布被证明服从柯尔莫哥洛夫分布。在备择假设下，我们建立了所提检验方法的一致性。我们还应用所提方法检测了每小时风浪方向（环面）测量数据以及2023年6月6日至19日发生在印度西海岸阿拉伯海上空的气旋风暴“比帕乔伊”路径（球面）的平均方向变化。