Stationarity of Manifold Time Series
本文首次定义了流形时间序列的一阶和二阶平稳性，并开发了相应的统计检验方法，通过数值模拟和细胞比例时间序列数据验证了其有效性。
In modern interdisciplinary research, manifold time series data have been garnering more attention. A critical question in analyzing such data is “stationarity”, which reflects the underlying dynamic behavior and is crucial across various fields like cell biology, neuroscience and empirical finance. Yet, there has been an absence of a formal definition of stationarity that is tailored to manifold time series. This work bridges this gap by proposing the first definitions of first-order and second-order stationarity for manifold time series. Additionally, we develop novel statistical procedures to test the stationarity of manifold time series and study their asymptotic properties. Our methods account for the curved nature of manifolds, leading to a more intricate analysis than that in Euclidean space. The effectiveness of our methods is evaluated through numerical simulations and their practical merits are demonstrated through analyzing a cell-type proportion time series dataset from a paper recently published in Cell. The first-order stationarity test result aligns with the biological findings of this article, while the second-order stationarity test provides numerical support for a critical assumption made therein. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.