Test of Weak Separability for Spatially Stationary Functional Field
针对空间相关函数型数据，提出弱可分离性概念并构建检验方法，通过滞后协方差估计推导检验统计量的渐近分布，模拟和两个真实数据（中国PM2.5、哈佛森林）验证了有效性。
For spatially dependent functional data, a generalized Karhunen-Loève expansion is commonly used to decompose data into an additive form of temporal components and spatially correlated coefficients. This structure provides a convenient model to investigate the space-time interactions, but may not hold for complex spatio-temporal processes. In this work, we introduce the concept of weak separability, and propose a formal test to examine its validity for non-replicated spatially stationary functional field. The asymptotic distribution of the test statistic that adapts to potentially diverging ranks is derived by constructing lag covariance estimation, which is easy to compute for practical implementation. We demonstrate the efficacy of the proposed test via simulations and illustrate its usefulness in two real examples: China PM 2.5 data and Harvard Forest data. Supplementary materials for this article are available online.