For spatially dependent functional data, a generalized Karhunen-Loeve 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 PM2.5 data and Harvard Forest data.
Papers Published in 2021:
 Decai Liang, Hui Huang, Yongtao Guan and Fang Yao*. Test of weak separability for spatially stationary functional field. Journal of the American Statistical Association.Published online,https://doi.org/10.1080/01621459.2021.2002156.
 Ying Yang and Fang Yao* . Online estimation for functional data.Journal of the American Statistical Association. Published online,https://doi.org/10.1080/01621459.2021/2002158.
 Hui Chen, Haojie Ren, Fang Yao* and Changliang Zou . Data-driven selection of the number of change-points via error rate control. Journal of the American Statistical Association.Published online, https://doi.org/10.1080/01621459.2021.1999820.
Prof. Yao Fang