Test of Weak Separability for Spatially Stationary Functional Field

Abstract

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.

Keywords:

• Lag covariance
• Spatial correlation
• Spatial functional data
• Weak separability

Supplementary Material

For space economy, we collect more discussions and additional results on spatial stationarity and Gaussian assumption, some implementation issues for the choices of truncation parameter and spatial lag, and some technical proofs of the propositions and lemmas in the supplementary material.

Acknowledgments

Decai Liang is the first author, and Fang Yao is the corresponding author. The authors thank the Editor, Associate Editor and three anonymous referees for their many helpful comments that have resulted in significant improvements in the article.

Additional information

Funding

This research is supported by National Key R&D Program of China (No. 2020YFE0204200), National Natural Science Foundation of China Grants (No. 11931001, 11871080, 11871485 and 12101332), the China's National Key Research Special Program Grant 2016YFC0207702, the LMAM, the LMEQF, the LPMC, and the KLMDASR. The China PM2.5 data were provided by the Institute of Atmospheric Physics, Chinese Academy of Science. We thank Giles Hooker for his kindness in sharing with us the Harvard Forest EVI data.