Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency

Abstract

We consider spatially dependent functional data collected under a geostatistics setting, where locations are sampled from a spatial point process. The functional response is the sum of a spatially dependent functional effect and a spatially independent functional nugget effect. Observations on each function are made on discrete time points and contaminated with measurement errors. Under the assumption of spatial stationarity and isotropy, we propose a tensor product spline estimator for the spatio-temporal covariance function. When a coregionalization covariance structure is further assumed, we propose a new functional principal component analysis method that borrows information from neighboring functions. The proposed method also generates nonparametric estimators for the spatial covariance functions, which can be used for functional kriging. Under a unified framework for sparse and dense functional data, infill and increasing domain asymptotic paradigms, we develop the asymptotic convergence rates for the proposed estimators. Advantages of the proposed approach are demonstrated through simulation studies and two real data applications representing sparse and dense functional data, respectively.

Keywords:

• Covariance estimation
• Dimension deduction
• Infill asymptotics
• Nugget effect
• Spatio-temporal
• Tensor product splines

Acknowledgments

We thank the two anonymous referees for their constructive comments and helpful suggestions, which lead to significant improvement of our article.

Supplemental Materials

The online supplementary material contains detailed proofs of the theoretical results, additional figures and tables for the simulation studies and real data analysis, and the codes implementing the proposed methods.

Additional information

Funding

Li’s research was partially supported by National Institute on Aging, grant 5R21AG058198.