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

TL;DR

This paper introduces a unified approach for analyzing spatially dependent functional data, combining principal component analysis and covariance estimation to handle both sparse and dense data under spatial dependence.

Contribution

It develops a new tensor product spline estimator for spatio-temporal covariance and a functional PCA method that leverages neighboring information, applicable to both sparse and dense data.

Findings

The method achieves consistent estimation under various asymptotic regimes.

Simulation studies demonstrate superior performance over existing methods.

Real data applications validate the practical utility of the approach.

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.