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

Area of Science:

• Geostatistics
• Functional Data Analysis
• Spatio-temporal Statistics

Background:

• Functional data analysis often involves spatially dependent observations.
• Existing methods may struggle with the complexity of spatio-temporal functional data and measurement errors.
• Geostatistical settings require specialized techniques for spatial point process data.

Purpose of the Study:

• To develop a robust method for estimating spatio-temporal covariance functions from spatially dependent functional data.
• To introduce a functional principal component analysis (FPCA) method that leverages neighboring function information.
• To provide nonparametric estimators for spatial covariance functions applicable to functional kriging.

Main Methods:

• Tensor product spline estimation for spatio-temporal covariance functions.
• A novel FPCA method assuming a coregionalization covariance structure.
• Asymptotic convergence rate analysis under sparse/dense and infill/increasing domain paradigms.

Main Results:

• The proposed tensor product spline estimator accurately models spatio-temporal covariance.
• The new FPCA method effectively borrows information from neighboring functions.
• Nonparametric spatial covariance estimators are derived for functional kriging.

Conclusions:

• The developed approach offers a unified framework for analyzing sparse and dense functional data in geostatistics.
• The method demonstrates advantages in simulation studies and real-world applications.
• This work advances the analysis of complex spatio-temporal functional data.