Multi-dimensional Functional Principal Component Analysis

TL;DR

This paper extends functional principal component analysis to multi-dimensional domains, addressing computational challenges with innovative smoothing, parallel processing, and random projection techniques, validated through simulations and real data.

Contribution

It introduces a novel multi-dimensional FPCA method with efficient computation strategies, achieving optimal convergence rates for high-dimensional functional data.

Findings

Achieves classical nonparametric rates for longitudinal data

Attains optimal convergence rates for functional data

Demonstrates effectiveness on real PM2.5 data

Abstract

Functional principal component analysis is one of the most commonly employed approaches in functional and longitudinal data analysis and we extend it to analyze functional/longitudinal data observed on a general $d$-dimensional domain. The computational issues emerging in the extension are fully addressed with our proposed solutions. The local linear smoothing technique is employed to perform estimation because of its capabilities of performing large-scale smoothing and of handling data with different sampling schemes (possibly on irregular domain) in addition to its nice theoretical properties. Besides taking the fast Fourier transform strategy in smoothing, the modern GPGPU (general-purpose computing on graphics processing units) architecture is applied to perform parallel computation to save computation time. To resolve the out-of-memory issue due to large-scale data, the random projection procedure is applied in the eigendecomposition step. We show that the proposed estimators can achieve the classical nonparametric rates for longitudinal data and the optimal convergence rates for functional data if the number of observations per sample is of the order $(n/ \log n)^{d/4}$. Finally, the performance of our approach is demonstrated with simulation studies and the fine particulate matter (PM 2.5) data measured in Taiwan.