Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains

TL;DR

This paper introduces a novel multivariate functional principal component analysis method that handles data on different, possibly high-dimensional, domains, extending existing techniques to more diverse data types like images and functions.

Contribution

It develops a theoretical framework based on the Karhunen-Loève Theorem for multivariate functional PCA across different domains, including estimation strategies and asymptotic properties.

Findings

Method is competitive with existing approaches on common domains.

Applicable to sparse data and data with measurement error.

Provides a flexible R implementation on CRAN.

Abstract

Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, e.g. functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Lo\`eve Theorem. For the practically relevant case of a finite Karhunen-Lo\`eve representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error. A flexible R-implementation is available on CRAN. The new method is shown to be competitive to existing approaches for data observed on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. Supplementary material, including detailed proofs, additional simulation results and software is available online.