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

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-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève 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.

现有的多元函数主成分分析方法仅适用于同一一维区间上的数据。本文提出的方法则针对不同定义域上的多元函数数据——这类数据的维度可能存在差异，例如函数与图像数据。多元函数主成分分析的理论基础基于卡亨南-洛维定理（Karhunen-Loève Theorem）构建。针对实际应用中常见的有限维卡亨南-洛维表示场景，本文建立了单变量与多变量函数主成分分析之间的关联关系。该关联为基于单变量函数主成分分析结果计算多变量函数主成分及其得分提供了估计策略。针对所得到的估计量，本文推导了其渐近性质。该方法可推广至一般的有限单变量展开场景，无需限定于标准正交基。该方法同样适用于稀疏函数数据或带测量误差的数据。该方法已在CRAN上提供了灵活的R语言实现。在公共一维定义域上的观测数据场景中，新方法的性能可与现有方法相媲美。本研究的应用背景源自一项神经影像学研究：该研究旨在探究神经心理测试得分的纵向轨迹与基线FDG-PET脑扫描图像之间的协变关系。包括详细证明、额外仿真结果与软件在内的补充材料均可在线获取。