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Title: Multivariate functional principal component analysis for data observed on different (dimensional) domains Authors:  Clara Happ - LMU Munich (Germany) [presenting]
Sonja Greven - LMU Munich (Germany)
Abstract: Existing approaches for multivariate functional principal component analysis are restricted to data on a single interval $\mathcal{T} \subset \mathbb{R}$. The presented approach focuses on multivariate functional data $X = (X^{(1)}, \ldots, X^{(p)})$ observed on different domains $\mathcal{T}_1 , \ldots ,\mathcal{T}_p$ 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-Loeve theorem. For the practically relevant case of a finite, possibly truncated, Karhunen-Loeve representation, a direct theoretical relationship between univariate and multivariate functional principal component analysis is established. This offers a simple estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. The approach can be extended to univariate components $X^{(j)}$ that have a finite expansion in a general, not necessarily orthonormal basis and is applicable for sparse data or data with measurement error. A flexible software implementation for representing multivariate functional data and estimating the multivariate functional PCA is made available in two R-packages. The approach is applied to a neuroimaging study to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline.