Functional Principal Component Analysis for Multiple Variables on Different Riemannian Manifolds

Functional principal component analysis (FPCA) is a very important dimension reduction tool for functional data analysis. The conventional FPCA procedures can only model multiple functional variables on the same Riemannian manifold. We propose a new multivariate FPCA method to jointly analyze multiple variables on different Riemannian manifolds. We introduce a mapping procedure to project these variables to the same Euclidean space. The mapped variables are able to preserve temporal variations from the original ones, and their covariance functions can be defined and estimated as the same as those of Euclidean variables. A normalization method is also employed to mitigate the effects of different units of variables and scales of variation. The proposed method allows us to reconstruct curves for non-Euclidean variables with missing observations by using functional principal components and their scores. Several simulation studies are conducted to investigate the finite sample performance of the proposed method. We demonstrate our method by jointly analyzing the wind direction and speed data in three Canadian cities. It shows that the proposed multivariate FPCA method is a versatile tool and provides useful insights for variables on different manifolds. Supplementary materials accompanying this paper appear on-line.