Multivariate functional principal component analysis: A normalization approach

Abstract

This study proposes a modified version of the classical Karhunen-Loeve expansion for a vector-valued random process, called a normalized multivariate functional principal component (mFPCn) approach, as a general stochastic representation for multivariate random functions. The mFPCn approach takes the varying extent of variations between the components of multivariate random functions into account and takes advantage of component dependency through the pairwise cross-covariance functions. The multivariate approach leads to a single set of multivariate functional principal component scores, which serves well as the proxy of multivariate functional data. We derive the consistency properties for the estimates of the mFPCn model components, and the asymptotic distributions for statistical inferences. We illustrate the finite sample performance of the mFPCn approach through the analysis of a traffic flow data set, including an application to clustering multivariate functional data derived from the mFPCn approach and a simulation study. The mFPCn approach serves as a basic and useful statistical tool for multivariate functional data analysis.