Abstract: We propose an extended version of the classical Karhunen-Loève expansion of a multivariate random process, termed a normalized multivariate functional principal component (mFPC_n) representation. This takes variations between the components of the process into account and takes advantage of component dependencies through the pairwise cross-covariance functions. This approach leads to a single set of multivariate functional principal component scores, which serve well as a proxy for multivariate functional data. We derive the consistency properties for the estimates of the mFPC_n, and the asymptotic distributions for statistical inferences. We illustrate the finite sample performance of this approach through the analysis of a traffic flow data set, including an application to clustering and a simulation study. The mFPC_n approach serves as a basic and useful statistical tool for multivariate functional data analysis.

Key words and phrases: Karhunen-Loève expansion, Mercer’s theorem, multivariate functional data, normalization, traffic flow.