Functional singular component analysis

Abstract

Aiming at quantifying the dependence of pairs of functional data (X,Y), we develop the concept of functional singular value decomposition for covariance and functional singular component analysis, building on the concept of 'canonical expansion' of compact operators in functional analysis. We demonstrate the estimation of the resulting singular values, functions and components for the practically relevant case of sparse and noise-contaminated longitudinal data and provide asymptotic consistency results. Expanding bivariate functional data into singular functions emerges as a natural extension of the popular functional principal component analysis for single processes to the case of paired processes. A natural application of the functional singular value decomposition is a measure of functional correlation. Owing to the involvement of an inverse operation, most previously considered functional correlation measures are plagued by numerical instabilities and strong sensitivity to the choice of smoothing parameters. These problems are exacerbated for the case of sparse longitudinal data, on which we focus. The functional correlation measure that is derived from the functional singular value decomposition behaves well with respect to numerical stability and statistical error, as we demonstrate in a simulation study. Practical feasibility for applications to longitudinal data is illustrated with examples from a study on aging and on-line auctions. © 2011 Royal Statistical Society.