Functional correlation and dynamic relations for sparsely sampled random processes.

摘要

In longitudinal studies, it is common to observe repeated measurements data from a sample of subjects where noisy measurements are made at irregular times, with a random number of measurements per subject. Often a reasonable assumption is that the data are generated by the trajectories of a smooth underlying stochastic process.;Aiming at quantifying the dependency 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. A natural application of the functional singular value decomposition is a measure of functional correlation. Due 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 acerbated for the case of sparse longitudinal data, on which we focus. The functional correlation measure 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 online auctions.;To understand the nature of the underlying processes, it is also of interest to relate the values of a process at one time with the value it assumes at another time, and also to relate the values assumed by different components of a multivariate trajectory and its derivative at the same time or at specific times selected for each trajectory.;Reviewing and extending recent work, we demonstrate the estimation of corresponding empirical dynamical systems and demonstrate asymptotic consistency of predictions and dynamic transfer functions. We illustrate the resulting prediction procedures and empirical first order differential equations with a study of the dynamics of longitudinal functional data for the relationship of blood pressure and body mass index.