Collection of functional data is becoming increasingly common including longitudinal observations in many studies. For example, we use magnetic resonance (MR) spectra collected over a period of time from late stage HIV patients. MR spectroscopy (MRS) produces a spectrum which is a mixture of metabolite spectra, instrument noise and baseline profile. Analysis of such data typically proceeds in two separate steps: feature extraction and regression modeling. In contrast, a recently-proposed approach, called partially empirical eigenvectors for regression (PEER) (Randolph, Harezlak and Feng, 2012), for functional linear models incorporates a priori knowledge via a scientifically-informed penalty operator in the regression function estimation process. We extend the scope of PEER to the longitudinal setting with continuous outcomes and longitudinal functional covariates. The method presented in this paper: 1) takes into account external information; and 2) allows for a time-varying regression function. In the proposed approach, we express the time-varying regression function as linear combination of several time-invariant component functions; the time dependence enters into the regression function through their coefficients. The estimation procedure is easy to implement due to its mixed model equivalence. We derive the precision and accuracy of the estimates and discuss their connection with the generalized singular value decomposition. Real MRS data and simulations are used to illustrate the concepts.
Kundu, Madan G.; Harezlak, Jaroslaw; and Randolph, Timothy W., "LONGITUDINAL FUNCTIONAL MODELS WITH STRUCTURED PENALTIES" (November 2012). Johns Hopkins University, Dept. of Biostatistics Working Papers. Working Paper 248.