Comments on: Dynamic relations for sparsely sampled Gaussian processes

Abstract

The issues arising from Functional Data Analysis (FDA) continue to be a rich source of theoretical and application research. Professor Hans-Georg Muller and his collaborators are leading innovators in this research area and their contributions have enable others, including myself, to establish sufficient background for conducting research that resolves problems we encounter in our research areas. The current paper covers a wide range of FDA literature with emphasis on both the response and the predictors being curves. It well presents the concepts behind a unified estimation approach which is built on the estimated functional principle component (FPC) eigenfunctions and scores. The focus of the current paper is on Gaussian processes. The authors have briefly mentioned the possibility and potential benefits of directly extending the established system to non-Gaussian measurements. Following this line of thinking, this discussion focuses on three topics aiming at potential future research: (i) What could be the potential effects of adopting the normality based approach directly when the normality assumption is violated? (ii) What would be the pros and cons of a direct use of the normality based approach versus a method that models the FPC scores nonparametrically? and (iii) For either approach, would there be a need to take into the fact that the FPC eigenfunctions and scores are estimated? The first two directions have a close link to the measurement error literature (Carroll et al. 2006). As is well recognized by Professor Muller, there is a close link between the FDA research and that of longitudinal data analysis (LDA). We adopt an alternative form of (6) of the paper with the mean function μ(Tij) being subtracted from Wij: Wij*=Wij−μ(Tij)=∑k=1∞ξikϕk(Tij)+eij, (1) where ϕk(·) and ξik denote the kth eigenfunction and the corresponding functional principal component score for the ith subject, respectively, and eij denotes the zero-mean, i.i.d. measurement error. As is commonly done in the literature, the corresponding eigenvalues of ϕk(·) are in descending order. We will reserve the notation Y as the regression response variable as in Sect. 2 of the paper. The Principal Analysis by Conditional Expectation (PACE) estimation procedure is built on proper use of the estimated ϕk(t) or its derivatives and the best linear unbiased predictor (BLUP) of ξik under the normality assumption (see (9) in the paper). Once these predictors are constructed, they replace the role of unobserved ξik in the regression estimation procedure. This practice is equivalent to the regression calibration approach (or can be modified into a refined version) in the measurement-error literature when the consideration is relating response Y to either the normal processes themselves or their derivatives. This approach is very general and can be adopted into various more complex situations as being clearly illustrated in the current paper. One key question to be considered could be that whether the expression in (1) remains valid when the normality assumption on ξik fails. I support the authors’ point of view of adopting this structure and then investigate the robustness of the estimation outcomes against the violation of normality assumption.