In functional data the interest is to find a global mean pattern, but also to capture the individual curve differences in phase and amplitude. This can be done conveniently by building in random effects on two levels: in the warping functions to account for individual phase variations; and in the linear structure to deal with individual amplitude variations. Via an appropriate choice of the warping function and B-spline approximations, estimation in the nonlinear mixed effects functional model is feasible, and does not require any prior knowledge on landmarks for the functional data. Sufficient and necessary conditions for identifiability of the flexible model are provided. A theoretical study is conducted: we establish asymptotic normality and consistency of the estimators of the registration and amplitude models, convergence of the iterative process, and consistency of the final estimator provided by the iterative process. The finite-sample performance of the proposed estimation procedure is investigated in a simulation study, which includes comparisons with existing methods. The added value of the developed method is further illustrated via the analysis of a real data example.
The first and third author gratefully acknowledge support from the C1-project C16/20/002 of the KU Leuven Research Fund. The resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government. Part of this work was accomplished when the second author was a Postdoctoral Researcher at the KU Leuven.
The authors thank Professor Alois Kneip for helpful discussions, and Professor Daniel Gervini for help with his computer codes. E. Devijver also sincerely thanks R. Molinier for fruitful discussions.
"Nonlinear mixed effects modeling and warping for functional data using B-splines." Electron. J. Statist. 15 (2) 5245 - 5282, 2021. https://doi.org/10.1214/21-EJS1917