Abstract
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.
Highlights
Functional data are encountered in many fields, a multitude of examples can be found in the books by [10, 24, 25]
When analyzing functional data it is of particular interest to provide answers to the questions: (i) is there a common main functional pattern to be distinguished?; (ii) can we quantify the significant individual fluctuations with respect to such a mean pattern? While the common functional mean is capturing main features such as peaks and valleys, differences between individual curves are often exposed via differences in phase and in amplitude of the main features
In this paper we use a mixed effects model in which random effects enter on two levels: (1) a warping function with random effects describes the individual phase variability in a flexible manner, and (2) a second random effect is used to model the individual amplitude variability
Summary
Functional data are encountered in many fields, a multitude of examples can be found in the books by [10, 24, 25]. In this paper we use a mixed effects model in which random effects enter on two levels: (1) a warping function with random effects describes the individual phase variability in a flexible manner, and (2) a second random effect is used to model the individual amplitude variability This follows the approach of [13], but with two major differences: (i) the definition of the warping function does not depend on ‘landmarks’ (locations of peaks and valleys); (ii) the estimation procedure. The added value of the method is illustrated on the Pinch Force data, where our analysis provides a mean pattern, and allows to describe clearly where most individual differences occur with respect to either phase or amplitude.
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