On the Definition of Phase and Amplitude Variability in Functional Data Analysis

The problem of data registration is often encountered in the recent statistical literature concerning both longitudinal data analysis (e.g. Lindstrom and Bates 1990, Ke and Wang 2001, and Altman and Villarreal 2004) and functional data analysis (e.g. Ramsay and Li 1998, Kneip et al. 2005, Ramsay and Silverman 2005, James 2007, Kaziska and Srivastava 2007, and Sangalli et al. 2009). The present work develops within the latter line of research. Although many successful methods have been proposed in the literature, a clear theoretical analysis about the soundness and the meaningfulness of the problem of functional data registration is still missing. This work aims at being a first attempt to put in a sound and coherent mathematical framework this key problem of functional data analysis. In detail, we show that the introduction of a metric/semi-metric and of a compact group of warping functions, respect to which the metric/semi-metric is invariant, is the key to a clear and not ambiguous definition of phase and amplitude variability. This work does not claim to cover all possible approaches to functional data registration; it rather aims at providing a clear mind-set through which setting and comparing different registration methods