2017/02/12 by Anirvan Chakraborty, Chakraborty, Anirvan, Victor M. Panaretos +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Environmental Science · #Time Series Analysis and Forecasting #Complex Systems and Time Series Analysis #Isotope Analysis in Ecology
paper · pdf · doi:10.48550/arxiv.1702.03556
We develop theory and methodology for the problem of nonparametric\nregistration of functional data that have been subjected to random deformation\n(warping) of their time scale. The separation of this phase variation\n("horizontal" variation) from the amplitude variation ("vertical" variation) is\ncrucial in order to properly conduct further analyses, which otherwise can be\nseverely distorted. We determine precise nonparametric conditions under which\nthe two forms of variation are identifiable. These show that the\nidentifiability delicately depends on the underlying rank. By means of several\ncounterexamples, we demonstrate that our conditions are sharp if one wishes a\ngenuinely nonparametric setup; and in doing so we caution that popular remedies\nsuch as structural assumptions or roughness penalties can easily fail. We then\npropose a nonparametric registration method based on a "local variation\nmeasure", the main element in elucidating identifiability. A key advantage of\nthe method is that it is free of any tuning or penalisation parameters\nregulating the amount of alignment, thus circumventing the problem of\nover/under-registration often encountered in practice. We provide asymptotic\ntheory for the resulting estimators under the identifiable regime, but also\nunder mild departures from identifiability, quantifying the resulting bias in\nterms of the amplitude variation's spectral gap.\n