Registration of Functional Data Using the Fisher-Rao Metric

• Activity Number: 2 - Phase Variation (Curve Registration) in Functional Data Analyzing
• Type: Invited
• Date/Time: Sunday, July 30, 2017 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Sebastian Kurtek* and Anuj Srivastava and Wei Wu and Eric Klassen and J. S. (Steve) Marron
• Companies: The Ohio State University and Florida State University and Florida State University and Florida State University and University of North Carolina
• Keywords: amplitude and phase separation ; Fisher-Rao metric ; functional data alignment ; square-root slope function

Abstract

We present a geometric framework for aligning functional data, or alternatively, separating the phase and amplitude variabilities in functional data. This framework is based on extending the nonparametric version of the Fisher-Rao Riemannian metric to general function spaces, and relies on the fact that this metric is invariant to identical warpings of its arguments. A square-root slope function (SRSF) representation of functions transforms the extended Fisher-Rao metric into the standard L2 distance, simplifying computations. The resulting distance is then used to (1) define phase and amplitude components of pairs of functions, (2) compute a mean amplitude for a set of functions, and (3) compute individual phases/amplitudes with respect to this mean (multiple alignment). These ideas are demonstrated using simulated examples as well as real data from different application domains including the Berkeley growth study, handwritten signature curves, mass spectrometry data in proteomics, and electrocardiogram signals. Additionally, we empirically compare multiple alignment results based on our method to several other approaches.