Registration of Functional Data Using Fisher-Rao Metric

TL;DR

This paper presents a geometric framework using the Fisher-Rao metric and SRVF representation for aligning functional data, effectively separating phase and amplitude variability, and demonstrating superior performance on diverse real and simulated datasets.

Contribution

Introduces a Fisher-Rao based geometric approach with SRVF for functional data alignment, improving over existing methods in accuracy and computational simplicity.

Findings

Effective separation of phase and amplitude variability.

Superior alignment performance on multiple datasets.

Consistent estimator under complex warping scenarios.

Abstract

We introduce a novel geometric framework for separating the phase and the amplitude variability in functional data of the type frequently studied in growth curve analysis. This framework uses the Fisher-Rao Riemannian metric to derive a proper distance on the quotient space of functions modulo the time-warping group. A convenient square-root velocity function (SRVF) representation transforms the Fisher-Rao metric into the standard $\ltwo$ metric, simplifying the computations. This distance is then used to define a Karcher mean template and warp the individual functions to align them with the Karcher mean template. The strength of this framework is demonstrated by deriving a consistent estimator of a signal observed under random warping, scaling, and vertical translation. These ideas are demonstrated using both simulated and real data from different application domains: the Berkeley growth study, handwritten signature curves, neuroscience spike trains, and gene expression signals. The proposed method is empirically shown to be be superior in performance to several recently published methods for functional alignment.