Combining Registration and Fitting for Functional Models

A registration method can be defined as a process of aligning features of a sample of curves by monotone transformations of their domain. The aligned curves exhibit only amplitude variation, and the domain transformations, called warping functions, capture the phase variation in the original curves. In this article we precisely define a new type of registration process, in which the warping functions optimize the fit of a principal components decomposition to the aligned curves. The principal components are effectively the features that this process aligns. We discuss the relationship of registration to closure of a function space under convex operations, and define consistency for registration methods. We define an explicit decomposition of functional variation into amplitude and phase partitions, and develop an algorithm for combining registration with principal components analysis, and apply it to simulated and real data.