ABSTRACT
This chapter considers curve alignment, i.e. registration, in Functional Data Analysis. The main goal is to extend the fundamental concept of modes of variation into both amplitude and phase modes. The need for this is demonstrated with a toy phase shift example where conventional PCA modes are clearly very non-intuitive. Much more insightful decomposition is accomplished using the Fisher-Rao approach that improves upon most proposed methods in this area through directly tackling the problem using the notion of warp equivalence classes and development of the warp invariant Fisher-Rao metric. Careful handling of the quotient space structure, i.e. getting the mathematics really right, results in a fully automatic method, which does not need the heavy manual tuning required by other approaches. The especially strong peak alignment properties of this metric are demonstrated with a toy example. Deeper analysis comes from combining the Fisher Rao approach with a Principal Nested Spheres analysis, as developed in Chapter 8.
