Amplitude and Phase Variation of Generalized Functional Data

• Activity Number: 286
• Type: Invited
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Victor Michael Panaretos*
• Companies: EPFL
• Keywords: Fréchet mean ; Monge problem ; Optimal transportation ; Point process ; Registration ; Warping

Abstract

We develop a framework for the study of the problem of registration of point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function corresponds to its random oscillations in the y-axis, typically encapsulated by its (co)variation around a mean level. In contrast, its phase variation refers to fluctuations in the x-axis, often caused by random time changes. We consider the problem of identifiably formalising similar notions for a point process, and of nonparametrically separating them based on realisations of iid copies of the phase-varying point process. The key element to our approach is the extension of classical phase variation assumptions from the functional to the generalised functional case, and their subsequent interpretation through the prism of the theory of optimal transportation of measure. We demonstrate that this induces a natural geometry compatible with the warping problem and we prove that it allows for consistent separation of the two types of variation (based on joint work with Yoav Zemel, EPFL).