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research article

Amplitude And Phase Variation Of Point Processes

Panaretos, Victor M.  
•
Zemel, Yoav  
2016
Annals Of Statistics

We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function {Y(x) : x is an element of [0, 1]} 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 formalise similar notions for a point process, and nonparametrically separate them based on realisations of i.i.d. copies {Pi(i)} of the phase-varying point process. A key element in our approach is to demonstrate that when the classical phase variation assumptions of Functional Data Analysis (FDA) are applied to the point process case, they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure. We demonstrate that these induce a natural Wasserstein geometry tailored to the warping problem, including a formal notion of bias expressing over-registration. Within this framework, we construct nonparametric estimators that tend to avoid over-registration in finite samples. We show that they consistently estimate the warp maps, consistently estimate the structural mean, and consistently register the warped point processes, even in a sparse sampling regime. We also establish convergence rates, and derivev root n-consistency and a central limit theorem in the Cox process case under dense sampling, showing rate optimality of our structural mean estimator in that case.

  • Details
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Type
research article
DOI
10.1214/15-Aos1387
Web of Science ID

WOS:000372594300012

Author(s)
Panaretos, Victor M.  
Zemel, Yoav  
Date Issued

2016

Publisher

Institute of Mathematical Statistics

Published in
Annals Of Statistics
Volume

44

Issue

2

Start page

771

End page

812

Subjects

Doubly stochastic Poisson process

•

Frechet mean

•

geodesic variation

•

Monge problem

•

optimal transportation

•

length space

•

registration

•

warping

•

Wasserstein metric

Editorial or Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
SMAT  
Available on Infoscience
July 19, 2016
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/127365
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