## The Annals of Statistics

### Amplitude and phase variation of point processes

#### Abstract

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 $\{Y(x):x\in[0,1]\}$ 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 derive $\sqrt{n}$-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.

#### Article information

Source
Ann. Statist., Volume 44, Number 2 (2016), 771-812.

Dates
Revised: September 2015
First available in Project Euclid: 17 March 2016

https://projecteuclid.org/euclid.aos/1458245735

Digital Object Identifier
doi:10.1214/15-AOS1387

Mathematical Reviews number (MathSciNet)
MR3476617

Zentralblatt MATH identifier
1381.62261

Subjects
Primary: 62M
Secondary: 60G55: Point processes 62G

#### Citation

Panaretos, Victor M.; Zemel, Yoav. Amplitude and phase variation of point processes. Ann. Statist. 44 (2016), no. 2, 771--812. doi:10.1214/15-AOS1387. https://projecteuclid.org/euclid.aos/1458245735

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