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.

arXiv commentPublished at http://dx.doi.org/10.1214/15-AOS1387 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Fast Computation of Wasserstein Barycenters
• Generative Models for Functional Data using Phase and Amplitude Separation
• Registration of Functional Data Using Fisher-Rao Metric
• Amplitude and phase variation of point processes
• Curve alignment by moments
• Estimating deformations of isotropic Gaussian random fields on the plane
• Consistent estimates of deformed isotropic Gaussian random fields on the plane
• Geodesic PCA in the Wasserstein space

Cited by in corpus (28)

• Statistical Aspects of Wasserstein Distances
• Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration
• Amplitude and phase variation of point processes
• Unsupervised Alignment of Embeddings with Wasserstein Procrustes
• Gradient descent algorithms for Bures-Wasserstein barycenters
• Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters
• Screening Sinkhorn Algorithm for Regularized Optimal Transport
• Quantum Optimal Transport
• Statistical Analysis of Functions on Surfaces, with an application to Medical Imaging
• Penalized Barycenters in the Wasserstein Space
• Wasserstein Autoregressive Models for Density Time Series
• Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in ${\mathbb{R}}^d$
• Inconsistency of Template Estimation by Minimizing of the Variance/Pre-Variance in the Quotient Space
• Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo
• Wasserstein Regression with Empirical Measures and Density Estimation for Sparse Data
• Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent
• Randomised Wasserstein Barycenter Computation: Resampling with Statistical Guarantees
• Tensor optimal transport, distance between sets of measures and tensor scaling
• Fast and Smooth Interpolation on Wasserstein Space
• Autoregressive Optimal Transport Models
• Learning Registered Point Processes from Idiosyncratic Observations
• Registration for Incomplete Non-Gaussian Functional Data
• Minimax convergence rate for estimating the Wasserstein barycenter of random measures on the real line
• Modelling of functional profiles and explainable shape shifts detection: An approach combining the notion of the Fréchet mean with the shape invariant model
• Wasserstein Gradients for the Temporal Evolution of Probability Distributions
• Wasserstein Regression
• On the use of Wasserstein metric in topological clustering of distributional data
• On the separation of shape and temporal patterns in time series -Application to signature authentication-