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 ∈ [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 {Πi} 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 √ 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.

Functional Models for Time-Varying Random Objects

“…Another issue that requires further investigation is the fact that often objects are not fully observed but must be estimated from a random sample that is generated by the object, where one assumes that these within-object samples are independent from the functional random objects. This estimation step induces another error that will typically not satisfy assumption (27) but these errors will vanish as the within-object samples become larger and so under reasonable assumptions can be ignored in the asymptotic analysis; this kind of estimation error has been analysed in detail for the case of distributions (Panaretos and Zemel, 2016;Petersen and Müller, 2016), where the distributions are usually not known but must be estimated from samples.…”

Wasserstein covariance for multiple random densities

“…For a single density process F, the theoretical and practical properties of the Wasserstein mean have been thoroughly investigated (Bolstad et al, 2003;Zhang & Müller, 2011;Panaretos & Zemel, 2016;Bigot et al, 2017), and recently the Wasserstein variance was adopted to quantify variability explained when performing dimension reduction for densities . To quantify the dependence between two random densities, we propose here the extension of these concepts to a Wasserstein covariance measure.…”

“…While functional principal component analysis using cross-sectional averaging can be directly applied for samples of density functions (Kneip & Utikal, 2001), more recently techniques have been developed that incorporate the geometric constraints inherent to the space of density functions. A popular metric for data where each data atom corresponds to a randomly sampled distribution or density is the Wasserstein metric, both for its theoretical appeal and its convincing empirical performance in various applications (Bolstad et al, 2003;Zhang & Müller, 2011;Bigot et al, 2016Bigot et al, , 2017Panaretos & Zemel, 2016).…”

Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

“…A future interesting aspect is the exploration of the applicability of the Optimal Transport framework to the registration problem, as suggested in Panaretos and Zemel (2016) in a discrete context, and its links with the diffeomorphic deformation framework. This is of potential interest in the surface registration framework, where we usually lack physical models that can describe the phenomena, and thus a 'least action' approach could well be effective.…”