## Statistics Seminar Series

Thursday, October 30, 4:10pm, MSB 1147 (Colloquium Room)
Refreshments at 3:30pm in MSB 4110 (Statistics Lounge)

Speaker: Victor Panaretos (École Polytechnique Fédérale de Lausanne, Switzerland)

Title: “Separating Amplitude and Phase Variation in Point Processes

Abstract: We develop a framework for the study of the problem of registration of point processes subjected to warping, otherwise 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 consider the problem of identifiably formalising similar notions for a point process, and of nonparametrically separating them based on realisations of iid copies $\{\Pi_i\}$ of the phase-varying point process.

By analogy to functional data analysis, where one observes multiple iid copies $\{Y_i(x)\}$ of the random function $\{Y(x)\}$, one may think of this problem as one of generalised functional data analysis, where we observe multiple copies $\{\Pi_i\}$ of the random generalised function $\{\Pi\}$. A key element of our approach is interpreting classical phase variation assumptions 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 the consistent separation of the two types of variation for point processes over compact intervals.

(Based on joint work with my PhD student, Yoav Zemel).