Super-resolution estimation of cyclic arrival rates

TL;DR

This paper introduces a super-resolution method for estimating cyclic intensity functions in nonhomogeneous Poisson processes, leveraging window functions to improve resolution without complex optimization.

Contribution

It proposes a novel super-resolution estimation technique that avoids semidefinite programming, extending spectral estimation theory for cyclic arrival rates.

Findings

Achieves super-resolution without semidefinite programming

Provides finite sample guarantees under certain conditions

Expands theoretical understanding of spectral estimation in point processes

Abstract

Exploiting the fact that most arrival processes exhibit cyclic behaviour, we propose a simple procedure for estimating the intensity of a nonhomogeneous Poisson process. The estimator is the super-resolution analogue to Shao 2010 and Shao & Lii 2011, which is a sum of $p$ sinusoids where $p$ and the frequency, amplitude, and phase of each wave are not known and need to be estimated. This results in an interpretable yet flexible specification that is suitable for use in modelling as well as in high resolution simulations. Our estimation procedure sits in between classic periodogram methods and atomic/total variation norm thresholding. Through a novel use of window functions in the point process domain, our approach attains super-resolution without semidefinite programming. Under suitable conditions, finite sample guarantees can be derived for our procedure. These resolve some open questions and expand existing results in spectral estimation literature.