Multiresolution analysis of point processes and statistical thresholding for wavelet-based intensity estimation

TL;DR

This paper introduces a wavelet-based multiresolution framework for analyzing point processes, defining homogeneity and innovation at different scales, and proposes likelihood ratio tests and thresholding strategies that outperform existing methods.

Contribution

It formulates a multiresolution analysis for point processes, introduces new scale-based properties, and develops improved thresholding techniques for intensity estimation.

Findings

Likelihood ratio tests for homogeneity and innovation are asymptotically valid.

Proposed thresholding strategies outperform existing local hard thresholding.

Method effectively estimates intensity functions in simulation scenarios.

Abstract

We take a wavelet based approach to the analysis of point processes and the estimation of the first order intensity under a continuous time setting. A multiresolution analysis of a point process is formulated which motivates the definition of homogeneity at different scales of resolution, termed $J$-th level homogeneity. Further to this, the activity in a point processes' first order behavior at different scales of resolution is also defined and termed $L$-th level innovation. Likelihood ratio tests for both these properties are proposed with asymptotic distributions provided, even when only a single realization of the point process is observed. The test for $L$-th level innovation forms the basis for a collection of statistical strategies for thresholding coefficients in a wavelet based estimator of the intensity function. These thresholding strategies are shown to outperform the existing local hard thresholding strategy on a range of simulation scenarios.