A Haar-Fisz Algorithm for Poisson Intensity Estimation

• Published in: Journal of computational and graphical statistics Vol. 13; no. 3; pp. 621 - 638
• Main Authors: Fryzlewicz, Piotr, Nason, Guy P
• Format: Journal Article
• Language: English
• Published: Alexandria Taylor & Francis 01.09.2004; American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America; Taylor & Francis Ltd
• Subjects: Algorithms; Analytical estimating; College mathematics; Computer software; Cycle spinning; Data smoothing; Denoising; Earthquakes; Estimation methods; Mathematical analysis; Mathematical vectors; Poisson process; Statistical methods; Statistical variance; Studies; Transform to Gaussian; Variance stabilizing transform; Wavelet analysis; United States > US
• ISSN: 1061-8600; 1537-2715
• DOI: 10.1198/106186004X2697

This article introduces a new method for the estimation of the intensity of an inhomogeneous one-dimensional Poisson process. The Haar-Fisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the Haar-Fisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the Haar-Fisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed state-of-the-art competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current state-of-the-art.