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   -> Volume 6, Issue 3

Preprint: Wavelet Shrinkage of Poisson Data
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Eric Kolaczyk (

PostPosted: Tue Mar 04, 1997 9:02 pm    
Subject: Preprint: Wavelet Shrinkage of Poisson Data
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#10 Preprint: Wavelet Shrinkage of Poisson Data

The following three manuscripts, relating to de-noising Poisson data
with wavelets, are available via anonymous ftp. The file format is
compressed postscript (*.ps.Z)

1.) Title: ``Estimation of Intensities of Burst-Like Poisson
Processes Using Haar Wavelets."
Author: Eric D. Kolaczyk

I present a method for producing estimates of the intensity function
of certain `burst-like' inhomogeneous Poisson processes, based on the
shrinkage of Haar wavelet coefficients of the observed counts. The
Haar basis is a natural wavelet basis in which to work in this
context, and I derive thresholds for shrinkage estimation based on the
distribution of the coefficients. The translation-invariant
de-noising approach of Donoho and Coifman (1995) is used in
conjunction with these level-dependent thresholds to yield smooth
estimates, without the usual `staircase' structure associated with
Haar wavelets. This work is motivated by recent efforts in astronomy
to model the intensity functions underlying gamma-ray bursts. It is
demonstrated that the method proposed herein (TIPSH) yields sharper
estimates of the detail structure in these signals than those obtained
through an analogous version of the standard adaptation of wavelet
shrinkage for Poisson counts, based on the square-root transformation.

2.) Title: ``Non-Parametric Estimation of Gamma-Ray
Burst Intensities Using Haar Wavelets."
The Astrophysical Journal, 1997 (to appear)
Author: Eric D. Kolaczyk

In this article, I present a method for the non-parametric
(model-free) estimation of intensity profiles underlying gamma-ray
bursts. The method, TIPSH, is based on applying specially calibrated
thresholds to the Haar wavelet coefficients of binned counts gathered
from such bursts. As functions well-localized with respect to both
time and scale, wavelets are an ideal tool for working with the often
sharp, abrupt nature of gamma-ray burst signals. When applied to an
idealized signal in a small simulation study and a selection of actual
gamma-ray bursts, the TIPSH algorithm is found to be well capable of
simultaneously estimating the smooth, uniform background and the
pulse-like structure of gamma-ray burst signals.

3.) Title: ``A Method for Wavelet Shrinkage Estimation of Certain
Poisson Intensity Signals Using Corrected Thresholds."
Author: Eric D. Kolaczyk

Wavelet shrinkage estimation has been found to be a powerful tool for
the non-parametric estimation of spatially variable phenomena. Most
work in this area to date has concentrated primarily on the use of
wavelet shrinkage techniques in contexts where the data are modeled as
observations of a signal plus additive, Gaussian noise. When the data
instead take the form of Poisson counts, a common procedure is to
first pre-process the data using Anscombe's square root
transformation, thereby normalizing the data and stabilizing the
variance. However, this approach has a tendency to smooth away sharp,
brief structure in the underlying intensity function, especially in
situations involving very low levels of counts. In this paper, I
introduce an alternative approach to estimating intensity functions
for a certain class of `burst-like' Poisson processes using wavelet
shrinkage. The proposed method is based on the shrinkage of wavelet
coefficients of the original, un-transformed count data. `Corrected'
versions of the usual Gaussian-based shrinkage thresholds are used.
The corrections explicitly account for effects of the first few
cumulants of the Poisson distribution on the tails of the coefficient
distributions. A large deviations argument is used to justify these
corrections. The performance of the new method is examined, and
compared to that of the pre-processing approach, in the context of an
application to an astronomical gamma-ray burst signal.

Communications regarding these manuscripts may be directed to
Eric Kolaczyk at
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