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   -> Volume 6, Issue 10

Preprint: Preprints in wavelets and statistics from Guy Nason
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Guy Nason (

PostPosted: Thu Sep 18, 1997 3:46 pm    
Subject: Preprint: Preprints in wavelets and statistics from Guy Nason
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#4 Preprint: Preprints in wavelets and statistics from Guy Nason

Preprint 1:

Title: Density and hazard rate estimation for right censored data using
wavelet methods.

Authors: Anestis Antoniadis and Girard Grigoire
Laboratoire LMC - IMAG,
Universiti Joseph Fourier
BP 53, 38041 Grenoble Cedex 09,


Guy Nason
Department of Mathematics
University of Bristol
University Walk, Bristol, BS8 1TW

Abstract: This paper describes a wavelet method for the estimation of density
and hazard rate functions from randomly right censored data. We adopt
a nonparametric approach in assuming that the density and hazard rate
have no specific parametric form. The method is based on dividing
the time axis into a dyadic number of intervals and then counting the
number of events within each interval. The number of events and the
survival function of the observations are then separately smoothed
over time via linear wavelet smoothers, and then the hazard rate
function estimators are obtained by taking the ratio. We prove that
the estimators possess pointwise and global mean square consistency,
obtain the best possible asymptotic MISE convergence rate and are
also asymptotically normally distributed. We also describe simulation
experiments that show these estimators are reasonably reliable
in practice. The method is illustrated with two real examples. The
first uses survival time data for patients with liver metastases from
a colorectal primary tumour without other distant metastases.
The second is concerned with times of unemployment for women and the
wavelet estimate, through its flexibility, provides a new and
interesting interpretation.

Keywords: survival data; hazard rate; wavelet estimation

To obtain the compressed PostScript:

Preprint 2.

Title: Statistical modelling of time series using non-decimated wavelet

Authors: Guy Nason and Theofanis Sapatinas
Department of Mathematics, University of Bristol
University Walk, Bristol, BS8 1TW


Andrew Sawczenko
Institute of Child Health,
Royal Hospital for Sick Children,
Bristol, England.

Abstract: This article proposes the use of time-ordered non-decimated wavelet
or non-decimated wavelet packet coefficients to provide a
representation of a time series (explanatory). The resulting
representations are then used as variables in a statistical model
to provide predictions of another time series (response).
The statistical model provides valuable information about which
components in the explanatory time series drive the response time

To represent our time series we use a collection of basis functions
known as wavelet packets. Our methodology modifies the standard
wavelet packets representation by providing an over-determined
representation of shifted wavelet packets where the shifts are not
restricted to the usual wavelet grid. We introduce a fast algorithm
for carrying out the time-ordered non-decimated wavelet packet

The shifted wavelet packet bases can represent many classes of time
series sparsely. The sparsity provides an effective dimension
reduction which enables standard statistical modelling techniques to
select the components of an explanatory series that best explain
the variation in the response time series.

Our modelling methodology is illustrated using time series examples
from two different arenas: (a) a biomedical example shows how infant
sleep states can be successfully classified using the time-ordered
non-decimated wavelet packet transform of heart rate and linear
discriminant analysis; (b) a renewable energy example using a
generalized linear model relating wind speeds at 40m to a
time-ordered non-decimated wavelet packet transform of wind speeds
at 10m.

Keywords: time series modelling; wavelets; dimension reduction; biomedical
time series; wind time series; linear discriminant analysis;
generalized linear models.

To obtain the compressed PostScript:

See also

for other reports.
Guy Nason
School of Mathematics Phone: +44 (0117) 117 928 8633
University of Bristol FAX: +44 (0117) 117 928 7999
University Walk, Bristol, BS8 1TW, U.K.
All times are GMT + 1 Hour
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