Computer Intensive Methods in Statistics

184 Pages·1993·7.41 MB·English

by  A. F. M. Smith (auth.), Prof. Dr. Wolfgang Härdle, Prof. Léopold Simar (eds.)

Statistics and Computing
Léopold Simar
Computer
Intensive Methods
in Statistics
Statistics and Computing
Editorial Board
w. Hardie
D. W. Scott
S. Sheather
Wolfgang Hardie
Leopold Simar (Eds.)
Computer Intensive
Methods in Statistics
With 34 Figures
Springer-Verlag Berlin Heidelberg GmbH
Prof. Dr. Wolfgang Hardie
lnstitut ftir Statistik und Okonometrie
Wirtschaftswissenschaftliche Fakultat
Humboldt-Universitat zu Berlin
Spandauer StraBe 1
0-1020 Berlin, FRG
Prof. Leopold Simar
Institute de Statistique
Universite Catholique de Louvain
B-1348 Louvain-la-Neuve, Belgium
ISBN 978-3-7908-0677-9 ISBN 978-3-642-52468-4 (eBook)
DOI 10.1007/978-3-642-52468-4
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© Springer-Verlag Berlin Heidelberg 1993
Originally published by Physica-Verlag Heidelberg in 1993
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Preface
The computer has created new fields in statistic. Numerical and statistical problems
that were untackable five to ten years ago can now be computed even on portable
personal computers. A computer intensive task is for example the numerical
calculation of posterior distributions in Bayesian analysis. The Bootstrap and image
analysis are two other fields spawned by the almost unlimited computing power. It is
not only the computing power through that has revolutionized statistics, the graphical
interactiveness on modern statistical environments has given us the possibility for
deeper insight into our data.
On November 21 ,22 1991 a conference on computer Intensive Methods in Statistics
has been organized at the Universite Catholique de Louvain, Louvain-La-Neuve,
Belgium. The organizers were Jan Beirlant (Katholieke Universiteit Leuven), Wolfgang
Hardie (Humboldt-Universitat zu Berlin) and Leopold Simar (Universite Catholique de
Louvain and Facultes Universitaires Saint-Louis). The meeting was the Xllth in the
series of the Rencontre Franco-Beige des Statisticians. Following this tradition both
theoretical statistical results and practical contributions of this active field of statistical
research were presented. The four topics that have been treated in more detail were:
Bayesian Computing; Interfacing Statistics and Computers; Image Analysis;
Resampling Methods. Selected and refereed papers have been edited and collected
for this book.
1) Bayesian Computing.
In Bayesian analysis, the statistician is very often confronted with the problem of
computing numerically posterior or predictive distributions. In the field of image
analysis D.A. STEPHENS and A.F.M. SMITH show how the Bayesian idea of Gibbs
sampling helps to solve computational problems in Edge-Detection. F. KLEIBERGEN
and H.K. van DIJK analyse how the generation of matric variate t drawings may be
used to approximate the computation of posteriors in econometric problems. L.
BAUWENS and A. RASQUERO consider high posterior density regions to solve the
problem of testing residual autocorrelation.
2) Interfacing Statistics and Computers.
D. WOUTERS and L. VERMEIRE use the NAG library for the numerical construction
of optimal critical regions in the gamma family and show how the Mathematica
language can be used for the symbolic computation of a geodesic test in the Weibull
family. A. ANTONIADES, J. BERRUYER and R. CARMONA report on teachware
experiments centered around the use of a dedicated program in the learning of
mathematical statistics and the practice of data analysis. In this program, graphical
tools, random number generation, simulation techniques and bootstrap are integrated
with the idea of guide undergraduates to modern statistical techniques.
VI
3) Image Analysis.
In M. ROUSSIGNOL, V. JOUANE, M. MENVIELLE and P. TARITS a method for
finding rock conductivities from electromagnetic measurements is presented. The
method uses a stochastic algorithm to find a Bayesian estimator of the conductivities.
Markov random field models are used in V. GRANVILLE and J.P. RASSON in image
remote sensing. The technique is extended, in a unifying approach, in order to
analyse problems of image segmentation, noise filtering and discriminant analysis. In
A.P. KOROSTELEV and A.B. TSYBAKOV minimax linewise algorithms for image
reconstruction are studied.
4) Resampling Methods.
Ph. VIEU presents a survey on theoretical results for bandwidth selection for kernel
regression. The behavior of kernel estimates of a regression function depends heavily
on the smoothing parameter, so for practical purpose it is important to ensure good
properties of the estimates. M.A. GRUET, S. HUET and E. JOLIVET consider the
bootstrap in regression problems. Four specific problems are considered: confidence
intervals for parameters, calibration analysis in nonlinear situations, estimation of the
covariance matrix of the estimates and confidence intervals for the regression
function. Finally, Ph. BESSE and A. de FALGUEROLLES investigate the use of
resampling methods in data analysis problems. More specifically, they study the
problem of the choice of the dimension in principal components analysis. Different
bootstrap and jackknife estimates are presented.
Several institutions have made the Xllth Franco-Begian Meeting of Statisticians
possible; their financial help is gratefully acknowledged: CORE, the Center for
Operation Research and Econometrics (Universite Catholique de Louvain), SMASH,
Seminaire de Mathematiques Appliquees aux Sciences Humaines (Facultes
Universitaires Saint-Louis), KUL, Katholieke Universiteit Leuven, the Ministere de
!'Education de Ia Communaute Francaise de Belgique and the FRNS, Fonds National
de Ia Recherche Scientifique, all helped to make this meeting possible. We would like
to thank all them. The organisation would not have been possible without the staff of
CORE, in particular Sheila Verkaeren and Mariette Huysentruit. We cordially like to
thank them.
W. Hard 1e L. Sima r
Contents
Bayesian Computing
Bayesian Edge-Detection in Images via Changepoint Methods
D. A. Stephens and A. F. M. Smith .............................................................. .
Efficient Computer Generation of Matric-Variate t Drawings with an
Application to Bayesian Estimation of Simple Market Models
F. Kleibergen and H. K. van Dijk ................................................................... 30
Approximate HPD Regions for Testing Residual Autocorrelation
Using Augmented Regressions
L. Bauwens and A. Rasquero ........................................................................ 47
Interfacing Statistics and Computers
Intensive Numerical and Symbolic Computing in Parametric Test Theory
D. Wauters and L. Vermeire .......................................................................... 62
Learning Data Analysis and Mathematical Statistics with a Macintosh
A. Antoniadis, J. Berruyer and R. Carmona ................................................. 73
Image Analysis
Bayesian Electromagnetic Imaging
M. Roussignol, V. Jouanne, M. Menvielle and P. Tarits 85
Markov Random Field Models in Image Remote Sensing
V. Granville and J.-P. Rasson ...................................................................... 98
Minimax Linewise Algorithm for Image Reconstruction
A. P. Korostelev and A. B. Tsybakov .......................................................... 113
Resampling Methods
Bandwidth Selection for Kernel Regression: a Survey
P. Vieu .........................................................................................•................ 134
Practical Use of Bootstrap in Regression
M.-A. Gruet, S. Huet and E. Jolive .............................................................. 150
Application of Resampling Methods to the Choice of Dimension
in Principal Component Analysis
Ph. Besse and A. de Falguerolles ............................................................... 167
Bayesian Edge-Detection in Images via Change point Methods
D. A. Stephens and A. F. M. Smith
Department of Mathematics, Imperial College London.
180 Queen's Gate, London SW7 2AZ, United Kingdom.
Abstract
The problem of edge-detection in images will be formulated as a statistical changepoint problem using a
Bayesian approach. It will be shown that tbe Gibbs sampler provides an effective procedure for the required
Bayesian calculations. The use of the method for "quick and dirty" image segmentation will be illustrated.
Keywords : Image analysis: edge-detection: changepoint identification; Bayesian statistics: Gibbs sampler;
edge reconstruction: image reconstruction.
1. Introduction
1.1 Background
Practical applications of image analysis abound in agronomy (remote sensing), astronomy (study of
galaxies), industrial processing (automated manufacturing and quality control), medicine (internal body imag-
ing), and the military (intelligence, reconnaissance, defence/offence systems), relating variously to imaging
technologies such as photography. tomography, radiography etc. In this paper, we shall discuss a "quick and
diny" statistical approach to the problem of edge-detection in such images.
We shall denote the observed data image by Y, and true scene by e. Generally, Y and 9 will be vec-
tors of possibly different lengths, reflecting discretization involved in the image formation process. It is pos-
sible. in specific problems. to regard the true scene as having a continuous parameter space, but here 'we res-
trict ourselves to the discretized version. For definiteness. we shall focus in what follows on the situation
where Y and 9 have equal length, M, and relate to the same rectangular grid. Our interpretation of the image
formation process is a conventional statistical one, namely that
Data = Slruclure * Noise , (!)
2
where the terms "Daw " and "Structure " in (I) correspond, respectively, to "image" and "true scene" as
deli ned above, "Noise " corresponds to the inherent but undesirable stochastic element, and • is an operator
de lining precisely how the Structure and /liaise interact. In common terminology. a noise-process is regarded
as acting w corrupt the underlying signal. Hence, we denote the noise-process by E, and thus we may for-
mally think of (I) in the image processing context as
Y =/(9,E) (2)
where f is merely some function involving the operation • and the assumed Y. e pixel correspondence
described above.
We shall adopt a Bayesian approach to inference about aspects of the unobservable e of interest (in our
case edges). First. we provide a general background and motivation.
1.2 Problems in image processing
In image segmentation. given the observed (image) data Y, the objective is to allocate or classify each
of the elements in the unobservable true scene vector e. to one of a set of "textures". The segmentation
problem may be approached in several ways, such as estimation, probabilistic classification etc., and is often
the prime objective in many fields of application, where the unobservable true scene is thought to comprise
of broadly homogeneous texture regions separated by thin boundaries or edges.
It is often of interest to be able w identify the positions of these edges, either as an end in itself. or as a
preliminary stage in segmentation. We note below that. despite the considerable literature concerned with
edge-detection methodology and applications. little has been done to formulate the problem in a formal pro-
babilistic framework. This Iauer task is the primary concern of this paper.
In many problems. for example in medical imaging: or military reconnaissance, the inference problem
involves discovery or the location of a high intensity object of small dimension relative to the background
image field. This problem of object detection is a somewhat specialized version of the segmentation problem
described above. Although edge-detection is also relevant here, we have the additional knowledge that edges
of the object are "close together".
A pattern is a particular configuration of texture regions or features in the true scene, having a charac-
teristic quality that allows discrimination between it and other configurations. Again. edge detection has at
least a preliminary role to play in many pattern recognition applications.