N. Balakrishnan, H. K. Ng
Precedence-Type Tests and Applications
ISBN: 0-471-45720-5
Hardcover
324 pages
April 2006
Precedence-Type Tests and Applications provides a comprehensive
overview of theoretical and applied approaches to a variety of
problems in which precedence-type test procedures can be used.
The authors clearly demonstrate the effectiveness of these tests
in life-testing situations designed for making quick and reliable
decisions in the early stages of an experiment. Most of the
text's examples use life-time data; however, theoretical
properties are also discussed in the context of precedence
testing. Monte Carlo studies are used to illustrate important
results.
Following the authors' careful step-by-step instructions and
guidance, readers master the wide range of statistical techniques
involved in the development and implementation of precedence-type
tests. The book covers the foundations of precedence testing
research from the early 1960s up to the most recent theory and
applications, including the authors' current contributions to the
field. The book features the following parts:
Part A deals with the original precedence test and some
properties of precedence and related test procedures
Part B explores alternatives to precedence testing, including
maximal precedence, weighted forms of precedence and maximal
precedence, and Wilcoxon-type rank-sum precedence tests and their
properties
Part C compares the extension of precedence, maximal precedence,
and Wilcoxon-type rank-sum precedence tests to situations in
which the sample arising from the life-testing experiment is
progressively Type-II censored
Part D examines precedence-type tests in multi-sample situations
and selection problems
Tables are presented throughout the book to facilitate the
application of the tests to practical problems. Helpful examples
illustrate all of the precedence-type procedures, and an
extensive bibliography enables readers to explore specialized
topics in greater depth.
This book is a recommended reference for researchers and
practitioners in reliability and life-time data analysis, applied
probabilists, and engineers. It also serves as a supplemental
text for courses in nonparametric statistics and reliability.
Table of Contents
List of Tables.
List of Figures.
Preface.
1. Introduction.
2. Preliminaries.
3. Precedence Test.
4. Maximal Precedence Test.
5. Weighted Precedence and Weighted Maximal Precedence Tests.
6. Wilcoxon-type Rank-sum Precedence Tests.
7. Extension to Progressive Censoring.
8. Generalization to k-Sample Situation.
9. Selecting the Best Population Using a Test for Equality Based
on Precedence Statistic.
10. Selecting the Best Population Using a Test for Equality Based
on Minimal Wilcoxon Rank-sum Precedence Statistic.
Appendix.
Bibliography.
Author Index.
Subject Index.
Carl J. Huberty, Stephen Olejnik
Applied MANOVA and Discriminant Analysis, 2nd Edition
ISBN: 0-471-46815-0
Hardcover
560 pages
May 2006
This Second Edition of the classic book, Applied Discriminant
Analysis, reflects and references current usage with its new
title, Applied MANOVA and Discriminant Analysis. Thoroughly
updated and revised, this book continues to be essential for any
researcher or student needing to learn to speak, read, and write
about discriminant analysis as well as develop a philosophy of
empirical research and data analysis. Its thorough introduction
to the application of discriminant analysis is unparalleled.
Offering the most up-to-date computer applications, references,
terms, and real-life research examples, the Second Edition also
includes new discussions of MANOVA, descriptive discriminant
analysis, and predictive discriminant analysis. Newer SASR macros
are included, and graphical software with data sets and programs
are provided on the book's related Web site.
The book features:
Detailed discussions of multivariate analysis of variance and
covariance
An increased number of chapter exercises along with selected
answers
Analyses of data obtained via a repeated measures design
A new chapter on analyses related to predictive discriminant
analysis
Basic SPSSR and SASR computer syntax and output integrated
throughout the book
Applied MANOVA and Discriminant Analysis enables the reader to
become aware of various types of research questions using MANOVA
and discriminant analysis; to learn the meaning of this field's
concepts and terms; and to be able to design a study that uses
discriminant analysis through topics such as one-factor MANOVA/DDA,
assessing and describing MANOVA effects, and deleting and
ordering variables.
Table of Contents
PART I: INTRODUCTION.
1. Discriminant Analysis in Research.
2. Preliminaries.
PART II: ONE-FACTOR MANOVA/DDA.
3. Group Separation.
4. Assessing MANOVA Effects.
5. Describing MANOVA Effects.
6. Deleting and Ordering Variables.
7. Reporting DDA Results.
PART III: FACTORIAL MANOVA, MANCOVA, AND REPEATED MEASURES.
8. Factorial MANOVA.
9. Analysis of Covariance.
10. Repeated-Measures Analysis.
11. Mixed-Model Analysis.
PART IV: GROUP-MEMBERSHIP PREDICTION.
12. Classification Basics.
13. Multivariate Normal Rules.
14. Classification Results.
15. Hit-Rate Estimation.
16. Effectiveness of Classification Rules.
17. Deleting and Ordering Predictors.
18. Two-Group Classification.
19. Nonnormal Rules.
20. Reporting PDA Results.
21. PDA-Related Analyses.
PART V: ISSUES AND PROBLEMS.
22. Issues in PDA and DDA.
23. Problems in PDA and DDA.
Appendix A: Data Set Descriptions.
Appendix B: Some DA-Related Originators.
Appendix C: List of Computer Syntax.
Appendix D: Contents of Wiley Web Site.
Edited by: Dominique Bakry, University of Toulouse, France, Lucian Beznea, Romanian Academy, Bucharest,
Romania, Gheorghe Bucur, University of Bucharest, Romania, and Michael
Rockner, Bielefeld University, Germany
Current Trends in Potential Theory
2006; 174 pp; hardcover
ISBN: 973-85432-6-6
Expected publication date is April 30, 2006.
This is the proceedings volume of two mathematical meetings on
Potential Theory organized in Bucharest, Romania, in September
2002 and September 2003. It includes six survey articles and
seven selected research papers, covering the main topics of the
conferences: geometric aspects in potential theory, Dirichlet
structures, stochastic analysis, potential theory, and Markov
processes.
A publication of the Theta Foundation. Distributed worldwide,
except in Romania, by the AMS.
Readership
Graduate students and research mathematicians interested in
potential theory.
Table of Contents
Survey articles
N. Arcozzi, E. C. Tarabusi, F. Di Biasse, and M. Picardello -- A
potential theoretic approach to twisting
D. Feyel -- A survey of the Monge transport problem
B. Fuglede -- Harmonic maps from Riemann polyhedra to spaces of
nonpositive curvature
F. Hirsch -- Measurable metrics, intrinsic metrics and Lipschitz
functions
A. Lejay and T. Lyons -- On the importance of the Levy area for
studying the limits of functions of stochastic processes.
Application to homogenization
V. Metz -- Superadditive Perron-Frobenius theory
Research papers
I. Bachar -- Estimates for the Green function and existence of
positive solutions of nonlinear equations with Navier boundary
conditions
D. Bakry and Z. Qian -- Volume comparison theorems without Jacobi
fields
N. B. Rhouma and M. Bezzarga -- On a singular value problem and
the boundary Harnack principle for fractional Laplacian
M. Biroli and P. G. Vernole -- Brelot property for the sheaf of
harmonics relative to a Dirichlet form
K. Janssen -- Factorization of excessive kernels
E. Popescu -- Pseudo differential operators in the context of
Feller semigroups and Dirichlet forms
C. Udrea -- Resolvent and nonlinear potential theory
Written by Dan Rockmore
Stalking the Riemann Hypothesis
The Quest to Find the Hidden Law of Prime Numbers
(0-375-72772-8) | 304 pages
Prologue?It All Begins with Zero
Itfs one of those slate-gray summer days that more properly
belong to mid-August than late May, one of those days in New York
City when it is barely clear where the city ends and the sky
begins. The hard-edged lines and Euclidean-inspired shapes that
are building, sidewalk, and pavement all seem to fuse into one
huge melted mass that slowly dissolves into the humid,
breezeless, torpid air. On mornings like this, even this
irrepressible metropolis seems to have slowed a notch, a muffled
cacophony more bass than treble, as the city that never sleeps
stumbles and shuffles to work.
But here in Greenwich Village, at the corner of Mercer and West
Fourth streets, where we find New York Universityfs Warren
Weaver Hall, the hazy torpor is interrupted by a localized high-energy
eddy. Here, deep in the heart of the artistic rain forest that is
gthe Village,h just across the street from the rock fnf
rolling nightclub the Bottom line, a stonefs throw from the
lofts and galleries that gave birth to
Jackson Pollock, Andy Warhol, and the Velvet Underground, is the
home of the Courant Institute of Mathematical Sciences, where at
this moment there is an excitement worthy of any gallery opening
in SoHo, or any new wave, next wave, or crest-of-the-wave musical
performance.
The lobby and adjacent plaza are teeming with mathematicians, a
polyglot and international group, abuzz with excitement. Listen
closely, and amid the multilingual, every-accent mathematical
jibber-jabber youfll hear a lot of talk about nothing, or more
properly a lot of talk about zero.
Zero is not an uncommon topic of conversation in New York, but
more often than not itfs the gplaceholder zerosh that are
on the tip of the New Yorkerfs tongue. These are the zeros that
stand in for the orders of magnitude by which we measure the
intellectual, cultural, and financial abundance that is New York:
one zero to mark the tens of ethnic neighborhoods, two for the
hundreds of entertainment options, three for the thousands of
restaurants, six for the millions of people, and, of course, the
zeros upon zeros that mark the billions or even trillions of
dollars that churn through the city every day. These are not the
zeros of void, but the zeros of plenty.
But, today, just one week past Memorial Day 2002, itfs a zero
of a different flavor which has attracted this eclectic group to
downtown New York City. Here some of the worldfs greatest
mathematicians are meeting to discuss and possibly, just
possibly, witness the resolution of the most important unsolved
problem in mathematics, a problem that holds the key to
understanding the basic mathematical elements that are the prime
numbers. The zeros that tip the tongues of these mathematical
adventurers are zeta zeros,* and the air is electric with the
feeling that perhaps this will be the day when we lay to rest the
mystery of these zeros, which constitutes the Riemann hypothesis.
For over a century mathematicians have been trying to prove the
Riemann hypothesis: that is, to settle once and for all a gently
asserted conjecture of Bernhard Riemann (1826?1866), who was a
professor of mathematics at the University of Gottingen in
Germany. Riemann is perhaps best known as the mathematician
responsible for inventing the geometrical ideas upon which
Einstein built his theory of general relativity. But in 1859, for
one brief moment, Riemann turned his attention to a study of the
long-familiar prime numbers. These are numbers like two, three,
five, and seven, each divisible only by one and itself,
fundamental numerical elements characterized by their
irreducibility. Riemann took up the age-old problem of trying to
find a rule which would explain the way in which prime numbers
are distributed among the whole numbers, indivisible stars
scattered without end throughout a boundless numerical universe.
In a terse eight-page gmemoirh delivered upon the occasion of
his induction into the prestigious Berlin Academy, Riemann would
revolutionize the way in which future mathematicians would
henceforth study the primes. He did this by connecting a law of
the primes to the understanding of a seemingly completely
unrelated complex collection of numbers?numbers characterized by
their common behavior under a sequence of mathematical
transformations that add up to the Riemann zeta function. Like a
Rube Goldbergesque piece of mathematical machinery, Riemannfs
zeta function takes in a number as raw material and subjects it
to a complicated sequence of mathematical operations that results
in the production of a new number. The relation of input to
output for Riemannfs zeta function is one of the most studied
processes in all of mathematics. This attention is largely due to
Riemannfs surprising and mysterious discovery that the numbers
which seem to hold the key to understanding the primes are
precisely the somethings which Riemannfs zeta function turns
into nothing, those inputs into Riemannfs number cruncher that
cause the production of the number zero. These are the zeta
zeros, or more precisely the zeros of Riemannfs zeta function,
and they are the zeros that have attracted a stellar cast of
mathematicians to New York.
In his memoir, Riemann had included, almost as an aside, that it
seemed ghighly likelyh that the zeta zeros have a
particularly beautiful and simple geometric description. This
offhand remark, born of genius and supported by experiment, is
the Riemann hypothesis. It exchanges the confused jumble of the
primes for the clarity of geometry, by proposing that a graphical
description of the accumulation of the primes has a beautiful and
surprisingly simple and precise shape. The resolution of Riemannfs
hypothesis holds a final key to our understanding of the primes.
Wefll never know if Riemann had in mind a proof for this
assertion. Soon after his brief moment of public glory, the
ravages of tuberculosis began to take their toll on his health,
leaving him too weak to work with the intensity necessary to tie
up the loose ends of his Berlin memoir. Just eight years later,
at the all too young age of thirty-nine, Riemann was dead,
cheated of the opportunity to settle his conjecture.
Since then, this puzzling piece of Riemannfs legacy has stumped
the greatest mathematical minds, but in recent years frustration
has begun to give way to excitement, for the pursuit of the
Riemann hypothesis has begun to reveal astounding connections
among nuclear physics, chaos, and number theory. This unforeseen
confluence of mathematics and physics, as well as certainty and
uncertainty, is creating a frenzy of activity that suggests that
after almost 150 years, the hunt might be over.
This is the source of the buzz filling the Courant Institutefs
entryway. It is a buzz amplified by the fact that whoever settles
the question of the zeta zeros can expect to acquire several new
zeros of his or her own, in the form of a reward offered by the
Clay Institute of Mathematics, which has included the Riemann
hypothesis as one of seven gMillennium Prize Problems,h each
worth $1 million. So the jungle of abstractions that is
mathematics is now full of hungry hunters. They are out stalking
big game?the resolution of the Riemann hypothesis?and it seems to
be in their sights.
The Riemann hypothesis stands in relation to modern mathematics
as New York City stands to the modern world, a crossroads and
nexus for many leading figures and concepts, rich in unexpected
and serendipitous conjunctions. The story of the quest to settle
the Riemann hypothesis is one of scientific exploration and
discovery. It is peopled with starry-eyed dreamers and moody
aesthetes, gregarious cheerleaders and solitary hermits, cool
calculators and wild-eyed visionaries. It crisscrosses the
Western world and includes Nobel laureates and Fields medalists.
It has similarities with other great scientific journeys but also
has its own singular hallmarks, peculiar to the fascinating world
of mathematics, a subject that has intrigued mankind since the
beginning of thought.
2The God-Given Natural Numbers
The great German mathematician Leopold Kronecker (1823?1891) said
that gGod created the natural numbers.h And it is true that
the natural numbers?one, two, three, four; on and on they
go?appear to have been present from the beginning, coming into
existence with the birth of the universe, part and parcel of the
original material from which was knit the ever-expanding
continuum of space-time.
The natural numbers are implicit in the journey of life, which is
a nesting of cycles imposed upon cycles, wheels within wheels.
One is the instant. Two is the breathing in and out of our lungs,
or the beat of our hearts. The moon waxes and wanes; the tides
ebb and flow. Day follows night, which in turn is followed once
again by day. The cycle of sunrise, noon, and sunset give us
three. Four describes the circle of seasons.
These natural numbers help us to make sense of the world by
finding order, in this case an order of temporal patterns, that
lets us know what to expect and when. We notice the rising and
setting of the sun, and that cycle of two is given a more
detailed structure as we follow the sun through the sky over the
course of a day. We turn the temporal telescope around and also
see day as part of the larger cycle of the phases of the moon,
whose steady progress is situated within the cycle of the seasons
that makes up the year. Patterns within patterns within patterns;
numbers within numbers within numbers?all working together to
create a celestial symphony of time.
Armed with this new understanding we make tentative, tiny forays
out into the Jamesian gbooming, buzzing worldh and shape a
life within and around it. Embedded in the recognition of the
cycle is the ability to predict, and thus to prepare, and then to
direct the world to our advantage. We coax and bend an
unflinching, steady march of time; and in a subtle jujitsu of
nature, technologies are born. We learn when to sow and when to
reap, when to hunt and when to huddle. We exploit that which we
cannot change. We discover the cycle and ride it as an eagle
rides an updraft.
In the absence of a natural cycle we may impose one, for in
routine we find a sense of control over the unwieldy mess that is
life. We relish the comfort of being a regular at a local diner
or a familiar face at the coffee cart on the street, and the
rhythm of the daily morning dog walk. We dream of options, if
only to choose our own routines, our own patterns, our own
numbers.
But as befits that which is part and parcel of space-time, number
is not only a synecdoche of temporal organization but also the
most basic and elementary means of quantifying a spatial
organization of the world. Nature gives us few, if any, truly
straight lines or perfect circles. But there is one moon; there
is one sun; the animals go two by two. We organize, we count, and
therefore we are.
In this way, number is presented to us in the world in both time
and space, instances upon instances, but this is only the
beginning. Kronecker said not only that the natural numbers are
God-given, but also that gall else is the work of man.h What
first appear as singular phenomena are eventually unified,
gathered into a collective that is then recognized as a pattern.
Soon, the pattern is itself familiar, and so it becomes less a
pattern and more a particular. The game is then repeated, and we
find a new superpattern to explain what had once seemed disparate
patterns. So on and so on we go, building the discipline that
will come to be known as mathematics.
Beginning gthe workh
Suppose that I walk past a restaurant and catch a glimpse of a
perfectly set square table, place settings at each edge, each
side of the table providing a resting place for a full complement
of plates, glasses, and silverware. As I approach the entrance to
the restaurant, a group of women arrive and I imagine them seated
at that beautiful table, one at each side, continuing their
animated conversation. As I pass by again some time later, I see
the women leave the restaurant. They stand outside, say their
good-byes, and one by one are whisked away by taxicabs.
What is it that the group of women has in common with the
collection of place settings, the chairs at the table, the very
sides of the table, and the taxicabs that finally take them away?
It is the correspondence that they engender. It is a
correspondence that I make mentally and visually as I watch the
women, one that you make as you read this story, seeing each
woman paired with a chair, a plate, or a taxi. Any other grouping
of objects that could be paired with them
in this way has this same property, this same basic pattern. This
pattern is one of gnumerosity.h These groups all share the
property of gfourness.h
Each collection, whether it be the chairs, the place settings, or
the taxis, is such that its component objects can be put into a
one-to-one correspondence with the group of women. We say, as an
abbreviation for this property, that the group of women has a
size of four, and this is a property shared among each of the
sets of objects that may be put into a correspondence with the
women. If you had in your possession a collection of hats and I
inquired if you had one for each of the women, you might have me
list the women, or show you a picture of the group, but even
better, you could ask me ghow manyh women need hats. My
answer, gfour,h would be enough for you to check to see if
you had one hat for each.
The self-contained nature of the correspondence?there is no
object left unpaired?is perhaps what underlies the other
classification of the number four, or for that matter any natural
number, as an integer, and in particular a positive integer. The
totality of the integers consists of the natural numbers, their
negatives, and zero.
Thus four becomes an agreed-upon name for a pattern that we
recognize in the world. At Christmas, four are the calling birds;
at Passover, four are the matriarchs, symbols that are
simultaneously iconic and generic. We wind our way back through
numerical history. Four are the fingers proudly displayed by a
protonumerate toddler, a set of scratchings on a Sumerian
cuneiform, or the bunch of beads or pebbles lying at the feet of
a Greek philosopher. The last of these universal physical
numerical proxies, which the Greeks called calculi, gave birth to
our words calculus and calculate, and mark the mathematician as
both the forefather and the child of the first gbean countersh:
the Pythagoreans.
Ulf Grenander and Michael Miller
Pattern Theory
From representation to inference
(Paperback)
ISBN-10: 0-19-929706-1
Publication date: 31 August 2006
656 pages, 175 b/w line, 93 b/w halftone, 11 colour line, 56
colour halftone, 246mm x 189mm
Description
Grenander is the founding father of pattern theory and his work
has had major applications in computer science, medical imaging,
tomography etc.
Additional resources including extended proofs, selected
solutions and examples are available on a companion website.
Highly illustrated, includes colour plates
Pattern Theory: From Representation to Inference provides a
comprehensive and accessible overview of the modern challenges in
signal, data and pattern analysis in speech recognition,
computational linguistics, image analysis and computer vision.
Aimed at graduate students in biomedical engineering,
mathematics, computer science and electrical engineering with a
good background in mathematics and probability, the text includes
numerous exercises and an extensive bibliography. Additional
resources including extended proofs, selected solutions and
examples are available on a companion website.
The book commences with a short overview of pattern theory and
the basics of statistics and estimation theory. Chapters 3-6
discuss the role of representation of patterns via conditioning
structure and Chapters 7 and 8 examine the second central
component of pattern theory: groups of geometric transformation
applied to the representation of geometric objects. Chapter 9
moves into probabilistic structures in the continuum, studying
random processes and random fields indexed over subsets of Rn,
and Chapters 10, 11 continue with transformations and patterns
indexed over the continuum. Chapters 12-14 extend from the pure
representations of shapes to the Bayes estimation of shapes and
their parametric representation. Chapters 15 and 16 study the
estimation of infinite dimensional shape in the newly emergent
field of Computational Anatomy, and finally Chapters 17 and 18
look at inference, exploring random sampling approaches for
estimation of model order and parametric representing of shapes.
Readership: Graduate students in biomedical engineering,
mathematics, computer science and electrical engineering with a
good background in mathematics and probability.
Contents
1 Introduction
2 The Bayes paradigm, estimation and information measures
3 Probabilistic directed acyclic graphs and their entropies
4 Markov random fields on undirected graphs
5 Gaussian random fields on undirected graphs
6 The canonical representations of general pattern theory
7 Matrix group actions transforming patterns
8 Manifolds, active modes, and deformable templates
9 Second order and Gaussian fields
10 Metrics spaces for the matrix groups
11 Metrics spaces for the infinite dimensional diffeomorphisms
12 Metrics on photometric and geometric deformable templates
13 Estimation bounds for automated object recognition
14 Estimation on metric spaces with photometric variation
15 Information bounds for automated object recognition
16 Computational anatomy: shape, growth and atrophy comparison
via diffeomorphisms
17 Computational anatomy: hypothesis testing on disease
18 Markov processes and random sampling
19 Jump diffusion inference in complex scenes