Discriminant Analysis and Applications

435 Pages·1973·16.264 MB·English

by  T. Cacoullos

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION
Proceedings of a NA TO Advanced Study Institute
on Discriminant Analysis and Applications
Athens, Greece
June 8-20, 1972
DISCRIMINANT ANALYSIS
AND
APPLICATIONS
Edited by
T. CACOULLOS
UNIVERSITY OF ATHENS
ATHENS, GREECE
Academic Press New York and London 1973
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1973, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute on Discriminant Analysis
and Applications, Athens, 1972.
Discriminant analysis and applications.
Proceedings of the conference held June 8-20, 1972.
Bibliography: p.
1. Discriminant analysis-Congresses. I. Cacoullos,
Theophilos, ed. II. Title.
QA278.65.N37 1972 519.5'3 73-806
ISBN 0-12-154050-2
PRINTED IN THE UNITED STATES OF AMERICA
In memory of
PRASANTA CHANDRA MAHALANOBIS
(1893-1972)
CONTRIBUTORS
J.A. Anderson, Department of Biomathematics, University of Oxford,
Oxford, England
T.W. Anderson, Department of Statistics, Stanford University, Stanford,
California
D.F. Andrews, Department of Mathematics, University of Toronto, Toronto,
Ontario, Canada
T. Cacoullos, Statistical Unit, University of Athens, Athens, Greece
Somesh Das Gupta, School of Statistics, University of Minnesota,
Minneapolis, Minnesota
M.M. Desu, Department of Statistics, State University of New York, Buffalo,
New York
W.J. Dixon, Department of Biomathematics, University of California, Los
Angeles, California
Daniel Dugue, Institut de Statistique, Universite de Paris, Paris, France
S. Geisser, School of Statistics, University of Minnesota, Minneapolis,
Minnesota
R.I. Jennrich, Department of Biomathematics, University of California, Los
Angeles, California
M.G. Kendall, Scientific Control Systems Ltd., London, England
Peter A. Lachenbruch, Department of Biostatistics, University of North
Carolina, Chapel Hill, North Carolina
Kameo Matusita, Institute of Statistical Mathematics, Tokyo, Japan
J. Michaelis, Institut für Medizinische Statistik und Dokumentation
der Universität Mainz, Mainz, West Germany
Elliott Nebenzahl, Department of Statistics, California State University,
Hay ward, California
ix
CONTRIBUTORS
R.A. Reyment, Paleontologiska Institut, Uppsala Universitet, Uppsala,
Sweden
Willem Schaafsma, Mathematisch Instituut, Risj Universiteit, Groningen, The
Netherlands
Rosedith Sitgreaves, Department of Statistics, California State University,
Hay ward, California
Milton Sobel, School of Statistics, University of Minnesota,
Minneapolis, Minnesota
George P. H. Styan, Department of Mathematics, McGill University,
Montreal, Quebec, Canada
J. Tiago de Oliveira, Faculdade de Ciências Matemâtica, Lisbon, Portugal
X
PREFACE
This book comprises the proceedings of the NATO Advanced Study
Institute on Discriminant Analysis and Applications held in Kifissia, Athens,
Greece, June 8-20, 1972. Participants in the Institute came from almost all
NATO countries; there were 19 lecturers and 60 additional participants. It
was unfortunate that Professor C. R. Rao was unable to participate due to
the serious illness of P. C. Mahalanobis during those days; Mahalanobis died
on June 28, 1972. In memoriam and appreciation of his pioneering contribu-
tions in the field of discriminant analysis, this volume is dedicated to
Prasanta Chandra Mahalanobis (1893-1972), founder of the Indian Statistical
Institute and of Sankhyä.
The term discriminant analysis in the title of these Proceedings is an
alternative to the term classification which is used as a synonym for discrimi-
nation by many statisticians. This choice is to avoid ambiguity, since classifi-
cation is often also used in the sense of taxonomy or construction of classes.
For a discussion of these differences the reader is referred to the paper by
Professor M. G. Kendall in this volume.
Both theory and applications are discussed. A comprehensive survey
and a historical development of discriminant analysis methods are given in
the review paper by Professor Somesh Das Gupta. Here the reader will also
find a fairly extensive up-to-date bibliography arranged by subject matter
and then chronologically within subject matter. This bibliography contains
articles and books published up to 1972 and some unpublished papers as
well. At the end of this book, there is a bibliography, presumably exhaustive,
with authors in alphabetical order, prepared by Professor G. P. H. Styan
(McGill University) and the Editor. It includes papers published up to 1971
and books up to 1972.
Several papers deal with the exact, asymptotic, and empirical (Monte
Carlo) evaluation of the probabilities of misclassification, especially under
the assumption of normal alternative populations with the same or different
dispersion matrices. Logistic and quasi-linear discrimination are also covered.
Distance functions are considered especially as regards discrimination in the
Gaussian case. Applications, particularly in medicine and biology, are also
discussed in some of these papers. Moreover, computer graphical analysis and
XI
PREFACE
graphical techniques for multidimensional data are discussed and illustrated
by examples in two papers. It is regretted that final versions of the papers by
Professors Ingram Olkin (Stanford University) and M. S. Srivastava (Univer-
sity of Toronto) did not arrive in time to be included in these Proceedings.
I wish to thank the Scientific Directors of the Institute, Professors T. W.
Anderson and M. G. Kendall, for their encouragement and suggestions
throughout the organization of the Symposium. Thanks are also due to my
Assistants at the University of Athens, Ch. Charalambides, Urania Chrys-
aphinou, A. Dallas, and Amalia Veini, for their assistance with the local
arrangements and for carrying out the correspondence for the Institute.
The editing of this book was completed during my visit at McGill
University. Special thanks go to my colleague G. P. H. Styan for valuable
suggestions; to Leslie Ann Hathaway for helping with proofreading and the
compilation of the bibliography at the end of this volume, I express my
sincere appreciation. I am particularly grateful to Maria Lam for patiently
preparing the difficult typescript; she did a superb job.
Thanks are also due to the contributors for their cooperation and to
Academic Press whose interest in publishing these Proceedings was expressed
in the early stages of the Institute.
It goes without saying that the Institute would not have materialized
without the financial support of the NATO Science Committee. I also
acknowledge with gratitute the support of the Greek Ministry of Culture and
Science for sponsoring the social activities of the Institute, by no means
less contributory to its success.
T. Cacoullos
Xll
INTRODUCTORY ADDRESS*
T. Cacoullos
Director of the NATO Advanced Study Institute
on Discriminant Analysis and Applications
In a NATO report on nonmilitary cooperation issued in 1956, discuss-
ing science and technology, the three foreign ministers of Canada, Italy, and
Norway recommended the creation of an ad hoc working group to study
topics of special importance to NATO and specifically the way in which the
Organization might support or supplement the scientific activities of other
organizations. A Task Force was formed in June 1957 to consider these
recommendations. The psychological shock of the launching of the first
Soviet Sputnik in October 1957 resulted in the immediate presentation of
the Task Force's Report to the Heads of Governments of the Alliance at
their meeting in December 1957. The recommendations of the former were
accepted and both a Science Adviser and a Science Committee were ap-
pointed to follow up the proposals of the Task Force.
The Science Committee met first in March 1958 and ever since its
activities have grown to significant proportions. Its programs of Fellowships,
Research Grants, and Advanced Study Institutes have contributed a great
deal to the promotion of common understanding and cooperation between
the scientific communities of NATO members.
The Advanced Study Institutes Program has been sponsoring about 50
Advanced Study Institutes annually since 1959. The Study Institutes, or
Summer Schools as they are called, range in subject from mathematics and
physics through biological and medical topics to such areas as psychological
measurement theory and programming languages for computers. The prin-
cipal feature of such meetings is the bringing together of research workers,
often in secluded surroundings, for about two weeks to study a specific
topic. It is exactly within this framework that the present Institute on
Discriminant Analysis and Applications was organized. It is therefore with
pleasure that we acknowledge the support of the NATO Science Committee
which made this meeting possible.
*Read at the Opening Ceremony, June 8, 1972.
xiii
INTRODUCTORY ADDRESS
On behalf of the participants I should also express our appreciation to
the Greek Ministry of Culture and Science for its cosponsoring of the
Institute; in particular, most of its social activities which include tonight's
cocktails and buffet dinner, the Delphi excursion this coming Sunday, and
the Hydra-Aegina cruise next weekend. We are honored by the presence of
his Exellency, the Minister of Culture and Science, Mr. C. A. Panayiotakis,
who has accepted my invitation to deliver the opening address of the
Institute.
Before I introduce his Excellency, the Minister, allow me to spend a
few minutes introducing the Institute's topic. Discriminant analysis, other-
wise known as classification theory, is one of the most important areas of
multivariate statistical analysis.
A statistical discrimination or classification problem consists in assign-
ing or classifying an individual or group of individuals to one of several
known or unknown alternative populations on the basis of several measure-
ments on the individual and samples from the unknown populations. In the
simplest case of two normal populations, for example, a linear combination
of the measurements, called linear discriminator or discriminant function, is
constructed, and on the basis of its value the individual is assigned to one or
the other of the two populations. This decision process involves two kinds of
error, namely, that the individual is assigned to population No. 2 while it
really belongs to population No. 1 and vice versa. The researcher wishes to
minimize in some sense the chances of a wrong decision. R. A. Fisher was
the first to suggest the use of a linear discriminant function in 1936. Its early
applications led to some important anthropométrie discoveries such as sex
differences in mandibles, the extraction from a dated series of the particular
compound of cranial measurements showing secular trends, and solutions of
taxonomic problems in general.
At about the same time that Fisher introduced the discriminant func-
tion, Mahalanobis (in India) proposed his generalized distance which was
applied with success to craniometric and other anthropological studies.
Let us mention a few of the early applications of discriminant analysis.
Professor C. R. Rao, Director of Research and Training at the Indian
Statistical Institute, considered the assignment of an individual to one of the
three Indian castes, Brahmin, Artisan, or Korwa, on the basis of four
measurements, namely height, sitting height, nasal depth, and nasal height.
More details about this and other applications can be found in the well-
known multivariate analysis textbooks of two distinguished participants of
the present Institute, Professors T. W. Anderson of Stanford University and
M. G. Kendall, formerly of the University of London, now Chairman of
Scientific Control Systems, whom we are happy to have with us in the role
of Scientific Directors and Lecturers for the Institute.
xiv