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Discriminant Analysis and Clustering (1988)
Commission on Physical Sciences, Mathematics, and Applications (CPSMA)

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Discriminant Analysis and Clustering Panel on Discriminant Analysis, Classification, and Clustering Committee on Applied and Theoretical Statistics Board on Mathematical Sciences Commission on Physical Sciences, Mathematics, and Resources National Research Council NATIONAL ACADEMY PRESS Washington, D.C. 1988

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NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineenng, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance. This report has been renewed by a group other than the authors accord- ing to procures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Ins1 itute of Medicine. The National Academy of Sciences is a private, nonprofit, self- perpetuating somet3r of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Frank Press is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of out- standing engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibil- ity for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Robert M. White is president of the National Academy of Engineenng. The Institute of Medicine was established in 1970 by the National Academy of Smences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an advisor to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Samuel O. Thier is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy~s purposes of furthering knowledge and advising the federal government. Functioning in accordance with the general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Frank Press and Dr. Robert M. White are chairman and vice chairman, respectively, of the National Research Council. Support for this project was provided by the Office of Naval Research under contract N00014-82-0652 and by core funds from the Board on Mathematical Sciences, National Research Council, prodded by several agen- cies of the Interagency Commission for Extramural Mathematics Program (ICEMAP). Available frolic Board on Mathematical Sciences 2101 Constitution Avenue Washington, D.C. 20418 Printed in the United States of Inca

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PANEL ON DISCRIM~IANT ANALYSIS, CLASSIFICATION, AND CLUSTERING Ramanathan Gnanadesikan, Bell Communications Research, Cha~rnum Roger K Blashfield, University of Florida Leo Breiman, University of California, Berkeley Olive J. Dunn, University of California, Los Ar~geles Jerome H. Fnedman, Stanford University King-Sun Fu, Purdue University (deceased) John A. Hartigan, Yale University Jon R. Kettenring, Bell Commurucations Research (appointed August, 19435) Peter A. Lachenb~uch, University of California, Los Angeles Richard A. Olshen, University of California, San Diego F. James Rohlf, SUNY, Stony Brook · · ~ 111

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COMMITTEE ON APPLIED AND THEORETICAL STATISTICS Ronald Pyke, University of Washington, Chairman Ralph A. Bradley, University of Georgia Morris De Groot, Carnegie Mellon University William Eddly, Carnegie Mellon University John A. Hartigan, Yale University Gary C. McDonald, General Motors Research Laboratories John Rice, University of California, San Diego Joan R. Rosenblatt, National Bureau of Standards Jerome Sacks, University of Illinois J. Sethuraman, Florida State University Michael J. Steele, Princeton University George C. Tiao, University of Chicago 1V

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BOARD ON MATHEMATICAL SCIENCES Phillip A. Griffiths, Duke University, Chairman Peter J. Bickel, University of California, Berkeley Herman Chernoff, Harvard University Ronald G. Douglas, SUNY Stony Brook E. F. Infante, University of Minnesota William Jaco, Oklahoma State University Joseph J. Kohn, Princeton University Cathleen S. Morawetz, New York University Alan Newell, University of Arizona Guido Weiss, Washington University Shmue] Winograd, IBM Corporation E:x-Officio Member Ronald Pyke (Chairman CATS) NRC Staff Lawrence H. Cox, StaH Director Seymour Selig, Staff Officer Robert Smythe, Staff Officer William G. Rosen, Staff Officer Rose M. Kopera, Staff Assistant Cathy Richardson, Secretary v

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CO~IISSION ON PHYSICAL SCIENCES, MATHEMATICS, AND RESOURCES Norman Hackerman, Robert A. Welch Foundation George F. Camer, Harvard University Dean E. Eastman, IBM Corporation Marye Anne Fox, University of Texas Gerhart Friediander, Brookhaven National Laboratory Lawrence W. Funlihouser, Chevron Corporation (retired) Phillip A. Griffiths, Dulre University J. Ross Macdonald, University of North Carolina at Chapel Hill Charles J. Mankin, Oklahoma Geological Survey Perry L. McCarthy, Stanford University Jack E. Oliver, Cornell University Jeremiah P. Ostriker, Princeton University William D. Phillips, Mallinckr~t, Inc. Denis J. Prager, MacArthur Foundation David M. Raup, University of Chicago Richard J. Reed, University of Washington Robert E. Sievers, University of Colorado Larry L. Smarr, National Center for Supercomputing Applications Edward C. Stone, Jr., California Institute of Technology Karl K TureMan, Yale University (George W. Wetherill, Carnegie Institution of Washington Irving Wladawsky-Berger, IBM Corporation Raphael G. Kasper, Executive Director Lawrence E. McCray, Associate Executive Director V1

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PREFACE This report is the output of a panel created under the auspices of the Committee on Applied and Theoretical Statistics (CATS). The Panel on Discriminant Analysis and Clustenng was organized in May 1981 with 11 members, including the author of this preface as Chairman. It was agreed that the general objective of the report would be both to provide a summary of the state-of-th~art in discrim- inant analysis and clustering and also to identify key research and unsolved problems that need to be addressed in these two areas. The goal was to have four chapters in the report: Theoretical Aspects, Methods, Algorithms & Software, and Applications. [A fifth possible chapter to address pedagogical aspects was initially considered but dismissed.] The intended audience for this report was assumed to be rea- sonably familiar with statistical concepts and terminology although not necessarily to possess expertise in the methodologies considered here. Also, the report could be useful to people in many fields of applications of statistical methods of classification and clustering who may be generally familiar with the issues and pro- cedures but not be professional statisticians. The tradeoff between the experience gathered as a user and the technical knowledge of a statistical expert would hopefully enable a wider audience to benefit from the report. As a summary of state-of-the- art methods (with references for pursuing details) the report was intended to be useful to users, while as an attempt to indicate additional directions for methodological research it would be of interest to professional statisticians. The first meeting of the pane] was held in Washington, D.C., on November 19 and 20, 1982, and was attended by all but one member of the panel. At this meeting, the objectives of the pane] were discussed and sharpened, and by the end of it, an outline for a possible report had emerged. Specific assignments to individual committee members for providing first drafts were agreed upon and, in addition, certain individuals undertook the responsibility for coordinating the efforts so as to result in particular chapters of · ~ V11

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the intended report. The general commitment was to have all of the individual contributions in to the coordinators by the end of the summer of 1983. For a variety of reasons, the initial goals had to be pared down. Specifically, the chapter on Applications had to abandoned. While the intended scope had to be cut back, the present report would not have been possible without the particular help of some of the pane] members. Drs. R. K Blashfield, O. J. Dunn, J. A. Har- tigan, P. A. Lachenbruch and R. A. Olshen all provided draft for the pieces they undertook. The chapter on Theoretical Aspects is the work of Drs. R. A. Olshen and J. A. Hartigan. The chapter on Methods was pulled together by Drs. R. Gnanadesikan and J. R. Kettennng based on useful inputs supplied by various members of the panel mentioned above The chapter on Algorithms and Software is a blending of material from Drs P. A Lachenbruch and R K Blashfield While some editorial efforts were expended at putting in references to material across chapters written by dif- ferent people, inevitably, there remains some duplication of cover- age and inconsistency of notation, which we hope are not too dis- tracting A draft of the full report was circulated to all members of the panel and was also made available to all current and a few past members of CATS Helpful comments received from Drs R A Olshen and F J. Rohlf, members of the panel, and from Drs R A Bradley and R Poke (respectively former and present chairmen of CATS) led to the first version of the report Additional comments received from reviewers were useful in developing this final ver sion Aside from the above-mentioned panel and CATS members, it is a pleasure to thank Nancy K Davidson and Hester A Glynn for their major help in the word processing effort for this report Thanks are also due to Bell Communications Research for their facilities and support in this process R Gnanadesikan November 1987 · · ~ V111