Modern Multivariate Statistical Techniques.pdf

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Izenman-2-lrg

Izenman A.J. Modern multivariate statistical techniques.. Regression, classification, and manifold learning (Springer, 2008)(ISBN 0387781889)(756s)_MVsa_

front-matter

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back-matter

Springer Texts in Statistics Series Editors: G. Casella S. Fienberg I. Olkin

Springer Texts in Statistics For other titles published in this series, go to www.springer.com/series/417

Alan Julian Izenman Modern Multivariate Statistical Techniques Regression, Classiﬁcation, and Manifold Learning 123

Alan J. Izenman Department of Statistics Temple University Speakman Hall Philadelphia, PA 19122 USA alan@temple.edu Editorial Board George Casella Department of Statistics University of Florida Gainesville, 8545 USA FL 32611- Stephen Fienberg Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213-3890 USA l Ingram O kin Department of Statistics Stanford University Stanford, CA 94305 USA ISSN: 1431-875X ISBN: 978-0-387-78188-4 DOI: 10.1007/978-0-387-78189-1 e-ISBN: 978-0-387-78189-1 Library of Congress Control Number: 2008928720. c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

This book is dedicated to the memory of my parents, Kitty and Larry, and to my family, Betty-Ann and Kayla

Preface Not so long ago, multivariate analysis consisted solely of linear methods illustrated on small to medium-sized data sets. Moreover, statistical com- puting meant primarily batch processing (often using boxes of punched cards) carried out on a mainframe computer at a remote computer facil- ity. During the 1970s, interactive computing was just beginning to raise its head, and exploratory data analysis was a new idea. In the decades since then, we have witnessed a number of remarkable developments in local computing power and data storage. Huge quantities of data are being col- lected, stored, and eﬃciently managed, and interactive statistical software packages enable sophisticated data analyses to be carried out eﬀortlessly. These advances enabled new disciplines called data mining and machine learning to be created and developed by researchers in computer science and statistics. As enormous data sets become the norm rather than the exception, sta- tistics as a scientiﬁc discipline is changing to keep up with this development. Instead of the traditional heavy reliance on hypothesis testing, attention is now being focused on information or knowledge discovery. Accordingly, some of the recent advances in multivariate analysis include techniques from computer science, artiﬁcial intelligence, and machine learning theory. Many of these new techniques are still in their infancy, waiting for statistical theory to catch up. The origins of some of these techniques are purely algorithmic, whereas the more traditional techniques were derived through modeling, optimiza-

viii Preface tion, or probabilistic reasoning. As such algorithmic techniques mature, it becomes necessary to build a solid statistical framework within which to embed them. In some instances, it may not be at all obvious why a partic- ular technique (such as a complex algorithm) works as well as it does: When new ideas are being developed, the most fruitful approach is often to let rigor rest for a while, and let intuition reign — at least in the beginning. New methods may require new concepts and new approaches, in extreme cases even a new language, and it may then be impossible to describe such ideas precisely in the old language. — Inge S. Helland, 2000 It is hoped that this book will be enjoyed by those who wish to under- stand the current state of multivariate statistical analysis in an age of high- speed computation and large data sets. This book mixes new algorithmic techniques for analyzing large multivariate data sets with some of the more classical multivariate techniques. Yet, even the classical methods are not given only standard treatments here; many of them are also derived as spe- cial cases of a common theoretical framework (multivariate reduced-rank regression) rather than separately through diﬀerent approaches. Another major feature of this book is the novel data sets that are used as examples to illustrate the techniques. I have included as much statistical theory as I believed is necessary to understand the development of ideas, plus details of certain computational algorithms; historical notes on the various topics have also been added wherever possible (usually in the Bibliographical Notes at the end of each chapter) to help the reader gain some perspective on the subject matter. References at the end of the book should be considered as extensive without being exhaustive. Some common abbreviations used in this book should be noted: “iid” means independently and identically distributed; “wrt” means with respect to; and “lhs” and “rhs” mean left- and right-hand side, respectively. Audience This book is directed toward advanced undergraduate students, gradu- ate students, and researchers in statistics, computer science, artiﬁcial in- telligence, psychology, neural and cognitive sciences, business, medicine, bioinformatics, and engineering. As prerequisites, readers are expected to have had previous knowledge of probability, statistical theory and methods, multivariable calculus, and linear/matrix algebra. Because vectors and ma- trices play such a major role in multivariate analysis, Chapter 3 gives the matrix notation used in the book and many important advanced concepts in matrix theory. Along with a background in classical statistical theory

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