Exploring Multivariate Data with the Forward Search

ISBN-13: 9781441923530 / Angielski / Miękka / 2010 / 624 str.

Anthony C. Atkinson; Marco Riani; Andrea Cerioli

Opis

Why We Wrote This Book This book is about using graphs to explore and model continuous multi variate data. Such data are often modelled using the multivariate normal distribution and, indeed, there is a literatme of weighty statistical tomes presenting the mathematical theory of this activity. Our book is very dif ferent. Although we use the methods described in these books, we focus on ways of exploring whether the data do indeed have a normal distribution. We emphasize outlier detection, transformations to normality and the de tection of clusters and unsuspected influential subsets. We then quantify the effect of these departures from normality on procedures such as dis crimination and duster analysis. The normal distribution is central to our book because, subject to our exploration of departures, it provides useful models for many sets of data. However, the standard estimates of the parameters, especially the covari ance matrix of the observations, are highly sensitive to the presence of outliers. This is both a blessing and a curse. It is a blessing because, if we estimate the parameters with the outliers excluded, their effect is appre ciable and apparent if we then include them for estimation. It is however a curse because it can be hard to detect which observations are outliers. We use the forward search for this purpose."

Recenzje

From the reviews:

"The book requires knowledge of multivariate statistical methods, because it provides only basic background information on the methods considered (although with excellent references for futher reading at the end of each chapter). Each chapter also includes exercises with solutions...This book could serve as an excellent text for an advanced course on modern multivariate statistics, as it is intended." Technometrics, November 2004

"This book is full of interest for anyone undertaking multivariate analyses, clearly emphasizing that uncritical use of standard methods can be misleading." Short Book Reviews of the International Statistical Institute, December 2004

"This book is an interesting complement to various textbooks on multivariate statistics." Biometrics, December 2005

"This book discusses multivariate data from a different perspective. ... it is an excellent book for researchers with interests in multivariate data and cluster analysis. It may also be a good reference for students of advanced statistics and practitioners working with large volumes of data ... ." (Kassim S. Mwitondi, Journal of Applied Statistics, Vol. 32 (4), 2005)

"This is a companion to an earlier book ... both of which feature many informative graphs. Here, the forward search has been applied in detail to classical multivariate approaches used with Gaussian data. ... One valuable feature of the book is the way that the illustrations concentrate on a relatively small number ... . This makes it easy to concentrate on the application ... . The implications of this book also strengthen the importance of data visualization, as well as providing a valuable approach to visualization." (Paul Hewson, Journal of the Royal Statistical Society Series A, Vol. 168 (2), 2005)

"This book is a companion to Atkinson ... . The objective is to identify outliers, appreciate their influence ... which would result in an overall improvement. ... Graphical tools are widely used, resulting in three hundred and ninety figures. Each chapter is followed by extensive exercises and their solutions, and the book could be used as an advanced textbook for multivariate analysis courses. Web-sites provide the relevant software ... . This book is full of interest for anyone undertaking multivariate analyses ... ." (B.J.T. Morgan, Short Book Reviews International Statistical Institute, Vol. 24 (3), 2004)

"This book discusses forward search (FS), a method using graphs to explore and model continuous multivariate data ... . Its viewpoint is toward applications, and it demonstrates the merits of FS using a variety of examples, with a thorough discussion of statistical issues and interpretation of results. ... This book could serve as an excellent text for an advanced course on modern multivariate statistics, as it is intended." (Tena Ipsilantis Katsaounis, Technometrics, Vol. 46 (4), November, 2004)

"The theoretical exercises with detailed solutions at the end of each chapter are extremely useful. I would recommend this book to practitioners who analyze moderately sized multivariate data. Of course, anyone associated with the application of statistics should find the book interesting to read." (Tathgata Banerjee, Journal of the American Statistical Association, March 2006)

"This book deals with procedures to analyze sets of multivariate observations of continuous variables. ... The procedures are described in general, and then applied to a collection of data sets. ... The data are shown through tables and figures, the latter numbering 390, which means that illustrations are abundant. ... This is an attractive and useful book. It has a wealth of suggestions. ... The results of the analysis are usually presented in graphical form, and as said, the book has 390 graphs." (Raúl Mentz, Zentralblatt MATH, Vol. 1049, 2004)

"This is a valuable book on explorative multivariate data analysis which makes particular use of plots. ... The book is intended to serve as a text for a postgraduate course on modern multivariate statistics. The theoretical material is complemented by exercises with detailed solutions. ... This is a ... carefully written book on explorative multivariate data analysis, with a lot of details and examples especially on effects caused by outliers and departures from normality. I recommend this book for any applied statistician." (Wolfgang Näther, Metrika, Vol. 64, 2006)

Spis treści

Contents Preface Notation 1 Examples of Multivariate Data 1.1 In.uence, Outliers and Distances 1.2 A Sketch of the Forward Search 1.3 Multivariate Normality and our Examples 1.4 Swiss Heads 1.5 National Track Records forWomen 1.6 Municipalities in Emilia-Romagna 1.7 Swiss Bank Notes 1.8 Plan of the Book 2 Multivariate Data and the Forward Search 2.1 The Univariate Normal Distribution 2.1.1 Estimation 2.1.2 Distribution of Estimators 2.2 Estimation and the Multivariate Normal Distribution 2.2.1 The Multivariate Normal Distribution 2.2.2 The Wishart Distribution 2.2.3 Estimation of Ó 2.3 Hypothesis Testing 2.3.1 Hypotheses About the Mean 2.3.2 Hypotheses About the Variance 2.4 The Mahalanobis Distance 2.5 Some Deletion Results 2.5.1 The Deletion Mahalanobis Distance 2.5.2 The (Bartlett)-Sherman-Morrison-Woodbury Formula 2.5.3 Deletion Relationships Among Distances 2.6 Distribution of the Squared Mahalanobis Distance 2.7 Determinants of Dispersion Matrices and the Squared Mahalanobis Distance 2.8 Regression 2.9 Added Variables in Regression 2.10 TheMean Shift OutlierModel 2.11 Seemingly Unrelated Regression 2.12 The Forward Search 2.13 Starting the Search 2.13.1 The Babyfood Data 2.13.2 Robust Bivariate Boxplots from Peeling 2.13.3 Bivariate Boxplots from Ellipses 2.13.4 The Initial Subset 2.14 Monitoring the Search 2.15 The Forward Search for Regression Data 2.15.1 Univariate Regression 2.15.2 Multivariate Regression 2.16 Further Reading 2.17 Exercises 2.18 Solutions 3 Data from One Multivariate Distribution 3.1 Swiss Heads 3.2 National Track Records for Women 3.3 Municipalities in Emilia-Romagna 3.4 Swiss Bank Notes 3.5 What Have We Seen? 3.6 Exercises 3.7 Solutions 4 Multivariate Transformations to Normality 4.1 Background 4.2 An Introductory Example: the Babyfood Data 4.3 Power Transformations to Approximate Normality 4.3.1 Transformation of the Response in Regression 4.3.2 Multivariate Transformations to Normality 4.4 Score Tests for Transformations 4.5 Graphics for Transformations 4.6 Finding a Multivariate Transformation with the Forward Search 4.7 Babyfood Data 4.8 Swiss Heads 4.9 Horse Mussels 4.10 Municipalities in Emilia-Romagna 4.10.1 Demographic Variables 4.10.2 Wealth Variables 4.10.3 Work Variables 4.10.4 A Combined Analysis 4.11 National Track Records for Women 4.12 Dyestuff Data 4.13 Babyfood Data and Variable Selection 4.14 Suggestions for Further Reading 4.15 Exercises 4.16 Solutions 5 Principal Components Analysis 5.1 Background 5.2 Principal Components and Eigenvectors 5.2.1 Linear Transformations and Principal Components . 5.2.2 Lack of Scale Invariance and Standardized Variables 5.2.3 The Number of Components 5.3 Monitoring the Forward Search 5.3.1 Principal Components and Variances 5.3.2 Principal Component Scores 5.3.3 Correlations Between Variables and Principal Components 5.3.4 Elements of the Eigenvectors 5.4 The Biplot and the Singular Value Decomposition 5.5 Swiss Heads 5.6 Milk Data 5.7 Quality of Life 5.8 Swiss Bank Notes 5.8.1 Forgeries and Genuine Notes 5.8.2 Forgeries Alone 5.9 Municipalities in Emilia-Romagna 5.10 Further reading 5.11 Exercises 5.12 Solutions 6 Discriminant Analysis 6.1 Background 6.2 An Outline of Discriminant Analysis 6.2.1 Bayesian Discrimination 6.2.2 Quadratic Discriminant Analysis 6.2.3 Linear Discriminant Analysis 6.2.4 Estimation of Means and Variances 6.2.5 Canonical Variates 6.2.6 Assessment of Discriminant Rules 6.3 The Forward Search 6.3.1 Step 1: Choice of the Initial Subset 6.3.2 Step 2: Adding

Blurb

The forward search provides a method of revealing the structure of data through a mixture of model fitting and informative plots. The continuous multivariate data that are the subject of this book are often analyzed as if they come from one or more normal distributions. Such analyses, including the need for transformation, may be distorted by the presence of unidentified subsets and outliers, both individual and clustered. These important features are disguised by the standard procedures of multivariate analysis. The book introduces methods that reveal the effect of each observation on fitted models and inferences.

The powerful methods of data analysis will be of importance to scientists and statisticians. Although the emphasis is on the analysis of data, theoretical developments make the book suitable for a graduate statistical course on multivariate analysis. Topics covered include principal components analysis, discriminant analysis, cluster analysis and the analysis of spatial data. S-Plus programs for the forward search are available on a web site.

This book is a companion to Atkinson and Riani's Robust Diagnostic Regression Analysis of which the reviewer for The Journal of the Royal Statistical Society wrote "I read this book, compulsive reading such as it was, in three sittings."

Anthony Atkinson is Emeritus Professor of Statistics at the London School of Economics. He is also the author of Plots, Transformations, and Regression and coauthor of Optimum Experimental Designs. Professor Atkinson has served as Editor of The Journal of the Royal Statistical Society, Series B.