First published in France in 2003 by Hermes Science/Lavoisier entitled: Analyse des données © LAVOISIER, 2003

First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication Data

Analyse des données. English.

Data analysis / edited by Gérard Govaert.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-098-1

1. Mathematical statistics. I. Govaert, Gérard. II. Title.

QA276.D325413 2009

519.5--dc22

2009016228

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-098-1

Preface

Statistical analysis has traditionally been separated into two phases: an exploratory phase, drawing on a set of descriptive and graphical techniques, and a decisional phase, based on probabilistic models. Some of the tools employed as part of the exploratory phase belong to descriptive statistics, whose elementary exploratory methods consider only a very limited number of variables. Other tools belong to data analysis, the subject matter of this book. This topic comprises more elaborate exploratory methods to handle multidimensional data, and is often seen as stepping beyond a purely exploratory context.

The first part of this book is concerned with methods for obtaining the pertinent dimensions from a collection of data. The variables so obtained provide a synthetic description, often leading to a graphical representation of the data. A considerable number of methods have been developed, adapted to different data types and different analytical goals. Chapters 1 and 2 discuss two reference methods, namely Principal Components Analysis (PCA) and Correspondence Analysis (CA), which we illustrate with examples from statistical process control and sensory analysis. Chapter 3 looks at a family of methods known as Projection Pursuit (less well known, but with a promising future), that can be seen as an extension of PCA and CA, which makes it possible to specify the structures that are being sought. Multidimensional positioning methods, discussed in Chapter 4, seek to represent proximity matrix data in low-dimensional Euclidean space. Chapter 5 is devoted to functional data analysis where a function such as a temperature or rainfall graph, rather than a simple numerical vector, is used to characterize individuals.

The second part is concerned with methods of clustering, which seek to organize data into homogenous classes. These methods provide an alternative means, often complementary to those discussed in the first part, of synthesizing and analyzing data. In view of the clear link between clustering and discriminant analysis – in pattern recognition the former is termed unsupervised and the latter supervised learning – Chapter 6 gives a general introduction to discriminant analysis. Chapter 7 then provides an overall picture of clustering. The statistical interpretation of clustering in terms of mixtures of probability distributions is discussed in Chapter 8 and Chapter 9 looks at how this approach can be applied to spatial data.

I would like to express my heartfelt thanks to all the authors who were involved in this publication. Without their expertise, their professionalism, their invaluable contributions and the wealth of their experience, it would not have been possible.

Gérard GOVAERT

Chapter 1

Principal Component Analysis: Application to Statistical Process Control ¹

1.1. Introduction

Principal component analysis (PCA) is an exploratory statistical method for graphical description of the information present in large datasets. In most applications, PCA consists of studying p variables measured on n individuals. When n and p are large, the aim is to synthesize the huge quantity of information into an easy and understandable form.

Unidimensional or bidimensional studies can be performed on variables using graphical tools (histograms, box plots) or numerical summaries (mean, variance, correlation). However, these simple preliminary studies in a multidimensional context are insufficient since they do not take into account the eventual relationships between variables, which is often the most important point.

Principal component analysis is often considered as the basic method of factor analysis, which aims to find linear combinations of the p variables called components used to visualize the observations in a simple way. Because it transforms a large number of correlated variables into a few uncorrelated principal components, PCA is a dimension reduction method. However, PCA can also be used as a multivariate outlier detection method, especially by studying the last principal components. This property is useful in multidimensional quality control.

1.2. Data table and related subspaces

1.2.1.Data and their characteristics

Data are generally represented in a rectangular table with n rows for the individuals and p columns corresponding to the variables. Choosing individuals and variables to analyze is a crucial phase which has an important influence on PCA results. This choice has to take into account the aim of the study; in particular, the variables have to describe the phenomenon being analyzed.

Usually PCA deals with numerical variables. However, ordinal variables such as ranks can also be processed by PCA. Later in this chapter, we present the concept of supplementary variables which afterwards integrates nominal variables.

1.2.1.1. Data table

Let X be the (n,p)matrix of observations:

where is the value of individual i for variable j (denoted xj ) which is identified with a vector of n components (, . . . , ). In a similar way, an individual i is identified to a vector xi of p components with xi = (, . . . ,).

Table 1.1 is an example of such a data matrix. Computations have been carried out using SPAD 5 software, version 5 ¹, kindly provided by J.-P. Gauchi.

The data file contains 57 brands of mineral water described by 11 variables defined in Table 1.2. The data come from the bottle labels. Numerical variables are homogenous; they are all active variables (see section 1.4.3). A variable of a different kind such as price would be considered as a supplementary variable. On the other hand, qualitative variables such as country, type and whether still or sparkling (PG) are necessarily supplementary variables.

Table 1.1. Data table

1.2.1.2. Summaries

1.2.1.2.1. Centroid

Let be the vector of arithmetic means of each of the p variables, defining the centroid:

Table 1.2. Variable description

Name | Complete water name as labeled on the bottle
Country | Identified by the official car registration letters; sometimes it is necessary to add a letter, for example Crete: GRC (Greece Crete)
Type | M for mineral water, S for spring water
PG | P for still water, G for sparkling water
CA | Calcium ions (mg/litre)
MG | Magnesium ions (mg/litre)
NA | Sodium ions (mg/litre)
K | Potassium ions (mg/litre)
SUL | Sulfate ions (mg/litre)
NO3 | Nitrate ions (mg/litre)
HCO3 | Carbonate ions (mg/litre)
CL | Chloride ions (mg/litre)

where .

If the data are collected following a random sampling, the n individuals all have the same importance in the computations of the sample characteristics. The same weight pi = 1/n is therefore allocated to each observation.

However, it can be useful for some applications to use weight pi varying from one individual to another as grouped data or a reweighted sample. These weights, which are positive numbers summing to 1, can be viewed as frequencies and are stored in a diagonal matrix of size n:

We then have the matrix expression = XDp1n where 1n represents the vector of n with all its components equal to 1. The centered data matrix associated with X is then Y with and Y = X 1n = (In 1n1n Dp)X, where In is the unity matrix of dimension n.

1.2.1.2.2. Covariance matrix and correlation matrix

Let and , the variance of variable j and the covariance between variables k and , respectively. They are stored in the covariance matrix S = XDpX = YDpY.

We define the linear correlation coefficient between variables k and by:

If Z is the standardized data table associated with X, , we have Z = YD1/s where D1/s the diagonal matrix of the inverse of standard deviations:

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