Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

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5. Graphical Representation of Data Using Principal Components 79by Chambers et al. (1983)). A more recent thorough review of graphicsfor multivariate data is given by Carr (1998). A major advance has beenthe development of dynamic multivariate graphics, which Carr (1998) describesas part of ‘the visual revolution in computer science.’ The techniquesdiscussed in the present chapter are almost exclusively static, althoughsome could be adapted to be viewed dynamically. Only those graphicsthat have links with, or can be used in conjunction with, PCA areincluded.Section 5.2 discusses principal coordinate analysis, which constructs lowdimensionalplots of a set of data from information about similarities ordissimilarities between pairs of observations. It turns out that the plotsgiven by this analysis are equivalent to plots with respect to PCs in certainspecial cases.The biplot, described in Section 5.3, is also closely related to PCA. Thereare a number of variants of the biplot idea, but all give a simultaneousdisplay of n observations and p variables on the same two-dimensionaldiagram. In one of the variants, the plot of observations is identical toa plot with respect to the first two PCs, but the biplot simultaneouslygives graphical information about the relationships between variables. Therelative positions of variables and observations, which are plotted on thesame diagram, can also be interpreted.Correspondence analysis, which is discussed in Section 5.4, again givestwo-dimensional plots, but only for data of a special form. Whereas PCAand the biplot operate on a matrix of n observations on p variables,and principal coordinate analysis and other types of scaling or ordinationtechniques use data in the form of a similarity or dissimilarity matrix, correspondenceanalysis is used on contingency tables, that is, data classifiedaccording to two categorical variables. The link with PCA is less straightforwardthan for principal coordinate analysis or the biplot, but the ideasof PCA and correspondence analysis have some definite connections. Thereare many other ordination and scaling methods that give graphical displaysof multivariate data, and which have increasingly tenuous links to PCA.Some of these techniques are noted in Sections 5.2 and 5.4, and in Section5.5 some comparisons are made, briefly, between PCA and the othertechniques introduced in this chapter.Another family of techniques, projection pursuit, is designed to find lowdimensionalrepresentations of a multivariate data set that best displaycertain types of structure such as clusters or outliers. Discussion of projectionpursuit will be deferred until Chapters 9 and 10, which include sectionson cluster analysis and outliers, respectively.The final section of this chapter describes some methods which have beenused for representing multivariate data in two dimensions when more thantwo or three PCs are needed to give an adequate representation of the data.The first q PCs can still be helpful in reducing the dimensionality in suchcases, even when q is much larger than 2 or 3.
80 5. Graphical Representation of Data Using Principal ComponentsFinally, we note that as well as the graphical representations describedin the present chapter, we have already seen, in Section 4.3, one other typeof plot that uses PCs. This type of plot is rather specialized, but is usedextensively in atmospheric science. Related plots are discussed further inChapter 12.5.1 Plotting Data with Respect to the First Two(or Three) Principal ComponentsThe idea here is simple: if a data set {x 1 , x 2 ,...,x n } has p variables, thenthe observations can be plotted as points in p-dimensional space. If we wishto plot the data in a ‘best-fitting’ q-dimensional subspace (q
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Principal ComponentAnalysis,Second
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viPreface to the Second Editionerty
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viiiPreface to the Second EditionA
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xPreface to the First Editionand in
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xiiPreface to the First EditionIn m
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xviAcknowledgmentsthese institution
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xviiiContents3.4.1 Example ........
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xxContents10 Outlier Detection, Inf
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xxivList of Figures5.2 Artistic qua
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xxviiiList of Tables6.1 First six e
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2 1. IntroductionFigure 1.1. Plot o
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4 1. IntroductionFigure 1.3. Studen
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1.2. A Brief History of Principal C
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1.2. A Brief History of Principal C
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2.1. Optimal Algebraic Properties o
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2.2. Geometric Properties of Popula
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2.3. Principal Components Using a C
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2.4. Principal Components with Equa
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Page 124 and 125: 5.3. Biplots 93ButandSubstituting i
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Page 132 and 133: 5.3. Biplots 101Figure 5.5. Biplot
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Page 142 and 143: 6Choosing a Subset of PrincipalComp
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.2. Choosing m, the Number of Comp
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6.2. Choosing m, the Number of Comp
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6.3. Selecting a Subset of Variable
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6.4. Examples Illustrating Variable
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7.1. Models for Factor Analysis 151
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7.2. Estimation of the Factor Model
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7.3. Comparisons Between Factor and
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7.4. An Example of Factor Analysis
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7.4. An Example of Factor Analysis
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7.5. Concluding Remarks 165To illus
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8Principal Components in Regression
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8.1. Principal Component Regression
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8.1. Principal Component Regression
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8.2. Selecting Components in Princi
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8.2. Selecting Components in Princi
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8.3. Connections Between PC Regress
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8.4. Variations on Principal Compon
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8.5. Variable Selection in Regressi
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8.5. Variable Selection in Regressi
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8.6. Functional and Structural Rela
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8.7. Examples of Principal Componen
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Table 8.3. Principal component regr
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9Principal Components Used withOthe
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9.1. Discriminant Analysis 201on th
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9.1. Discriminant Analysis 203Figur
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9.1. Discriminant Analysis 205Corbi
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9.1. Discriminant Analysis 207that
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9.1. Discriminant Analysis 209betwe
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9.2. Cluster Analysis 211dimensiona
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9.2. Cluster Analysis 213Before loo
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9.2. Cluster Analysis 215Figure 9.3
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9.2. Cluster Analysis 217demographi
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9.2. Cluster Analysis 219county clu
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9.2. Cluster Analysis 221choosing a
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9.3. Canonical Correlation Analysis
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10.1. Detection of Outliers Using P
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10.2. Influential Observations in a
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10.3. Sensitivity and Stability 259
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10.3. Sensitivity and Stability 261
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10.4. Robust Estimation of Principa
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11Rotation and Interpretation ofPri
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11.1. Rotation of Principal Compone
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oot of the corresponding eigenvalue
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11.2. Alternatives to Rotation 279w
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11.2. Alternatives to Rotation 281F
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11.2. Alternatives to Rotation 283F
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11.2. Alternatives to Rotation 285T
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11.2. Alternatives to Rotation 287T
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11.2. Alternatives to Rotation 289A
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11.2. Alternatives to Rotation 291
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11.3. Simplified Approximations to
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11.3. Simplified Approximations to
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11.4. Physical Interpretation of Pr
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12Principal Component Analysis forT
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12.1. Introduction 301series is alm
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12.2. PCA and Atmospheric Time Seri
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and a typical row of the matrix is1
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12.3. Functional PCA 317A key refer
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12.3. Functional PCA 319The sample
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12.3. Functional PCA 321speed (mete
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12.3. Functional PCA 323of the data
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12.3. Functional PCA 325subject to
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12.3. Functional PCA 327series than
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12.4. PCA and Non-Independent Data
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13.1. Principal Component Analysis
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13.2. Analysis of Size and Shape 34
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13.2. Analysis of Size and Shape 34
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13.5. Common Principal Components 3
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13.7. PCA in Statistical Process Co
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13.8. Some Other Types of Data 369A
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13.8. Some Other Types of Data 371d
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14Generalizations and Adaptations o
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14.1. Non-Linear Extensions of Prin
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14.1. Additive Principal Components
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14.2. Weights, Metrics, Transformat
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14.3. PCs in the Presence of Second
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14.4. PCA for Non-Normal Distributi
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14.5. Three-Mode, Multiway and Mult
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14.5. Three-Mode, Multiway and Mult
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14.6. Miscellanea 401• Linear App
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14.6. Miscellanea 40314.6.3 Regress
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14.7. Concluding Remarks 405space o
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Appendix AComputation of Principal
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A.1. Numerical Calculation of Princ
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ReferencesAguilera, A.M., Gutiérre
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References 417Apley, D.W. and Shi,
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References 419Benasseni, J. (1986b)
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References 421Boik, R.J. (1986). Te
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References 423Castro, P.E., Lawton,
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References 425Cook, R.D. (1986). As
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References 427Dempster, A.P., Laird
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References 429Feeney, G.J. and Hest
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References 431in Descriptive Multiv
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References 433Gunst, R.F. and Mason
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References 435Hocking, R.R., Speed,
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References 437Jeffers, J.N.R. (1978
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References 439Kazi-Aoual, F., Sabat
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References 441Krzanowski, W.J. (200
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References 443Mann, M.E. and Park,
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References 445Monahan, A.H., Tangan
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References 447Pack, P., Jolliffe, I
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References 449Richman M.B. (1993).
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References 451Soofi, E.S. (1988). P
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References 453Tenenbaum, J.B., de S
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References 455Vong, R., Geladi, P.,
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References 457regularities in multi
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Index 459116, 127-130, 133, 270, 27
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Index 461computationin (PC) regress
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Index 463discriminant principal com
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Index 465of correlations between va
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Index 467see also hypothesis testin
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Index 469PC algorithms with noise 4
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Index 471in functional and structur
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Index 473variance ellipsoids,S-esti
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Index 475spatial correlation/covari
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Index 477for covariance matrices 26
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Author Index 479Belsley, D.A. 169Be
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Author Index 481Fowlkes, E.B. 377Fr
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Author Index 483Krzanowski, W.J. 46
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Author Index 485Rencher, A.C. 64, 1
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Author Index 487Yaguchi, H. 371Yana
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