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

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5.1. Plotting Two or Three Principal Components 83Figure 5.1. (b). Student anatomical measurements: plot of the first two PCs for28 students with minimum spanning tree superimposed.Artistic Qualities of PaintersThe second data set described in this section was analysed by Davenportand Studdert-Kennedy (1972). It consists of a set of subjective measurementsof the artistic qualities ‘composition,’ ‘drawing,’ ‘colour’ and‘expression’ for 54 painters. The measurements, on a scale from 0 to 20,were compiled in France in 1708 by Roger de Piles for painters ‘of establishedreputation.’ Davenport and Studdert-Kennedy (1972) give data for56 painters, but for two painters one measurement is missing, so thesepainters are omitted from the analysis.Table 5.1 gives the variances and coefficients for the first two PCs basedon the correlation matrix for the 54 painters with complete data. The components,and their contributions to the total variation, are very similar tothose found by Davenport and Studdert-Kennedy (1972) for the covariancematrix. This strong similarity between the PCs for correlation andcovariance matrices is relatively unusual (see Section 3.3) and is due to thenear-equality of the variances for the four variables. The first component
84 5. Graphical Representation of Data Using Principal ComponentsTable 5.1. First two PCs: artistic qualities of painters.Component 1 Component 2⎫Composition 0.50 -0.49⎪⎬Drawing 0.56 0.27CoefficientsColour −0.35 −0.77⎪⎭Expression 0.56 −0.31Eigenvalue 2.27 1.04Cumulative percentage of total variation 56.8 82.8is interpreted by the researchers as an index of de Piles’ overall assessmentof the painters, although the negative coefficient for colour needs some additionalexplanation. The form of this first PC could be predicted from thecorrelation matrix. If the sign of the variable ‘colour’ is changed, then allcorrelations in the matrix are positive, so that we would expect the first PCto have positive coefficients for all variables after this redefinition of ‘colour’(see Section 3.8). The second PC has its largest coefficient for colour, butthe other coefficients are also non-negligible.A plot of the 54 painters with respect to the first two components isgiven in Figure 5.2, and this two-dimensional display represents 82.8% ofthe total variation. The main feature of Figure 5.2 is that painters of thesame school are mostly fairly close to each other. For example, the set ofthe ten ‘Venetians’ {Bassano, Bellini, Veronese, Giorgione, Murillo, PalmaVecchio, Palma Giovane, Pordenone, Tintoretto, Titian} are indicated onthe figure, and are all in a relatively small area at the bottom left of theplot. Davenport and Studdert-Kennedy (1972) perform a cluster analysison the data, and display the clusters on a plot of the first two PCs. Theclusters dissect the data in a sensible looking manner, and none of themhas a convoluted shape on the PC plot. However, there is little evidence ofa strong cluster structure in Figure 5.2. Possible exceptions are a group ofthree isolated painters near the bottom of the plot, and four painters at theextreme left. The first group are all members of the ‘Seventeenth CenturySchool,’ namely Rembrandt, Rubens, and Van Dyck, and the second groupconsists of three ‘Venetians,’ Bassano, Bellini, Palma Vecchio, together withthe ‘Lombardian’ Caravaggio. This data set will be discussed again in Sections5.3 and 10.2, and the numbered observations on Figure 5.2 will bereferred to there. Further examples of the use of PCA in conjunction withcluster analysis are given in Section 9.2.Throughout this section there has been the suggestion that plots of thefirst two PCs may reveal interesting structure in the data. This contradictsthe implicit assumption that the n observations are identically distributedwith a common mean and covariance matrix. Most ‘structures’ in the dataindicate that different observations have different means, and that PCA
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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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3Mathematical and StatisticalProper
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where3.1. Optimal Algebraic Propert
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Page 142 and 143: 6Choosing a Subset of PrincipalComp
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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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