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

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3.3. Covariance and Correlation Matrices: An Example 41Table 3.2. Principal components based on the correlation matrix for eight bloodchemistry variables.Component number 1 2 3 4CoefficientsRBLOOD 0.2 −0.4 0.4 0.6PLATE 0.4 −0.2 0.2 0.0WBLOOD 0.4 0.0 0.2 −0.2NEUT 0.4 0.4 −0.2 0.2LYMPH −0.4 −0.4 0.0 −0.2BILIR −0.4 0.4 −0.2 0.6SODIUM −0.2 0.6 0.4 −0.2POTASS −0.2 0.2 0.8 0.0Percentage of total variation explained 34.9 19.1 15.6 9.7Table 3.3. Principal components based on the covariance matrix for eight bloodchemistry variables.Component number 1 2 3 4CoefficientsRBLOOD 0.0 0.0 0.0 0.0PLATE 1.0 0.0 0.0 0.0WBLOOD 0.0 −0.2 0.0 1.0NEUT 0.0 0.0 0.0 0.0LYMPH 0.0 0.0 0.0 0.0BILIR 0.0 1.0 −0.2 0.2SODIUM 0.0 0.2 1.0 0.0POTASS 0.0 0.0 0.0 0.0Percentage of total variation explained 98.6 0.9 0.4 0.2
42 3. Properties of Sample Principal ComponentsBecause variable PLATE has a variance 100 times larger than any othervariable, the first PC accounts for over 98 percent of the total variation.Thus the first six components for the covariance matrix tell us almost nothingapart from the order of sizes of variances of the original variables. Bycontrast, the first few PCs for the correlation matrix show that certainnon-trivial linear functions of the (standardized) original variables accountfor substantial, though not enormous, proportions of the total variation inthe standardized variables. In particular, a weighted contrast between thefirst four and the last four variables is the linear function with the largestvariance.This example illustrates the dangers in using a covariance matrix to findPCs when the variables have widely differing variances; the first few PCswill usually contain little information apart from the relative sizes of variances,information which is available without a PCA. There are, however,circumstances in which it has been argued that using the covariance matrixhas some advantages; see, for example, Naik and Khattree (1996), althoughthese authors transform their data (track record times for Olympic eventsare transformed to speeds) in order to avoid highly disparate variances.Apart from the fact already mentioned in Section 2.3, namely, that itis more difficult to base statistical inference regarding PCs on correlationmatrices, one other disadvantage of correlation matrix PCs is that theygive coefficients for standardized variables and are therefore less easy to interpretdirectly. To interpret the PCs in terms of the original variables eachcoefficient must be divided by the standard deviation of the correspondingvariable. An example which illustrates this is given in the next section.It must not be forgotten, however, that correlation matrix PCs, when reexpressedin terms of the original variables, are still linear functions of xthat maximize variance with respect to the standardized variables and notwith respect to the original variables.An alternative to finding PCs for either covariance or correlation matricesis to calculate the eigenvectors of ˜X ′ ˜X rather than X ′ X, that is, measurevariables about zero, rather than about their sample means, when computing‘covariances’ and ‘correlations.’ This idea was noted by Reyment andJöreskog (1993, Section 5.4) and will be discussed further in Section 14.2.3.‘Principal component analysis’ based on measures of association of thisform, but for observations rather than variables, has been found useful forcertain types of geological data (Reyment and Jöreskog, 1993). Anothervariant, in a way the opposite of that just mentioned, has been used byBuckland and Anderson (1985), among others. Their idea, which is alsodiscussed further in Section 14.2.3, and which again is appropriate for aparticular type of data, is to ‘correct for the mean’ in both the rows andcolumns of ˜X. Further possibilities, such as the use of weights or metrics,are described in Section 14.2.
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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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Page 25 and 26: xxivList of Figures5.2 Artistic qua
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5.3. Biplots 91columns, L is an (r
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5.3. Biplots 93ButandSubstituting i
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5.3. Biplots 95The vector gi ∗ co
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5.3. Biplots 97Figure 5.3. Biplot u
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5.3. Biplots 99Table 5.2. First two
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5.3. Biplots 101Figure 5.5. Biplot
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5.4. Correspondence Analysis 103of
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5.4. Correspondence Analysis 105Fig
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5.6. Displaying Intrinsically High-
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5.6. Displaying Intrinsically High-
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6Choosing a Subset of PrincipalComp
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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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Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

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