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

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2.3. Principal Components Using a Correlation Matrix 23Figure 2.1. Contours of constant probability based on Σ 1 = ( 80444480).Figure 2.2. Contours of constant probability based on Σ 2 = ( 8000440)44080 .
24 2. Properties of Population Principal Componentsthe original variables rearranged in decreasing order of the size of theirvariances. Also, the first few PCs account for little of the off-diagonal elementsof Σ in this case (see Property A3) above. In most circumstances,such a transformation to PCs is of little value, and it will not occur if thecorrelation, rather than covariance, matrix is used.The example has shown that it is unwise to use PCs on a covariancematrix when x consists of measurements of different types, unless there is astrong conviction that the units of measurements chosen for each element ofx are the only ones that make sense. Even when this condition holds, usingthe covariance matrix will not provide very informative PCs if the variableshave widely differing variances. Furthermore, with covariance matrices andnon-commensurable variables the PC scores are difficult to interpret—whatdoes it mean to add a temperature to a weight? For correlation matrices, thestandardized variates are all dimensionless and can be happily combinedto give PC scores (Legendre and Legendre, 1983, p. 129).Another problem with the use of covariance matrices is that it is moredifficult than with correlation matrices to compare informally the resultsfrom different analyses. Sizes of variances of PCs have the same implicationsfor different correlation matrices of the same dimension, but not for differentcovariance matrices. Also, patterns of coefficients in PCs can be readilycompared for different correlation matrices to see if the two correlationmatrices are giving similar PCs, whereas informal comparisons are oftenmuch trickier for covariance matrices. Formal methods for comparing PCsfrom different covariance matrices are, however, available (see Section 13.5).The use of covariance matrices does have one general advantage overcorrelation matrices, and a particular advantage seen in a special case. Thegeneral advantage is that statistical inference regarding population PCsbased on sample PCs is easier for covariance matrices than for correlationmatrices, as will be discussed in Section 3.7. This is relevant when PCAis used in a context where statistical inference is important. However, inpractice, it is more common to use PCA as a descriptive, rather than aninferential, tool, and then the potential advantage of covariance matrixPCA is irrelevant.The second advantage of covariance matrices holds in the special casewhen all elements of x are measured in the same units. It can then beargued that standardizing the elements of x to give correlations is equivalentto making an arbitrary choice of measurement units. This argumentof arbitrariness can also be applied more generally to the use of correlationmatrices, but when the elements of x are measurements of different types,the choice of measurement units leading to a covariance matrix is evenmore arbitrary, so that the correlation matrix is again preferred.Standardizing the variables may be thought of as an attempt to removethe problem of scale dependence from PCA. Another way of doing this isto compute PCs of the logarithms of the original data (Flury, 1997, Section8.4), though this is only feasible and sensible for restricted types of data,
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Principal ComponentAnalysis,Second
Page 7 and 8: viPreface to the Second Editionerty
Page 9 and 10: viiiPreface to the Second EditionA
Page 11 and 12: xPreface to the First Editionand in
Page 13 and 14: xiiPreface to the First EditionIn m
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Page 17 and 18: xviAcknowledgmentsthese institution
Page 19 and 20: xviiiContents3.4.1 Example ........
Page 21 and 22: xxContents10 Outlier Detection, Inf
Page 25 and 26: xxivList of Figures5.2 Artistic qua
Page 29 and 30: xxviiiList of Tables6.1 First six e
Page 33 and 34: 2 1. IntroductionFigure 1.1. Plot o
Page 35: 4 1. IntroductionFigure 1.3. Studen
Page 38 and 39: 1.2. A Brief History of Principal C
Page 40 and 41: 1.2. A Brief History of Principal C
Page 42 and 43: 2.1. Optimal Algebraic Properties o
Page 50 and 51: 2.2. Geometric Properties of Popula
Page 52 and 53: 2.3. Principal Components Using a C
Page 56 and 57: 2.3. Principal Components Using a C
Page 58 and 59: 2.4. Principal Components with Equa
Page 60 and 61: 3Mathematical and StatisticalProper
Page 62 and 63: where3.1. Optimal Algebraic Propert
Page 64 and 65: 3.2. Geometric Properties of Sample
Page 70 and 71: 3.3. Covariance and Correlation Mat
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Page 88 and 89: 3.8. Patterned Covariance and Corre
Page 90 and 91: 3.9. Models for Principal Component
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Page 94 and 95: 4Principal Components as a SmallNum
Page 96 and 97: 4.1. Anatomical Measurements 65Tabl
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4.3. Spatial and Temporal Variation
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4.4. Properties of Chemical Compoun
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4.5. Stock Market Prices 77Table 4.
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5. Graphical Representation of Data
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Anatomical Measurements5.1. Plottin
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5.1. Plotting Two or Three Principa
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5.2. Principal Coordinate Analysis
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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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