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

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2.4. Principal Components with Equal and/or Zero Variances 272.4 Principal Components with Equal and/or ZeroVariancesThe final, short, section of this chapter discusses two problems that mayarise in theory, but are relatively uncommon in practice. In most of thischapter it has been assumed, implicitly or explicitly, that the eigenvaluesof the covariance or correlation matrix are all different, and that none ofthem is zero.Equality of eigenvalues, and hence equality of variances of PCs, will occurfor certain patterned matrices. The effect of this occurrence is that for agroup of q equal eigenvalues, the corresponding q eigenvectors span a certainunique q-dimensional space, but, within this space, they are, apart frombeing orthogonal to one another, arbitrary. Geometrically (see PropertyG1), what happens for q = 2 or 3 is that the principal axes of a circle orsphere cannot be uniquely defined; a similar problem arises for hypersphereswhen q>3. Thus individual PCs corresponding to eigenvalues in a group ofequal eigenvalues are not uniquely defined. A further problem with equalvariancePCs is that statistical inference becomes more complicated (seeSection 3.7).The other complication, variances equal to zero, occurs rather more frequently,but is still fairly unusual. If q eigenvalues are zero, then the rankof Σ is (p − q) rather than p, and this outcome necessitates modificationsto the proofs of some properties given in Section 2.1 above. Any PC withzero variance defines an exactly constant linear relationship between theelements of x. If such relationships exist, then they imply that one variableis redundant for each relationship, as its value can be determined exactlyfrom the values of the other variables appearing in the relationship. Wecould therefore reduce the number of variables from p to (p − q) withoutlosing any information. Ideally, exact linear relationships should be spottedbefore doing a PCA, and the number of variables reduced accordingly. Alternatively,any exact or near-exact linear relationships uncovered by thelast few PCs can be used to select a subset of variables that contain mostof the information available in all of the original variables. This and relatedideas are more relevant to samples than to populations and are discussedfurther in Sections 3.4 and 6.3.There will always be the same number of zero eigenvalues for a correlationmatrix as for the corresponding covariance matrix, since an exactlinear relationship between the elements of x clearly implies an exact linearrelationship between the standardized variables, and vice versa. There isnot the same equivalence, however, when it comes to considering equal variancePCs. Equality of some of the eigenvalues in a covariance (correlation)matrix need not imply that any of the eigenvalues of the correspondingcorrelation (covariance) matrix are equal. A simple example is when the pvariables all have equal correlations but unequal variances. If p>2, then
28 2. Properties of Population Principal Componentsthe last (p−1) eigenvalues of the correlation matrix are equal (see Morrison,1976, Section 8.6), but this relationship will not hold, in general, for thecovariance matrix. Further discussion of patterns in covariance or correlationmatrices, and their implications for the structure of the correspondingPCs, is given in Section 3.8.
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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
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Page 40 and 41: 1.2. A Brief History of Principal C
Page 42 and 43: 2.1. Optimal Algebraic Properties o
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Page 52 and 53: 2.3. Principal Components Using a C
Page 60 and 61: 3Mathematical and StatisticalProper
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Page 64 and 65: 3.2. Geometric Properties of Sample
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Page 94 and 95: 4Principal Components as a SmallNum
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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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