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

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5.2. Principal Coordinate Analysis 89structed in the first stage of the principal coordinate analysis are∆ 2 hi = t hh + t ii − 2t hip∑ p∑= x 2 hj + x 2 ij − 2=j=1j=1p∑(x hj − x ij ) 2j=1= d 2 hi,p∑x hj x ijthe Euclidean distance between the observations using the original p variables.As before, the PCA in the second stage of principal coordinateanalysis gives the same results as a PCA on the original data. Note, however,that XX ′ is not a very obvious similarity matrix. For a ‘covariancematrix’ between observations it is more natural to use a row-centred, ratherthan column-centred, version of X.Even in cases where PCA and principal coordinate analysis give equivalenttwo-dimensional plots, there is a difference, namely that in principalcoordinate analysis there are no vectors of coefficients defining the axesin terms of the original variables. This means that, unlike PCA, theaxes in principal coordinate analysis cannot be interpreted, unless thecorresponding PCA is also done.The equivalence between PCA and principal coordinate analysis in thecircumstances described above is termed a duality between the two techniquesby Gower (1966). The techniques are dual in the sense that PCAoperates on a matrix of similarities between variables, whereas principal coordinateanalysis operates on a matrix of similarities between observations(individuals), but both can lead to equivalent results.To summarize, principal coordinate analysis gives a low-dimensional representationof data when the data are given in the form of a similarity ordissimilarity matrix. As it can be used with any form of similarity or dissimilaritymatrix, it is, in one sense, ‘more powerful than,’ and ‘extends,’ PCA(Gower, 1967). However, as will be seen in subsequent chapters, PCA hasmany uses other than representing data graphically, which is the overridingpurpose of principal coordinate analysis.Except in the special cases discussed above, principal coordinate analysishas no direct relationship with PCA, so no examples will be given of thegeneral application of the technique. In the case where principal coordinateanalysis gives an equivalent representation to that of PCA, nothing newwould be demonstrated by giving additional examples. The examples givenin Section 5.1 (and elsewhere) which are presented as plots with respectto the first two PCs are, in fact, equivalent to two-dimensional principalcoordinate plots if the ‘dissimilarity’ between observations h and i isproportional to the Euclidean squared distance between the hth and ithobservations in p dimensions.j=1
90 5. Graphical Representation of Data Using Principal ComponentsIn most cases, if the data are available in the form of an (n × p) matrixof p variables measured for each of n observations, there is no advantagein doing a principal coordinate analysis instead of a PCA, unless for somereason a dissimilarity measure other than Euclidean distance is deemedto be appropriate. However, an exception occurs when n
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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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3.2. Geometric Properties of Sample
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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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Page 108 and 109: 4.5. Stock Market Prices 77Table 4.
Page 110 and 111: 5. Graphical Representation of Data
Page 112 and 113: Anatomical Measurements5.1. Plottin
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Page 122 and 123: 5.3. Biplots 91columns, L is an (r
Page 124 and 125: 5.3. Biplots 93ButandSubstituting i
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Page 130 and 131: 5.3. Biplots 99Table 5.2. First two
Page 132 and 133: 5.3. Biplots 101Figure 5.5. Biplot
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Page 138 and 139: 5.6. Displaying Intrinsically High-
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Page 142 and 143: 6Choosing a Subset of PrincipalComp
Page 144 and 145: 6.1. How Many Principal Components?
Page 164 and 165: 6.2. Choosing m, the Number of Comp
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Page 168 and 169: 6.3. Selecting a Subset of Variable
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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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