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

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13.3. Principal Component Analysis for Compositional Data 349replaced by their ranks, which range from 1 to p, from1ton, orfrom1tonp, respectively, depending on the type of ranking. These ranks are thenscaled within each row so that each row sum equals 1, as is true for theoriginal data.Bacon-Shone (1992) does not use PCA on these rank-transformed data,but Baxter (1993) does. He looks at several approaches for a number ofcompositional examples from archaeology, and demonstrates that for typicalarchaeological data, which often include zeros, Bacon-Shone’s (1992)procedure is unsatisfactory because it is too sensitive to the ranked zeros.Baxter (1993) also shows that both Aitchison’s and Bacon-Shone’sapproaches can be misleading when there are small but non-zero elementsin the data. He claims that simply ignoring the compositional nature of thedata and performing PCA on the original data is often a more informativealternative in archaeology than these approaches.Kaciak and Sheahan (1988) advocate the use of uncentred PCA(see Section 14.2.3), apparently without a log transformation, for theanalysis of compositional data, and use it in a market segmentationexample.13.3.1 Example: 100 km Running DataIn Sections 5.3 and 12.3.3, a data set was discussed which consisted oftimes taken for each of ten 10 km sections by 80 competitors in a 100 kmrace. If, instead of recording the actual time taken in each section, we lookat the proportion of the total time taken for each section, the data thenbecome compositional in nature. A PCA was carried out on these compositionaldata, and so was a modified analysis as proposed by Aitchison(1983). The coefficients and variances for the first two PCs are given forthe unmodified and modified analyses in Tables 13.1, 13.2, respectively.It can be seen that the PCs defined in Tables 13.1 and 13.2 have verysimilar coefficients, with angles between corresponding vectors of coefficientsequal to 8 ◦ for both first and second PCs. This similarity continueswith later PCs. The first PC is essentially a linear contrast between timesearly and late in the race, whereas the second PC is a ‘quadratic’ contrastwith times early and late in the race contrasted with those in themiddle.Comparison of Tables 13.1 and 13.2 with Table 5.2 shows that convertingthe data to compositional form has removed the first (overall time)component, but the coefficients for the second PC in Table 5.2 are verysimilar to those of the first PC in Tables 13.1 and 13.2. This correspondencecontinues to later PCs, with the third, fourth, ...PCs for the ‘raw’data having similar coefficients to those of the second, third,. . . PCs for thecompositional data.
350 13. Principal Component Analysis for Special Types of DataTable 13.1. First two PCs: 100 km compositional data.Coefficients CoefficientsComponent 1 Component 2First 10 km 0.42 0.19Second 10 km 0.44 0.18Third 10 km 0.44 0.00Fourth 10 km 0.40 −0.23Fifth 10 km 0.05 −0.56Sixth 10 km −0.18 −0.53Seventh 10 km −0.20 −0.15Eighth 10 km −0.27 −0.07Ninth 10 km −0.24 0.30Tenth 10 km −0.27 0.41Eigenvalue 4.30 2.31Cumulative percentageof total variation 43.0 66.1Table 13.2. First two PCs: Aitchison’s (1983) technique for 100 km compositionaldata.Coefficients CoefficientsComponent 1 Component 2First 10 km 0.41 0.19Second 10 km 0.44 0.17Third 10 km 0.42 −0.06Fourth 10 km 0.36 −0.31Fifth 10 km −0.04 −0.57Sixth 10 km −0.25 −0.48Seventh 10 km −0.24 −0.08Eighth 10 km −0.30 −0.01Ninth 10 km −0.24 0.30Tenth 10 km −0.25 0.43Eigenvalue 4.38 2.29Cumulative percentageof total variation 43.8 66.6
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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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3.3. Covariance and Correlation Mat
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3.3. Covariance and Correlation Mat
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3.4. Principal Components with Equa
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show that X = ULA ′ .⎡ULA ′ =
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3.6. Probability Distributions for
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3.7. Inference Based on Sample Prin
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3.7.2 Interval Estimation3.7. Infer
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3.8. Patterned Covariance and Corre
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3.9. Models for Principal Component
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3.9. Models for Principal Component
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4Principal Components as a SmallNum
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4.1. Anatomical Measurements 65Tabl
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4.1. Anatomical Measurements 67spac
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4.2. The Elderly at Home 69Table 4.
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4.3. Spatial and Temporal Variation
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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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Page 332 and 333: 12.1. Introduction 301series is alm
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Page 354 and 355: 12.3. Functional PCA 323of the data
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Page 370 and 371: 13.1. Principal Component Analysis
Page 374 and 375: 13.2. Analysis of Size and Shape 34
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Page 404 and 405: 14Generalizations and Adaptations o
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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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