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

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11.2. Alternatives to Rotation 279which are described in Section 11.2. The normalization of loadings and theshading in Figures 11.2–11.5 follow the same conventions as in Figure 11.1,except that any loadings which are exactly zero are unshaded in the figures.For the autumn data, the first two SCoT components may be viewedas slightly simpler versions of PC1 and PC2. The largest loadings becomelarger and the smallest become smaller, as is often observed during rotation.The strong similarity between SCoT and PCA components is reflected bythe fact that the first two SCoT components account for 78.0% of thetotal variation compared to 78.2% for the first two PCs. Turning to thewinter data, the first two PCs and RPCs in Figures 11.4, 11.5 are not toodifferent from those in autumn (Figures 11.2, 11.3), but there are biggerdifferences for SCoT. In particular, the second SCoT component is verymuch dominated by a single grid box in the Eastern Mediterranean. Thisextreme simplicity leads to a reduction in the total variation accounted forby the first two SCoT components to 55.8%, compared to 71.0% for thefirst two PCs.Results are presented for only one value of the tuning parameter ψ, andthe choice of this value is not an easy one. As ψ increases, there is oftena rapid jump from components that are very close to the correspondingPC, like those in autumn, to components that are dominated by a singlevariable, as for SC2 in winter. This jump is usually accompanied by a largedrop in the proportion of variation accounted for, compared to PCA. Thisbehaviour as ψ varies can also be seen in a later example (Tables 11.3,11.4), and is an unsatisfactory feature of SCoT. Choosing different valuesψ 1 ,ψ 2 ,... of ψ for SC1, SC2, ..., only partially alleviates this problem.The cause seems to be the presence of a number of local maxima for thecriterion defined by (11.1.6). As ψ changes, a different local maximum maytake over as the global maximum, leading to a sudden switch betweentwo quite different solutions. Further discussion of the properties of SCoT,together with additional examples, is given by Jolliffe and Uddin (2000)and Uddin (1999).Filzmoser (2000) argues that sometimes there is simple structure in aplane but not in any single direction within that plane. He derives a wayof finding such simply structured planes—which he calls principal planes.Clearly, if a data set has such structure, the methods discussed in thissection and the next are unlikely to find simple single components withouta large sacrifice in variance.11.2 Alternatives to RotationThis section describes two ideas for constructing linear functions of the poriginal variables having large variances; these techniques differ from PCAin imposing additional constraints on the loadings or coefficients of the
280 11. Rotation and Interpretation of Principal ComponentsFigure 11.2. Loadings of first autumn components for PCA, RPCA, SCoT,SCoTLASS and simple component analysis.
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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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Page 260 and 261: 9.3. Canonical Correlation Analysis
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Page 264 and 265: 10.1. Detection of Outliers Using P
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Page 290 and 291: 10.3. Sensitivity and Stability 259
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Page 300 and 301: 11Rotation and Interpretation ofPri
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Page 328 and 329: 11.4. Physical Interpretation of Pr
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Page 332 and 333: 12.1. Introduction 301series is alm
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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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