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

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11.2. Alternatives to Rotation 283Figure 11.5. Loadings of second winter components for PCA, RPCA, SCoT,SCoTLASS and simple component analysis.
284 11. Rotation and Interpretation of Principal Componentsvariables in the functions. The constraints are designed to make the resultingcomponents simpler to interpret than PCs, but without sacrificing toomuch of the variance accounted for by the PCs. The first idea, discussedin Section 11.2.1, is a simple one, namely, restricting coefficients to a set ofintegers, though it is less simple to put into practice. The second type oftechnique, described in Section 11.2.2, borrows an idea from regression, thatof the LASSO (Least Absolute Shrinkage and Selection Operator). By imposingan additional constraint in the PCA optimization problem, namely,that the sum of the absolute values of the coefficients in a component isbounded, some of the coefficients can be forced to zero. A technique fromatmospheric science, empirical orthogonal teleconnections, is described inSection 11.2.3, and Section 11.2.4 makes comparisons between some of thetechniques introduced so far in the chapter.11.2.1 Components with Discrete-Valued CoefficientsA fairly obvious way of constructing simpler versions of PCs is to successivelyfind linear functions of the p variables that maximize variance, as inPCA, but with a restriction on the values of coefficients in those functionsto a small number of values. An extreme version of this was suggested byHausmann (1982), in which the loadings are restricted to the values +1,−1 and 0. To implement the technique, Hausman (1982) suggests the useof a branch-and-bound algorithm. The basic algorithm does not includean orthogonality constraint on the vectors of loadings of successive ‘components,’but Hausmann (1982) adapts it to impose this constraint. Thisimproves interpretability and speeds up the algorithm, but has the implicationthat it may not be possible to find as many as p components. In the6-variable example given by Hausmann (1982), after 4 orthogonal componentshave been found with coefficients restricted to {−1, 0, +1} the nullvector is the only vector with the same restriction that is orthogonal to allfour already found. In a unpublished M.Sc. project report, Brooks (1992)discusses some other problems associated with Hausmann’s algorithm.Further information is given on Hausmann’s example in Table 11.2. Herethe following can be seen:• The first component is a straightforward average or ‘size’ componentin both analyses.• Despite a considerable simplication, and a moderately different interpretation,for the second constrained component, there is very littleloss in variance accounted by the first two constrained componentscompared to first two PCs.A less restrictive method is proposed by Vines (2000), in which the coefficientsare also restricted to integers. The algorithm for finding so-calledsimple components starts with a set of p particularly simple vectors of
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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 264 and 265: 10.1. Detection of Outliers Using P
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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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Page 350 and 351: 12.3. Functional PCA 319The sample
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Page 354 and 355: 12.3. Functional PCA 323of the data
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Page 358 and 359: 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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