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

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Index 473variance ellipsoids,S-estimatorsrotationof factors 116, 153–156, 159,162–165of functional PCs 316, 327of PCs 43, 74, 151, 154, 162,163, 165, 166, 182, 185, 188,191, 213, 238, 248, 269–279,291, 295, 297, 298, 370, 396,407of subsets of PCs with similarvariances 276, 277rotation/switching of PCs 259to simple components 285, 291to simple structure 153, 154,182, 185, 270, 271, 276, 277,369see also oblique factors,orthogonal factors,quartimin rotation, varimaxrotationrounded PC coefficients 40, 42–44,67, 259, 263, 292, 293see also discrete PC coefficients,simplified PC coefficientsrow-centered data 89rules for selecting PCs 54, 111–137,159, 162, 217ad hoc rules 112–118, 130–135,136, 138, 147, 149, 238based on cross-validation 112,120–127, 130–132, 135–137based on cumulative variances ofPCs 112–114, 117, 130–136,138, 147, 147based on gaps betweeneigenvalues 126, 127, 129,133based on individual variances ofPCs 114–118, 123, 130–136,138, 147, 149, 238based on partial correlation 127from atmospheric science 112,116, 118, 127–130, 132–136statistically based rules 112,118–137see also broken stick model,equal variance PCs, howmany PCs, Kaiser’s rule,log eigenvalue diagram,parallel analysis, screegraph, selection of a subsetof PCsRV-coefficient 38, 143–145, 147,252S-estimators 267S-mode analysis 308, 398sample sizeseffective/equivalent 129, 148,299large see large data setsmoderate 249, 252small 65, 68, 148, 235, 257smaller than number of variables90, 148, 207, 413sample surveys 49, 328, 335, 336,353, 353stratified survey design 336, 353sampling variationPC coefficients 65PC variances 115, 123scale dependence of covariancematrices 24, 26see also invariance (scaleinvariance)scaling or ordination techniques85–90, 102, 106, 107, 200classical scaling 85dual scaling 103, 343non-metric multidimensionalscaling 86, 372reciprocal averaging 103, 343see also principal co-ordinateanalysisscatter, definitions of 395scores for PCs, see PC scoresSCoT (simplified componenttechnique) 278–279, 287,
474 Index289, 290, 291SCoTLASS (simplified componenttechnique - LASSO)280–283, 287–291scree graph 115–118, 125, 126,130–132, 134, 135selection of subsets of PCsin discriminant analysis 201,202, 204–206in latent root regression 180, 181in PC regression 168, 170–177,196–198, 202, 205, 245see also how many PCs, rulesfor selecting PCsselection of variablesin non-regression contexts 13,27, 38, 111, 137–149, 186,188, 191, 198, 220, 221, 260,270, 286, 288, 290, 293–295,376stepwise selection/backwardelimination algorithms 142,144, 145, 147see also principal variables,regression analysis (variableselection)self-consistency 20, 378, 379sensible PCA 60sensitivity matrix 240sensitivity of PCs 232, 252,259–263, 278shape and size PCs, see size andshape PCsShapiro-Wilk test 402shrinkage methods 167, 178–181,264, 288signal detection 130, 304, 332signal processing 303, 317, 395signal to noise ratio 337, 388, 401SIMCA 207–208, 239similarity measuresbetween configurations 38between observations 79, 89,106, 210-212, 339, 390between variables 89, 213, 391see also distance/dissimilaritymeasuressimple components 280–287, 291simplicity/simplification 269–271,274, 277–286, 403, 405simplified PC coefficients 66, 67,76, 77see also approximations to PCs,discrete PC coefficients,rounded PC coefficientssimultaneous components 361singular spectrum analysis (SSA)302–308, 310, 316singular value decomposition(SVD) 7, 29, 44–46, 52, 59,101, 104, 108, 113, 120, 121,129, 172, 173, 226, 229, 230,253, 260, 266, 273, 353, 365,366, 382, 383comparison of SVDs 362computation based on SVD 46,173, 412, 413generalized SVD 46, 342, 383,385, 386multitaper frequency domainSVD (MTM-SVD) 302, 311,314, 316size and shape PCs 53, 57, 64, 67,68, 81, 104, 297, 298, 338,343–346, 355, 356, 388, 393,401see also contrasts betweenvariables, interpretationof PCs, patternedcorrelation/covariancematricesskewness 219, 372smoothing and interpolation 274,316, 318, 320, 322, 324–326,334, 335, 377–379of spatial data 334, 335, 364, 365lo(w)ess 326splines 320, 322, 331, 377, 378,387sparse data 331
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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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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
Page 454 and 455: References 423Castro, P.E., Lawton,
Page 456 and 457: References 425Cook, R.D. (1986). As
Page 458 and 459: References 427Dempster, A.P., Laird
Page 460 and 461: References 429Feeney, G.J. and Hest
Page 462 and 463: References 431in Descriptive Multiv
Page 464 and 465: References 433Gunst, R.F. and Mason
Page 466 and 467: References 435Hocking, R.R., Speed,
Page 468 and 469: References 437Jeffers, J.N.R. (1978
Page 470 and 471: References 439Kazi-Aoual, F., Sabat
Page 472 and 473: References 441Krzanowski, W.J. (200
Page 474 and 475: References 443Mann, M.E. and Park,
Page 476 and 477: References 445Monahan, A.H., Tangan
Page 478 and 479: References 447Pack, P., Jolliffe, I
Page 480 and 481: References 449Richman M.B. (1993).
Page 482 and 483: References 451Soofi, E.S. (1988). P
Page 484 and 485: References 453Tenenbaum, J.B., de S
Page 486 and 487: References 455Vong, R., Geladi, P.,
Page 488 and 489: References 457regularities in multi
Page 490 and 491: Index 459116, 127-130, 133, 270, 27
Page 492 and 493: Index 461computationin (PC) regress
Page 494 and 495: Index 463discriminant principal com
Page 496 and 497: Index 465of correlations between va
Page 498 and 499: Index 467see also hypothesis testin
Page 500 and 501: Index 469PC algorithms with noise 4
Page 502 and 503: Index 471in functional and structur
Page 506 and 507: Index 475spatial correlation/covari
Page 508 and 509: Index 477for covariance matrices 26
Page 510 and 511: Author Index 479Belsley, D.A. 169Be
Page 512 and 513: Author Index 481Fowlkes, E.B. 377Fr
Page 514 and 515: Author Index 483Krzanowski, W.J. 46
Page 516 and 517: Author Index 485Rencher, A.C. 64, 1
Page 518: Author Index 487Yaguchi, H. 371Yana
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