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Index 467see also hypothesis testing forequality of PC variances,near-constant relationships,residuals after fitting firstfew PCs, zero variance PCslatent root regression 168, 178,180–182, 185–187, 190, 191,197, 239latent semantic indexing 90latent variables 151, 165, 226, 230,231latent variable multivariateregression 230, 231least squares estimation/estimators32, 34, 59, 157, 167–173,175–179, 181, 184, 185, 189,208, 229, 286, 288, 294, 304,326, 382, 385see also partial least squaresleverage points 240see also influential observationslikelihood ratio test 54, 55, 120,353, 356, 360linear approximation asymmetricPCA 401loadings see factor loadings, PCcoefficientslocal authoritiesBritish towns 71, 215England and Wales 195–198,245–247English counties 108–110,215–219local PCA 381log-eigenvalue (LEV) diagram115–118, 128, 134-136log transform see transformedvariableslongitudinal data 328, 330, 331,355lower (or upper) triangularmatrices 182, 411, 412lower rank approximations tomatrices 38, 46, 113, 120,342, 365, 383, 385LR algorithm 411M-estimators 264, 265, 267Mahalanobis distances 33, 93, 94,104, 203, 204, 209, 212, 237,264, 265manufacturing processes 366–368matrix correlation 96, 140, 141matrix-valued data 370maximum covariance analysis 225,226, 229, 401maximum likelihood estimation220, 264for common PCs 355for covariance matrices 50, 336,363, 364for factor loadings 155–157for functional and structuralrelationships 189for PC coefficients and variances8, 50, 365in PC models 60, 222, 267, 364,386measurement errors 151, 188, 189medical applicationsbiomedical problems 395clinical trials 40, 239epidemiology 248, 336opthalmology 266see also chemistry (bloodchemistry)meteorology and climatology 8, 9,90, 183, 213, 381atmospheric pressure 71–73, 401cloudbase heights 211cloud-seeding 339monsoon onset date 174satellite meteorology 358wind data 369, 370see also atmospheric science,climate change/variation,ENSO, NAO, temperaturesmethod of moments estimationfor PC coefficients andvariances 50
468 Indexmetrics 42, 59, 60, 185, 189, 210,220, 260, 325, 331, 373, 382,386–388optimal metric 387minimax components 267minimum χ 2 test 120minimum description length 19,39, 395minimum spanning tree (MST)81–83, 130minimum variance ellipsoids 267misclassification probabilities, seediscriminant analysismissing data 60, 61, 83, 134, 339,363–366, 412estimating covariance/correlationmatrices 363–365estimating PCs 365, 385in biplots 103, 104in designed experiments 353, 365in regression 363mixtures of distributions 61, 165,200, 221, 222, 241, 364modal dispersion matrix 395models for PCA 50, 54, 59–61, 119,124–126, 132, 151, 158–160,220, 364, 369, 405see also fixed effects model forPCAmodified principal components 144most correlated components 26multichannel singular spectrumanalysis (MSSA) 302, 305,307, 308, 310, 311, 316, 329multicollinearities 167, 168,170–173, 177, 180, 181, 185,188, 196, 286, 378predictive and non-predictivemulticollinearities 180, 181,185, 188variance inflation factors (VIFs)173, 174see also ill-conditioningmultidimensional scaling seescaling or ordinationtechniquesmultilevel models 353multiple correlation coefficient 25,141, 143, 174, 177, 191, 197,198, 403multiple correspondence analysis,see correspondence analysismultiple regression, see regressionanalysismultivariate analysis of variance(MANOVA) 102, 351, 353multivariate normal distribution 8,16, 18, 20, 22, 33, 39, 47–55,60, 69, 119, 152, 155–157,160, 201, 207, 220–222, 236,239, 244, 254, 264, 267, 276,299, 338, 339, 365, 367, 368,379, 386, 388curvature 395see also contours of constantprobability, inference forPCsmultivariate regression 17, 183,223, 228–230, 331, 352multiway PCA, see three-modePCAnear-constant relationshipsbetween variables 3, 13, 27,28, 42–44, 119, 138, 167,181, 182, 189, 235, 374, 377,378nearly equal eigenvalues 43, 262,263, 276, 277, 360, 408, 410see also stabilityneural networks 200, 266, 373,379–381, 388, 400, 401, 405,408, 412–414analogue/digital 414autoassociative 381biological plausibility 413first or last PCs 400, 413input training net 381
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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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2.1. Optimal Algebraic Properties o
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2.1. Optimal Algebraic Properties o
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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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2.3. Principal Components Using a C
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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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3.2. Geometric Properties of Sample
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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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3.7. Inference Based on Sample Prin
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3.7. Inference Based on Sample Prin
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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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5.2. Principal Coordinate Analysis
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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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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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6.1. How Many Principal Components?
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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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6.3. Selecting a Subset of Variable
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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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6.4. Examples Illustrating 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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7.2. Estimation of the Factor Model
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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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8.4. Variations on Principal Compon
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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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8.7. Examples of Principal Componen
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8.7. Examples of Principal Componen
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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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9.3. Canonical Correlation Analysis
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9.3. Canonical Correlation Analysis
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9.3. Canonical Correlation Analysis
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9.3. Canonical Correlation Analysis
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10.1. Detection of Outliers Using P
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10.1. Detection of Outliers Using P
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10.1. Detection of Outliers Using P
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10.1. Detection of Outliers Using P
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10.1. Detection of Outliers Using P
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10.1. Detection of Outliers Using P
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10.1. Detection of Outliers Using P
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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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10.2. Influential Observations in a
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10.2. Influential Observations in a
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10.2. Influential Observations in a
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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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10.4. Robust Estimation of Principa
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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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11.1. Rotation of Principal Compone
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11.1. Rotation of Principal Compone
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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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12.2. PCA and Atmospheric Time Seri
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and a typical row of the matrix is1
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12.2. PCA and Atmospheric Time Seri
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12.2. PCA and Atmospheric Time Seri
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12.2. PCA and Atmospheric Time Seri
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12.2. PCA and Atmospheric Time Seri
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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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12.4. PCA and Non-Independent Data
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12.4. PCA and Non-Independent Data
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12.4. PCA and Non-Independent Data
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12.4. PCA and Non-Independent Data
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13.1. Principal Component Analysis
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13.1. 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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13.3. Principal Component Analysis
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13.3. Principal Component Analysis
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13.4. Principal Component Analysis
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13.4. Principal Component Analysis
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13.5. Common Principal Components 3
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13.5. Common Principal Components 3
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13.5. Common Principal Components 3
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13.5. Common Principal Components 3
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13.6. Principal Component Analysis
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13.6. Principal Component Analysis
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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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14.1. Additive Principal Components
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14.1. Additive Principal Components
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14.2. Weights, Metrics, Transformat
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14.2. Weights, Metrics, Transformat
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14.2. Weights, Metrics, Transformat
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14.2. Weights, Metrics, Transformat
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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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A.1. Numerical Calculation of Princ
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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 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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