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

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7.4. An Example of Factor Analysis 163Table 7.1. Coefficients for the first four PCs: children’s intelligence tests.Component number 1 2 3 4⎧ 1 0.34 −0.39 0.09 −0.082 0.34 −0.37 −0.08 −0.233 0.35 −0.10 0.05 0.034 0.30 −0.24 −0.20 0.63Variable⎪⎨5 0.34 −0.32 0.19 −0.28number 6 0.27 0.24 −0.57 0.307 0.32 0.27 −0.27 −0.348 0.30 0.51 0.19 −0.27⎪⎩ 9 0.23 0.22 0.69 0.4310 0.36 0.33 −0.03 0.02Eigenvalue 4.77 1.13 0.96 0.71Cumulative percentageof total variation 47.7 59.1 68.6 75.7Table 7.2. Rotated factor loadings–four factors: children’s intelligence tests.Factor number 1 2 3 4Varimax⎧1 0.48 0.09 0.17 0.142 0.49 0.15 0.18 −0.033 0.35 0.22 0.24 0.224 0.26 −0.00 0.64 0.20Variable⎪⎨5 0.49 0.16 0.02 0.15number 6 0.05 0.34 0.60 −0.097 0.20 0.51 0.18 −0.078 0.10 0.54 −0.02 0.32⎪⎩ 9 0.10 0.13 0.07 0.8310 0.17 0.46 0.28 0.26Direct quartimin⎧1 0.51 −0.05 0.05 0.052 0.53 0.04 0.05 −0.143 0.32 0.13 0.16 0.154 0.17 −0.19 0.65 0.20Variable⎪⎨5 0.54 0.06 −0.13 0.05number 6 −0.07 0.28 0.67 −0.127 0.16 0.53 0.13 −0.178 0.03 0.62 −0.09 0.26⎪⎩ 9 0.00 0.09 0.02 0.8710 0.08 0.45 0.24 0.21
164 7. Principal Component Analysis and Factor AnalysisTable 7.3. Correlations between four direct quartimin factors: children’s intelligencetests.Factor number1 2 3Factor 2 0.349number 3 0.418 0.3064 0.305 0.197 0.112Table 7.4. Factor loadings—three factors, varimax rotation: children’s intelligencetests.Factor number1 2 3⎧1 0.47 0.09 0.142 0.47 0.17 0.053 0.36 0.23 0.244 0.37 0.23 0.00Variable⎪⎨5 0.45 0.08 0.23number 6 0.12 0.55 −0.057 0.17 0.48 0.178 0.05 0.36 0.52⎪⎩ 9 0.13 −0.01 0.6610 0.18 0.43 0.36The first factor in both methods has its highest loadings in variables 1, 2,3 and 5, with the next highest loadings on variables 4 and 7. In factors2, 3, 4 there is the same degree of similarity in the position of the highestloadings: for factor 2, the loadings for variables 7, 8, 10 are highest, with anintermediate value on variable 6; factor 3 has large loadings on variables 4and 6 and an intermediate value on variable 10; and factor 4 is dominated byvariable 9 with intermediate values on variables 8 and 10. The only notabledifference between the results for the two methods is that obliqueness allowsthe second method to achieve slightly higher values on the highest loadingsand correspondingly lower values on the low loadings, as indeed it is meantto.By contrast, the differences between the loadings before and after rotationare more substantial. After rotation, the ‘general factor’ withcoefficients of similar size on all variables disappears, as do most negativecoefficients, and the structure of the loadings is simplified. Again, thisis precisely what rotation is meant to achieve.
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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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Page 184 and 185: 7.2. Estimation of the Factor Model
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Page 198 and 199: 8Principal Components in Regression
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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
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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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Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

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