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

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Table 8.3. Principal component regression for the pitprop data: coefficients, variances, regression coefficients and t-statistics for eachcomponent.Coefficients8.7. Examples of Principal Components in Regression 193Principal component1 2 3 4 5 6 7 8 9 10 11 12 13x 1 −0.40 0.22 −0.21 −0.09 −0.08 0.12 −0.11 0.14 0.33 −0.31 0.00 0.39 −0.57x 2 −0.41 0.19 −0.24 −0.10 −0.11 0.16 −0.08 0.02 0.32 −0.27 −0.05 −0.41 0.58x 3 −0.12 0.54 0.14 0.08 0.35 −0.28 −0.02 0.00 −0.08 0.06 0.12 0.53 0.41x 4 −0.17 0.46 0.35 0.05 0.36 −0.05 0.08 −0.02 −0.01 0.10 −0.02 −0.59 −0.38x 5 −0.06 −0.17 0.48 0.05 0.18 0.63 0.42 −0.01 0.28 −0.00 0.01 0.20 0.12x 6⎫⎪ ⎬ −0.28 −0.01 0.48 −0.06 −0.32 0.05 −0.30 0.15 −0.41 −0.10 −0.54 0.08 0.06x 7 −0.40 −0.19 0.25 −0.07 −0.22 0.00 −0.23 0.01 −0.13 0.19 0.76 −0.04 0.00x 8 −0.29 −0.19 −0.24 0.29 0.19 −0.06 0.40 0.64 −0.35 −0.08 0.03 −0.05 0.02x 9 −0.36 0.02 −0.21 0.10 −0.10 0.03 0.40 −0.70 −0.38 −0.06 −0.05 0.05 −0.06x 10 −0.38 −0.25 −0.12 −0.21 0.16 −0.17 0.00 −0.01 0.27 0.71 −0.32 0.06 0.00x 11 0.01 0.21 −0.07 0.80 −0.34 0.18 −0.14 0.01 0.15 0.34 −0.05 0.00 −0.01x 12⎪ ⎭0.12 0.34 0.09 −0.30 −0.60 −0.17 0.54 0.21 0.08 0.19 0.05 0.00 0.00x 13 0.11 0.31 −0.33 −0.30 0.08 0.63 −0.16 0.11 −0.38 0.33 0.04 0.01 −0.01Variance 4.22 2.38 1.88 1.11 0.91 0.82 0.58 0.44 0.35 0.19 0.05 0.04 0.04% of totalvariance 32.5 18.3 14.4 8.5 7.0 6.3 4.4 3.4 2.7 1.5 0.4 0.3 0.3Regressioncoefficient γ k 0.13 −0.37 0.13 −0.05 −0.39 0.27 −0.24 −0.17 0.03 0.00 −0.12 −1.05 0.00t-value 68.6 14.39 4.38 1.26 9.23 6.19 4.50 2.81 0.46 0.00 0.64 5.26 0.01
194 8. Principal Components in Regression AnalysisTable 8.4. Variable selection using various techniques on the pitprop data. (Eachrow corresponds to a selected subset with × denoting a selected variable.)Variables1 2 3 4 5 6 7 8 9 10 11 12 13Five variablesJeffers (1981) × × × × ×× × × × ×McCabe (1982) × × × × ×× × × × ×× × × × ×× × × × ×Six variablesJeffers (1981) × × × × × ×McCabe (1982) × × × × × ×Jolliffe (1973) × × × × × ×× × × × × ×{× × × × × ×McCabe (1982)× × × × × ×Jolliffe (1973)× × × × × ×Eight variablesMansfield et al. × × × × × × × ×(1977)Boneh and × × × × × × × ×Mendieta (1994)ble. McCabe (1982), using a technique related to PCA (see Section 6.3),and with a similar purpose to Jolliffe’s (1973) methods, chose subsets ofvarious sizes. McCabe’s subsets in Table 8.4 are the best few with respectto a single criterion, whereas Jolliffe gives the single best subset butfor several different methods. The best subsets due to Jolliffe (1973) andMcCabe (1982) have considerable overlap with each other, but there aresubstantial differences from the subsets of Jeffers (1981) and Mansfield etal. (1977). This reflects the different aims of the different selection methods.It shows again that substantial variation within the set of regressor variablesdoes not necessarily imply any relationship with y and, conversely,that variables having little correlation with the first few PCs can still beimportant in predicting y. Interestingly, the subset chosen by Boneh andMendieta (1994) overlaps less with Jeffers’ (1981) selection than that ofMansfield et al. (1977), but more than do those of Jolliffe (1973) or McCabe(1982).
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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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Page 198 and 199: 8Principal Components in Regression
Page 200 and 201: 8.1. Principal Component Regression
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Page 204 and 205: 8.2. Selecting Components in Princi
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Page 232 and 233: 9.1. Discriminant Analysis 201on th
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Page 252 and 253: 9.2. Cluster Analysis 221choosing a
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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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