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

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A.1. Numerical Calculation of Principal Components 413observation on each PC, and hence all the information that is requiredto construct a biplot (see Section 5.3). The PC scores would otherwiseneed to be derived as an extra step after calculating the eigenvalues andeigenvectors of the covariance or correlation matrix S = 1n−1 X′ X.The values of the PC scores are related to the eigenvectors of XX ′ ,whichcan be derived from the eigenvectors of X ′ X (see the proof of Property G4in Section 3.2); conversely, the eigenvectors of X ′ X can be found fromthose of XX ′ . In circumstances where the sample size n is smaller than thenumber of variables p, XX ′ has smaller dimensions than X ′ X,sothatitcan be advantageous to use the algorithms described above, based on thepower method or QL method, on a multiple of XX ′ rather than X ′ X in suchcases. Large computational savings are possible when n ≪ p, as in chemicalspectroscopy or in the genetic example of Hastie et al. (2000), which isdescribed in Section 9.2 and which has n = 48, p = 4673. Algorithms alsoexist for updating the SVD if data arrive sequentially (see for exampleBerry et al. (1995)).Neural Network AlgorithmsNeural networks provide ways of extending PCA, including some non-lineargeneralizations (see Sections 14.1.3, 14.6.1). They also give alternative algorithmsfor estimating ‘ordinary’ PCs. The main difference between thesealgorithms and the techniques described earlier in the Appendix is thatmost are ‘adaptive’ rather than ‘batch’ methods. If the whole of a dataset is collected before PCA is done and parallel processing is not possible,then batch methods such as the QR algorithm are hard to beat (see Diamantarasand Kung, 1996 (hereafter DK96), Sections 3.5.3, 4.4.1). On theother hand, if data arrive sequentially and PCs are re-estimated when newdata become available, then adaptive neural network algorithms come intotheir own. DK96, Section 4.2.7 note that ‘there is a plethora of alternative[neural network] techniques that perform PCA.’ They describe a selectionof single-layer techniques in their Section 4.2, with an overview of these intheir Table 4.1. Different algorithms arise depending on• whether the first or last few PCs are of interest;• whether one or more than one PC is required;• whether individual PCs are wanted or whether subspaces spanned byseveral PCs will suffice;• whether the network is required to be biologically plausible.DK96, Section 4.2.7 treat finding the last few PCs as a different technique,calling it minor component analysis.In their Section 4.4, DK96 compare the properties, including speed, ofseven algorithms using simulated data. In Section 4.5 they discuss multilayernetworks.
414 Appendix A. Computation of Principal ComponentsNeural network algorithms are feasible for larger data sets than batchmethods because they are better able to take advantage of developmentsin computer architecture. DK96, Chapter 8, discuss the potential for exploitingparallel VSLI (very large scale integration) systems, where themost appropriate algorithms may be different from those for non-parallelsystems (DK96, Section 3.5.5). They discuss both digital and analogueimplementations and their pros and cons (DK96, Section 8.3). Classicaleigenvector-based algorithms are not easily parallelizable, whereas neuralnetwork algorithms are (DK96 pp. 205–207).
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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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Page 404 and 405: 14Generalizations and Adaptations o
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Page 432 and 433: 14.6. Miscellanea 401• Linear App
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Page 438 and 439: Appendix AComputation of Principal
Page 440 and 441: A.1. Numerical Calculation of Princ
Page 442 and 443: A.1. Numerical Calculation of Princ
Page 446 and 447: ReferencesAguilera, A.M., Gutiérre
Page 448 and 449: References 417Apley, D.W. and Shi,
Page 450 and 451: References 419Benasseni, J. (1986b)
Page 452 and 453: 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
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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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