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

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10.1. Detection of Outliers Using <strong>Principal</strong> <strong>Component</strong>s 245that this (male) student has the equal largest chest measurement, but thatonly 3 of the other 16 male students are shorter than him, and only twohave a smaller waist measurement—perhaps he was a body builder? Similaranalyses can be done for other observations in Table 10.1. For example,observation 20 is extreme on the fifth PC. This PC, which accounts for2.7% of the total variation, is mainly a contrast between height and forearmlength with coefficients 0.67, −0.52, respectively. Observation 20 is (jointlywith one other) the shortest student of the 28, but only one of the otherten women has a larger forearm measurement. Thus, observations 15 and20, and other observations indicated as extreme by the last few PCs, arestudents for whom some aspects of their physical measurements contradictthe general positive correlation among all seven measurements.Household Formation DataThese data were described in Section 8.7.2 and are discussed in detail byGarnham (1979) and Bassett et al. (1980). Section 8.7.2 gives the results ofa PC regression of average annual total income per adult on 28 other demographicvariables for 168 local government areas in England and Wales.Garnham (1979) also examined plots of the last few and first few PCs ofthe 28 predictor variables in an attempt to detect outliers. Two such plots,for the first two and last two PCs, are reproduced in Figures 10.4 and 10.5.An interesting aspect of these figures is that the most extreme observationswith respect to the last two PCs, namely observations 54, 67, 41 (and 47,53) are also among the most extreme with respect to the first two PCs.Some of these observations are, in addition, in outlying positions on plotsof other low-variance PCs. The most blatant case is observation 54, whichis among the few most extreme observations on PCs 24 to 28 inclusive, andalso on PC1. This observation is ‘Kensington and Chelsea,’ which must bean outlier with respect to several variables individually, as well as beingdifferent in correlation structure from most of the remaining observations.In addition to plotting the data with respect to the last few and first fewPCs, Garnham (1979) examined the statistics d 2 1i for q =1, 2,...,8usinggamma plots, and also looked at normal probability plots of the values ofvarious PCs. As a combined result of these analyses, he identified six likelyoutliers, the five mentioned above together with observation 126, which ismoderately extreme according to several analyses.The PC regression was then repeated without these six observations. Theresults of the regression were noticeably changed, and were better in tworespects than those derived from all the observations. The number of PCswhich it was necessary to retain in the regression was decreased, and theprediction accuracy was improved, with the standard error of predictionreduced to 77.3% of that for the full data set.
246 10. Outlier Detection, Influential Observations and Robust Estimation375 PC2250125-450-300-150150 300 450 600 750PC1-125-2506754-37541-5004753-625Figure 10.4. Household formation data: plot of the observations with respect tothe first two PCs.
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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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Page 300 and 301: 11Rotation and Interpretation ofPri
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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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