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

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8.7. Examples of Principal Components in Regression 195No.Table 8.5. Variables used in the household formation example.Description1. Population in non-private establishments2. Population age 0–143. Population age 15–444. Population age 60/65+5. Females currently married6. Married males 15–297. Persons born ex UK8. Average population increase per annum (not births and deaths)9. Persons moved in previous 12 months10. Households in owner occupation11. Households renting from Local Authority12. Households renting private unfurnished13. Vacant dwellings14. Shared dwellings15. Households over one person per room16. Households with all exclusive amenities17. Ratio households to rateable units18. Domestic rateable value (£) per head19. Rateable units with rateable value < £10020. Students age 15+21. Economically active married females22. Unemployed males seeking work23. Persons employed in agriculture24. Persons employed in mining and manufacturing25. Males economically active or retired in socio-economicgroup1,2,3,4,1326. Males economically active or retired in socio-economicgroup 5, 6, 8, 9, 12, 1427. With degrees (excluding students with degree)28. Economically active males socio-economic group 3, 429. Average annual total income (£) per adult8.7.2 Household Formation DataThis example uses part of a data set that arose in a study of householdformation. The subset of data used here has 29 demographic variables measuredin 1971 for 168 local government areas in England and Wales. Thevariables are listed in Table 8.5. All variables, except numbers 17, 18 and29, are expressed as numbers per 1000 of population; precise definitions ofeach variable are given in Appendix B of Bassett et al. (1980).
196 8. Principal Components in Regression AnalysisTable 8.6. Eigenvalues of the correlation matrix and order of importance inpredicting y for the household formation data.PC Eigenvalue Order of PC Eigenvalue Order ofnumber importance number importancein predicting yin predicting y1 8.62 1 15 0.24 172 6.09 4 16 0.21 253 3.40 2 17 0.18 164 2.30 8 18 0.14 105 1.19 9 19 0.14 76 1.06 3 20 0.10 217 0.78 13 21 0.10 288 0.69 22 22 0.07 69 0.58 20 23 0.07 1810 0.57 5 24 0.05 1211 0.46 11 25 0.04 1412 0.36 15 26 0.03 2713 0.27 24 27 0.02 1914 0.25 23 28 0.003 26Although this was not the purpose of the original project, the objectiveconsidered here is to predict the final variable (average annual total incomeper adult) from the other 28. This objective is a useful one, as informationon income is often difficult to obtain accurately, and predictions from other,more readily available, variables would be valuable. The results presentedbelow were given by Garnham (1979) in an unpublished M.Sc. dissertation,and further details of the regression analysis can be found in thatsource. A full description of the project from which the data are taken isavailable in Bassett et al. (1980). Most regression problems with as manyas 28 regressor variables have multicollinearities, and the current exampleis no exception. Looking at the list of variables in Table 8.5 it is clear,even without detailed definitions, that there are groups of variables thatare likely to be highly correlated. For example, several variables relate totype of household, whereas another group of variables considers rates ofemployment in various types of job. Table 8.6, giving the eigenvalues of thecorrelation matrix, confirms that there are multicollinearities; some of theeigenvalues are very small.Consider now PC regression and some of the strategies that can be usedto select a subset of PCs to be included in the regression. Deleting componentswith small variance, with a cut-off of about l ∗ =0.10, implies thatbetween seven and nine components can be left out. Sequential deletionof PCs with the smallest variances using t-statistics at each stage suggeststhat only six PCs can be deleted. However, from the point of view of R 2 ,the squared multiple correlation coefficient, deletion of eight or more might
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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 264 and 265: 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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10.3. 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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