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

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12.3. Functional PCA 317A key reference for functional PCA is the book by Ramsay and Silverman(1997), written by two researchers in the field who, together withBesse (see, for example, Besse and Ramsay, 1986), have been largely responsiblefor bringing these ideas to the attention of statisticians. We drawheavily on this book in the present section. However, the ideas of PCAin a continuous domain have also been developed in other fields, such assignal processing, and the topic is an active research area. The terminology‘Karhunen-Loève expansion’ is in common use in some disciplines to denotePCA in a continuous domain. Diamantaras and Kung (1996, Section 3.2)extend the terminology to cover the case where the data are discrete timeseries with a theoretically infinite number of time points.In atmospheric science the Karhunen-Loève expansion has been used inthe case where the continuum is spatial, and the different observationscorrespond to different discrete times. Preisendorfer and Mobley (1988,Section 2d) give a thorough discussion of this case and cite a number ofearlier references dating back to Obukhov (1947). Bouhaddou et al. (1987)independently consider a spatial context for what they refer to as ‘principalcomponent analysis of a stochastic process,’ but which is PCA for a (twodimensionalspatial) continuum of variables. They use their approach toapproximate both a spatial covariance function and the underlying spatialstochastic process, and compare it with what they regard as the less flexiblealternative of kriging. Guttorp and Sampson (1994) discuss similar ideasin a wider review of methods for estimating spatial covariance matrices.Durbin and Knott (1972) derive a special case of functional PCA in thecontext of goodness-of-fit testing (see Section 14.6.2).12.3.1 The Basics of Functional PCA (FPCA)When data are functions, our usual data structure x ij ,i=1, 2,...,n; j =1, 2,...,pis replaced by x i (t), i=1, 2,...,n where t is continuous in someinterval. We assume, as elsewhere in the book, that the data are centred,so that a mean curve ¯x = 1 ∑ nn i=1 ˜x i(t) has been subtracted from each ofthe original curves ˜x i (t). Linear functions of the curves are now integralsinstead of sums, that is z i = ∫ a(t)x i (t)dt rather than z i = ∑ pj=1 a jx ij .In‘ordinary’ PCA the first PC has weights a 11 ,a 21 ,...,a p1 , which maximizethe sample variance var(z i ), subject to ∑ pj=1 a2 j1 = 1. Because the data arecentred,var(z i1 )= 1n − 1n∑i=1z 2 i1 = 1n − 1n∑ [ ∑p ] 2.a j1 x iji=1j=1Analogously for curves, we find a 1 (t) which maximizes1n∑zi1 2 = 1 n∑[∫2a 1 (t)x i (t) dt],n − 1 n − 1i=1i=1
318 12. PCA for Time Series and Other Non-Independent Datasubject to ∫ a 2 1(t)dt = 1. Here, and elsewhere, the integral is over the rangeof values of t for which the data are observed. Subsequent FPCs are definedsuccessively, as with ordinary PCA, to maximisevar(z ik )= 1n − 1n∑[∫2a k (t)x i (t) dt],i=1subject to ∫ a 2 k (t)dt =1;∫ a k (t)a h (t)dt =0,k =2, 3,...; h =1, 2,...;h
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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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10.4. Robust Estimation of Principa
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Page 300 and 301: 11Rotation and Interpretation ofPri
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Page 332 and 333: 12.1. Introduction 301series is alm
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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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