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

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12.2. PCA and Atmospheric Time Series 311gether with the variance accounted for by each PC. Figure 12.5 displayssimilar plots for the real and imaginary parts of the first Hilbert EOF (labelledCEOFs on the plots). It can be seen that the real part of the firstHilbert EOF in Figure 12.5 looks similar to EOF1 in Figure 12.4. There arealso similarities between EOF3 and the imaginary part of the first HilbertEOF.Figures 12.6, 12.7 show plots of time series (scores), labelled temporalcoefficients in 12.7, for the first and third ordinary PCs, and the real andimaginary parts of the first HEOF, respectively. The similarity betweenthe first PC and the real part of the first HEOF is obvious. Both representthe same oscillatory behaviour in the series. The imaginary part of thefirst HEOF is also very similar, but lagged by π 2, whereas the scores on thethird EOF show a somewhat smoother and more regular oscillation. Cai andBaines (2001) note that the main oscillations visible in Figures 12.6, 12.7can be identified with well-known El Niño-Southern Oscillation (ENSO)events. They also discuss other physical interpretations of the results ofthe HEOF analysis relating them to other meteorological variables, andthey provide tests for statistical significance of HEOFs.The main advantage of HEOF analysis over ordinary PCA is its abilityto identify and reconstruct propagating waves, whereas PCA only findsstanding oscillations. For the first and second Hilbert EOFs, and for theirsum, this propagating behaviour is illustrated in Figure 12.8 by the movementof similar-valued coefficients from west to east as time progresses inthe vertical direction.12.2.4 Multitaper Frequency Domain-Singular ValueDecomposition (MTM SVD)In a lengthy paper, Mann and Park (1999) describe MTM-SVD, developedearlier by the same authors. It combines multitaper spectrum estimationmethods (MTM) with PCA using the singular value decomposition (SVD).The paper also gives a critique of several of the other techniques discussed inthe present section, together with frequency domain PCA which is coveredin Section 12.4.1. Mann and Park (1999) provide detailed examples, bothreal and artificial, in which MTM-SVD is implemented.Like MSSA, POP analysis, HEOF analysis and frequency domain PCA,MTM-SVD looks for oscillatory behaviour in space and time. It is closestin form to PCA of the spectral matrix F(λ) (Section 12.4.1), as it operatesin the frequency domain. However, in transferring from the time domainto the frequency domain MTM-SVD constructs a set of different taperedFourier transforms (hence the ‘multitaper’ in its name). The frequency domainmatrix is then subjected to a singular value decomposition, giving‘spatial’ or ‘spectral’ EOFs, and ‘principal modulations’ which are analogousto principal component scores. Mann and Park (1999) state that the
312 12. PCA for Time Series and Other Non-Independent DataFigure 12.4. The first four EOFs for Southern Hemisphere SST.Figure 12.5. Real and imaginary parts of the first Hilbert EOF for SouthernHemisphere SST.
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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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Page 294 and 295: 10.4. Robust Estimation of Principa
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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13.5. Common Principal Components 3
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13.6. Principal Component Analysis
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13.6. Principal Component Analysis
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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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