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

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