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

310 12. PCA for Time Series and Other Non-Independent DataSuppose that x t , t =1, 2,...,n is a p-variate time series, and lety t = x t + ix H t , (12.2.2)where i = √ −1andx H t is the Hilbert transform of x t , defined as∞∑x H 2t =(2s +1)π (x (t+2s+1) − x (t−2s−1) ).s=0The definition assumes that x t is observed at an infinite number of times t =...,−1, 0, 1, 2,.... Estimation of x H t for finite samples, to use in equation(12.2.2), is discussed by von Storch and Zwiers (1999, Section 16.2.4) andBloomfield and Davis (1994).If a series is made up of oscillatory terms, its Hilbert transform advanceseach oscillatory term by π 2 radians. When x t is comprised of asingle periodic oscillation, x H t is identical to x t , except that it is shiftedby π 2radians. In the more usual case, where x t consists of a mixture oftwo or more oscillations or pseudo-oscillations at different frequencies, theeffect of transforming to x H t is more complex because the phase shift of π 2is implemented separately for each frequency.A Hilbert EOF (HEOF) analysis is simply a PCA based on the covariancematrix of y t defined in (12.2.2). As with (M)SSA and POP analysis, HEOFanalysis will find dominant oscillatory patterns, which may or may not bepropagating in space, that are present in a standard meteorological dataset of p spatial locations and n time points.Similarly to POP analysis, the eigenvalues and eigenvectors (HEOFs) arecomplex, but for a different reason. Here a covariance matrix is analysed,unlike POP analysis, but the variables from which the covariance matrix isformed are complex-valued. Other differences exist between POP analysisand HEOF analysis, despite the similarities in the oscillatory structuresthey can detect, and these are noted by von Storch and Zwiers (1999,Section 15.1.7). HEOF analysis maximizes variances, has orthogonal componentscores and is empirically based, all attributes shared with PCA.In direct contrast, POP analysis does not maximize variance, has nonorthogonalPOP coefficients (scores) and is model-based. As in ordinaryPCA, HEOFs may be simplified by rotation (Section 11.1), and Bloomfieldand Davis (1994) discuss how this can be done.There is a connection between HEOF analysis and PCA in the frequencydomain which is discussed in Section 12.4.1. An example of HEOF analysisis now given.Southern Hemisphere Sea Surface TemperatureThis example and its figures are taken, with permission, from Cai andBaines (2001). The data are sea surface temperatures (SSTs) in the SouthernHemisphere. Figure 12.4 gives a shaded contour map of the coefficientsin the first four (ordinary) PCs for these data (the first four EOFs), to-