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

12.2. PCA and Atmospheric Time Series 309mixture of real and complex eigenvectors. The latter occur in pairs, witheach pair sharing the same eigenvalue and having eigenvectors that arecomplex conjugate pairs. The real eigenvectors describe non-oscillatory,non-propagating damped patterns, but the complex eigenvectors representdamped oscillations and can include standing waves and/or spatiallypropagating waves, depending on the relative magnitudes of the real andimaginary parts of each complex POP (von Storch et al., 1988).As with many other techniques, the data may be pre-processed usingPCA, with x in equation (12.2.1) replaced by its PCs. The description ofPOP analysis in Wu et al. (1994) includes this initial step, which providesadditional insights.Kooperberg and O’Sullivan (1996) introduce and illustrate a techniquewhich they describe as a hybrid of PCA and POP analysis. The analogousquantities to POPs resulting from the technique are called PredictiveOscillation Patterns (PROPs). In their model, x t is written as a lineartransformation of a set of underlying ‘forcing functions,’ which in turn arelinear functions of x t . Kooperberg and O’Sullivan (1996) find an expressionfor an upper bound for forecast errors in their model, and PROP analysisminimizes this quantity. The criterion is such that it simultaneouslyattempts to account for as much as possible of x t , as with PCA, and toreproduce as well as possible the temporal dependence in x t ,asinPOPanalysis.In an earlier technical report, Kooperberg and O’Sullivan (1994) mentionthe possible use of canonical correlation analysis (CCA; see Section 9.3) ina time series context. Their suggestion is that a second group of variablesis created by shifting the usual measurements at p locations by one timeperiod. CCA is then used to find relationships between the original andtime-lagged sets of variables.12.2.3 Hilbert (Complex) EOFsThere is some confusion in the literature over the terminology ‘complexEOFs,’ or ‘complex PCA.’ It is perfectly possible to perform PCA on complexnumbers, as well as real numbers, whether or not the measurements aremade over time. We return to this general version of complex PCA in Section13.8. Within the time series context, and especially for meteorologicaltime series, the term ‘complex EOFs’ has come to refer to a special type ofcomplex series. To reduce confusion, von Storch and Zwiers (1999) suggest(for reasons that will soon become apparent) referring to this procedureas Hilbert EOF analysis. We will follow this recommendation. Baines hassuggested removing ambiguity entirely by denoting the analysis as ‘complexHilbert EOF analysis.’ The technique seems to have originated withRasmusson et al. (1981), by whom it was referred to as Hilbert singulardecomposition.