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

12.2. PCA and Atmospheric Time Series 315April than in June. Similarly, the joint distribution for April and August,which have a four-month separation, may be the same in different years,but different from the joint distribution for June and October, even thoughthe latter are also separated by four months. Such behaviour is known ascyclo-stationarity, and both PCA and POP analysis have been modifiedfor such data.The modification is easier to explain for POP analysis than for PCA.Suppose that τ is the length of the cycle; for example τ = 12 for monthlydata and an annual cycle. Then for a time series of length n = n ′ τ equation(12.2.1) is replaced by(x (sτ+t+1) − µ (t+1) )=Υ t (x (sτ+t) − µ t )+ɛ (sτ+t) , (12.2.3)t =0, 1,...,(τ − 2); s =0, 1, 2,...,(n ′ − 1),with (t + 1) replaced by 0, and s replaced by (s + 1) on the left-hand sideof (12.2.3) when t =(τ − 1) and s =0, 1, 2,...,(n ′ − 2). Here the mean µ tand the matrix Υ t can vary within cycles, but not between cycles. CyclostationaryPOP analysis estimates Υ 0 , Υ 1 ,...,Υ (τ−1) and is based on aneigenanalysis of the product of those estimates ˆΥ 0 ˆΥ1 ... ˆΥ (τ−1) .The cyclo-stationary variety of PCA is summarized by Kim and Wu(1999) but is less transparent in its justification than cyclo-stationaryPOP analysis. First, vectors a 0t , a 1t ,...,a (τ−1)t are found such that x t =∑ τ−1j=0 a jte 2πijtτ, and a new vector of variables is then constructed byconcatenating a 0t , a 1t ,...,a (τ−1)t . Cyclo-stationary EOFs (CSEOFs) areobtained as eigenvectors of the covariance matrix formed from this vectorof variables. Kim and Wu (1999) give examples of the technique, and alsogive references explaining how to calculate CSEOFs.Kim and Wu (1999) describe an additional modification of PCA thatdeals with periodicity in a time series, and which they call a periodicallyextended EOF technique. It works by dividing a series of length n = n ′ τinto n ′ blocks of length τ. A covariance matrix, S EX , is then computed as⎡⎢⎣S EX11 S EX12 ··· S EX1n ′S EX21 S EX22 ··· S EX2n ′. ..S EXn ′ 1.S EXn ′ 2 ··· S EXn ′ n ′in which the (i, j)th element of S EXklis the sample covariance betweenthe measurements at the ith location at time k in each block and at thejth location at time l in the same block. The description of the techniquein Kim and Wu (1999) is sketchy, but it appears that whereas in ordinaryPCA, covariances are calculated by averaging over all time points, here theaveraging is done over times at the same point within each block acrossblocks. A similar averaging is implicit in cyclo-stationary POP analysis.Examples of periodically extended EOFs are given by Kim and Wu (1999).⎤⎥⎦ ,