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

308 12. PCA for Time Series and Other Non-Independent Data(2000), who give an example of the technique for five variables, and comparethe results to those of separate EEOFs (MSSAs) for each variable. Moteand coworkers note that it is possible that some of the dominant MEEOFpatterns may not be dominant in any of the individual EEOF analyses,and this may viewed as a disadvantage of the method. On the other hand,MEEOF analysis has the advantage of showing directly the connectionsbetween patterns for the different variables. Discussion of the properties ofMSSA and MEEOF analysis is ongoing (in addition to Mote et al. (2000),see Monahan et al. (1999), for example). Compagnucci et al. (2001) proposeyet another variation on the same theme. In their analysis, the PCA is doneon the transpose of the matrix used in MSSA, a so-called T-mode instead ofS-mode analysis (see Section 14.5). Compagnucci et al. call their techniqueprincipal sequence pattern analysis.12.2.2 Principal Oscillation Pattern (POP) AnalysisSSA, MSSA, and other techniques described in this chapter can be viewedas special cases of PCA, once the variables have been defined in a suitableway. With the chosen definition of the variables, the procedures performan eigenanalysis of a covariance matrix. POP analysis is different, but itis described briefly here because its results are used for similar purposesto those of some of the PCA-based techniques for time series includedelsewhere in the chapter. Furthermore its core is an eigenanalysis, albeitnot on a covariance matrix.POP analysis was introduced by Hasselman (1988). Suppose that we havethe usual (n×p) matrix of measurements on a meteorological variable, takenat n time points and p spatial locations. POP analysis has an underlyingassumption that the p time series can be modelled as a multivariate firstorderautoregressive process. If x ′ t is the tth row of the data matrix, wehave(x (t+1) − µ) =Υ(x t − µ)+ɛ t , t =1, 2,...,(n − 1), (12.2.1)where Υ is a (p × p) matrix of constants, µ is a vector of means for the pvariables and ɛ t is a multivariate white noise term. Standard results frommultivariate regression analysis (Mardia et al., 1979, Chapter 6) lead to estimationof Υ by ˆΥ = S 1 S −10 , where S 0 is the usual sample covariance matrixfor the p variables, and S 1 has (i, j)th element equal to the sample covariancebetween the ith and jth variables at lag 1. POP analysis then finds theeigenvalues and eigenvectors of ˆΥ. The eigenvectors are known as principaloscillation patterns (POPs) and denoted p 1 , p 2 ,...,p p . The quantitiesz t1 ,z t2 ,...,z tp which can be used to reconstitute x t as ∑ pk=1 z tkp k arecalled the POP coefficients. They play a similar rôle in POP analysis tothat of PC scores in PCA.One obvious question is why this technique is called principal oscillationpattern analysis. Because ˆΥ is not symmetric it typically has a