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

and a typical row of the matrix is12.2. PCA and Atmospheric Time Series 307x ′ i =(x i1 ,x (i+1)1 ,...,x (i+m−1)1 ,x i2 ,...,x (i+m−1)2 ,...,x (i+m−1)p ),i =1, 2,...,n ′ , where x ij is the value of the measured variable at the ithtime point and the jth spatial location, and m plays the same rôle in MSSAas p does in SSA. The covariance matrix for this data matrix has the form⎡⎤S 11 S 12 ··· S 1pS 21 S 22 ··· S 2p⎢⎣.⎥.⎦ ,S p1 S p2 ··· S ppwhere S kk is an (m × m) covariance matrix at various lags for the kthvariable (location), with the same structure as the covariance matrix inan SSA of that variable. The off-diagonal matrices S kl , k ≠ l, have(i, j)thelement equal to the covariance between locations k and l at time lag |i−j|.Plaut and Vautard (1994) claim that the ‘fundamental property’ of MSSAis its ability to detect oscillatory behaviour in the same manner as SSA, butrather than an oscillation of a single series the technique finds oscillatoryspatial patterns. Furthermore, it is capable of finding oscillations with thesame period but different spatially orthogonal patterns, and oscillationswith the same spatial pattern but different periods.The same problem of ascertaining ‘significance’ arises for MSSA as inSSA. Allen and Robertson (1996) tackle this problem in a similar mannerto that adopted by Allen and Smith (1996) for SSA. The null hypothesishere extends one-dimensional ‘red noise’ to a set of p independentAR(1) processes. A general multivariate AR(1) process is not appropriateas it can itself exhibit oscillatory behaviour, as exemplified in POP analysis(Section 12.2.2).MSSA extends SSA from one time series to several, but if the number oftime series p is large, it can become unmanageable. A solution, which is usedby Benzi et al. (1997), is to carry out PCA on the (n × p) data matrix, andthen implement SSA separately on the first few PCs. Alternatively for largep, MSSA is often performed on the first few PCs instead of the variablesthemselves, as in Plaut and Vautard (1994).Although MSSA is a natural extension of SSA, it is also equivalent toextended empirical orthogonal function (EEOF) analysis which was introducedindependently of SSA by Weare and Nasstrom (1982). Barnett andHasselmann (1979) give an even more general analysis, in which differentmeteorological variables, as well as or instead of different time lags, may beincluded at the various locations. When different variables replace differenttime lags, the temporal correlation in the data is no longer taken intoaccount, so further discussion is deferred to Section 14.5.The general technique, including both time lags and several variables,is referred to as multivariate EEOF (MEEOF) analysis by Mote et al.