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

12.2. PCA and Atmospheric Time Series 303Some time series have cyclic behaviour with fixed periods, such as anannual or diurnal cycle. Modifications of POP analysis and PCA that takesuch cycles into account are discussed in Section 12.2.5. A brief discussionof examples and comparative studies is given in Section 12.2.6.12.2.1 Singular Spectrum Analysis (SSA)Like a number of other statistical techniques, SSA appears to have been‘invented’ independently in a number of different fields. Elsner and Tsonis(1996) give references from the 1970s and 1980s from chaos theory, biologicaloceanography and signal processing, and the same idea is described inthe statistical literature by Basilevsky and Hum (1979), where it is referredto as the ‘Karhunen-Loève method,’ a term more often reserved for continuoustime series (see Section 12.3). Other names for the technique are‘Pisarenko’s method’ (Smyth, 2000) and ‘singular systems analysis’ (vonStorch and Zwiers, 1999, Section 13.6); fortunately the latter has the sameacronym as the most popular name. A comprehensive coverage of SSA isgiven by Golyandina et al. (2001).The basic idea in SSA is simple: a principal component analysis is donewith the variables analysed being lagged versions of a single time seriesvariable. More specifically, our p variables are x t ,x (t+1) ,...,x (t+p−1) and,assuming the time series is stationary, their covariance matrix is such thatthe (i, j)th element depends only on |i − j|. Such matrices are knownas Töplitz matrices. In the present case the (i, j)th element is the autocovarianceγ |i−j| . Because of the simple structure of Töplitz matrices,the behaviour of the first few PCs, and their corresponding eigenvaluesand eigenvectors (the EOFs) which are trigonometric functions (Brillinger,1981, Section 3.7), can be deduced for various types of time series structure.The PCs are moving averages of the time series, with the EOFs providingthe weights in the moving averages.Töplitz matrices also occur when the p-element random vector x consistsof non-overlapping blocks of p consecutive values of a single time series. Ifthe time series is stationary, the covariance matrix Σ for x has Töplitzstructure, with the well-known pattern of trigonometric functions for itseigenvalues and eigenvectors. Craddock (1965) performed an analysis ofthis type on monthly mean temperatures for central England for the periodNovember 1680 to October 1963. The p (= 12) elements of x are meantemperatures for the 12 months of a particular year, where a ‘year’ startsin November. There is some dependence between different values of x, butit is weaker than that between elements within a particular x; between-yearcorrelation was minimized by starting each year at November, when therewas apparently evidence of very little continuity in atmospheric behaviourfor these data. The sample covariance matrix does not, of course, haveexact Töplitz structure, but several of the eigenvectors have approximatelythe form expected for such matrices.