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

12.2. PCA and Atmospheric Time Series 305Figure 12.1. Plots of loadings for the first two components in an SSA with p =61of the Southern Oscillation Index data.Southern Oscillation IndexThe data considered here are monthly values of the Southern OscillationIndex (SOI) for the years 1876–2000, produced by the Australian Bureauof Meteorology’s National Climate Centre. The number of observations inthe series is therefore n =12× 125 = 1500. The index is a measure ofthe East-West pressure gradient between Tahiti in the mid-Pacific Oceanand Darwin, Australia. It is a major source of climate variation. SSA wascarried on the data with p = 61, and Figure 12.1 gives a plot of the loadingsfor the first two EOFs. Their eigenvalues correspond to 13.7% and 13.4%of the total variation. The closeness of the eigenvalues suggests a quasioscillatorypattern, and this is clearly present in the loadings of Figure 12.1.Note, however, that the relationship between the two EOFs is not a simpledisplacement by π 2. Figure 12.2 shows time series plots of the scores for thefirst two components (PCs) in the SSA. These again reflect the oscillatorynature of the components. A reconstruction of the series using only the firsttwo PCs shown in Figure 12.3 captures some of the major features of theoriginal series, but a large amount of other variability remains, reflecting thefact that the two components only account for 27.1% of the total variation.Multichannel SSAIn multichannel SSA (MSSA) we have the more usual atmospheric scienceset-up of p spatial locations and n time points, but rather than findinga covariance matrix directly from the (n × p) data matrix, the data arerearranged into a larger (n ′ × p ′ ) matrix, where n ′ = n − m +1,p ′ = mp