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

12.4. PCA and Non-Independent Data—Some Additional Topics 333Underlying the answer to this question is the fact that Σ e needs tobe estimated, usually from a ‘control run’ of the atmospheric model inwhich no changes are made to the potential causal variables. Because thedimension ml of Σ e is large, it may be nearly singular, and estimation ofΣ −1e , which is required in calculating ŵ, will be unstable. To avoid thisinstability, the estimate of Σ e is replaced by the first few terms in itsspectral decomposition (Property A3 of Section 2.1), that is by using itsfirst few PCs. Allen and Tett (1999) discuss the choice of how many PCsto retain in this context. ‘First few’ is perhaps not quite the right phrase insome of these very large problems; North and Wu (2001) suggest keepingup to 500 out of 3600 in one of their studies.Expressing the optimal fingerprint in terms of the PCs of the estimateof Σ e also has advantages in interpretation, as in principal componentregression (Chapter 8), because of the uncorrelatedness of the PCs (Zwiers,1999, equations (13), (14)).12.4.4 Spatial DataMany of the techniques discussed earlier in the chapter have a spatial dimension.In Section 12.2, different points in space mostly correspond todifferent ‘variables.’ For MSSA, variables correspond to combinations oftime lag and spatial position, and Plaut and Vautard (1994) refer to theoutput of MSSA as ‘space-time EOFs.’ North and Wu (2001) use the sameterm in a different situation where variables also correspond to space-timecombinations, but with time in separated rather than overlapping blocks.The possibility was also raised in Section 12.3 that the continuum of variablesunderlying the curves could be in space rather than time. The exampleof Section 12.3.3 is a special case of this in which space is one-dimensional,while Bouhaddou et al. (1987) and Guttorp and Sampson (1994) discussestimation of continuous (two-dimensional) spatial covariances. ExtendedEOF analysis (see Sections 12.2.1, 14.5) has several variables measuredat each spatial location, but each space × variable combination is treatedas a separate ‘variable’ in this type of analysis. Another example in whichvariables correspond to location is described by Boyles (1996). Here a qualitycontrol regime includes measurements taken on a sample of n partsfrom a production line. The measurements are made at p locations forminga lattice on each manufactured part. We return to this example inSection 13.7.In some situations where several variables are measured at each of nspatial locations, the different spatial locations correspond to different observationsrather than variables, but the observations are typically notindependent. Suppose that a vector x of p variables is measured at each ofn spatial locations. Let the covariance between the elements x j and x k ofx for two observations which are separated in space by a vector h be the(j, k)th element of a matrix Σ h . This expression assumes second order sta-