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

334 12. PCA for Time Series and Other Non-Independent Datationarity in the sense that the covariances do not depend on the location ofthe two observations, only on the vector joining them. On the other hand,it does not require isotropy—the covariance may depend on the directionof h as well as its length. The intrinsic correlation model (Wackernagel,1995, Chapter 22) assumes that Σ h = ρ h Σ. Because all terms in Σ aremultiplied by the same spatial factor, this factor cancels when correlationsare calculated from the covariances and the correlation between x j and x kdoes not depend on h. Wackernagel (1995, Chapter 22) suggests testingwhether or not this model holds by finding principal components based onsample covariance matrices for the p variables. Cross-covariances betweenthe resulting PCs are then found at different separations h. Fork ≠ l, thekth and lth PCs should be uncorrelated for different values of h under theintrinsic correlation model, because Σ h has the same eigenvectors for all h.An extension of the intrinsic correlation model to a ‘linear model of coregionalization’is noted by Wackernagel (1995, Chapter 24). In this modelthe variables are expressed as a sum of (S + 1) spatially uncorrelatedcomponents, and the covariance matrix now takes the formΣ h =S∑ρ uh Σ u .u=0Wackernagel (1995, Chapter 25) suggests that separate PCAs of the estimatesof the matrices Σ 0 , Σ 1 ,...,Σ S may be informative, but it is notclear how these matrices are estimated, except that they each representdifferent spatial scales.As well as dependence on h, the covariance or correlation between x j andx k may depend on the nature of the measurements made (point measurements,averages over an area where the area might be varied) and on the sizeof the domain. Vargas-Guzmán et al. (1999) discuss each of these aspects,but concentrate on the last. They describe a procedure they name growingscale PCA, in which the nature of the measurements is fixed and averaging(integration) takes place with respect to h, but the size of the domain isallowed to vary continuously. As it does, the PCs and their variances alsoevolve continuously. Vargas-Guzmán et al. illustrate this technique and alsothe linear model of coregionalization with a three-variable example. The basicform of the PCs is similar at all domain sizes and stabilizes as the sizeincreases, but there are visible changes for small domain sizes. The examplealso shows the changes that occur in the PCs for four different areal extentsof the individual measurements. Buell (1975) also discussed the dependenceof PCA on size and shape of spatial domains (see Section 11.4), but hisemphasis was on the shape of the domain and, unlike Vargas-Guzmán etal. (1999), he made no attempt to produce a continuum of PCs dependenton size.Kaplan et al. (2001) consider optimal interpolation and smoothing ofspatial field data that evolve in time. There is a basic first-order autore-