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

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

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

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13.5. Common <strong>Principal</strong> <strong>Component</strong>s 355different but closely related species of animals, then the same general ‘size’and ‘shape’ components (see Section 13.2) may be present for each species,but with varying importance. Similarly, if the same variables are measuredon the same individuals but at different times, so that ‘groups’ correspondto different times as in longitudinal studies (see Section 12.4.2), then thecomponents may remain the same but their relative importance may varywith time.One way of formally expressing the presence of ‘common PCs’ as justdefined is by the hypothesis that there is an orthogonal matrix A thatsimultaneously diagonalizes all the Σ g so thatA ′ Σ g A = Λ g , (13.5.1)where Λ g , g =1, 2,...,G are all diagonal. The kth column of A gives thecoefficients of the kth common PC, and the (diagonal) elements of Λ g givethe variances of these PCs for the gth population. Note that the order ofthese variances need not be the same for all g, so that different PCs mayhave the largest variance in different populations.In a series of papers in the 1980s Flury developed ways of estimating andtesting the model implied by (13.5.1). Much of this work later appearedin a book (Flury, 1988), in which (13.5.1) is the middle level of a 5-levelhierarchy of models for a set of G covariance matrices. The levels are these:• Σ 1 = Σ 2 = ...= Σ G (equality).• Σ g = ρ g Σ 1 for some positive constants ρ 2 ,ρ 3 ,...,ρ g (proportionality).• A ′ Σ g A = Λ g (the common PC model).• Equation (13.5.1) can also be written, using spectral decompositions(2.1.10) of each covariance matrix, asΣ g = λ g1 α 1 α ′ 1 + λ g2 α 2 α ′ 2 + ...+ λ gp α p α ′ p.Level 4 (the partial common PC model) replaces this byΣ g = λ g1 α 1 α ′ 1+...+λ gq α q α ′ q+λ g(q+1) α (g) (g)q+1 α′ q+1 +...+λ gpα (g)p α ′ (g)p(13.5.2)Thus, q of the p PCs have common eigenvectors in the G groups,whereas the other (p − q) do not. The ordering of the components in(13.5.2) need not in general reflect the size of the eigenvalues. Anysubset of q components can be ‘common.’• No restriction on Σ 1 , Σ 2 ,...,Σ g .The first and last levels of the hierarchy are trivial, but Flury (1988)devotes a chapter to each of the three intermediate ones, covering maximumlikelihood estimation, asymptotic inference and applications. The partialcommon PC model is modified to give an additional level of the hierarchy

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