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

342 13. Principal Component Analysis for Special Types of Datawhere V ′ ΩV = I, B ′ ΦB = I, Ω and Φ are (r × r) and(c × c) matrices,respectively, and M, I are diagonal and identity matrices whose dimensionsequal the rank of X (see Section 14.2.1). If Ω = D −1r , Φ = D −1c , where D r ,D c are diagonal matrices whose diagonal entries are the elements of r, c,respectively, then the columns of B define principal axes for the set of r ‘observations’given by the rows of X. Similarly, the columns of V define principalaxes for the set of c ‘observations’ given by the columns of X,andfromthe first q columns of B and V, respectively, we can derive the ‘coordinates’of the row and column profiles of N in q-dimensional space (see Greenacre,1984, p. 87) which are the end products of a correspondence analysis.A correspondence analysis is therefore based on a generalized SVD of X,and, as will be shown in Section 14.2.1, this is equivalent to an ‘ordinary’SVD of˜X = Ω 1/2 XΦ 1/2= D −1/2r XD −1/2c= D −1/2r( 1n N − rc′ )D −1/2c .The SVD of ˜X can be written˜X = WKC ′ (13.1.2)with V, M, B of (13.1.1) defined in terms of W, K, C of (13.1.2) asV = Ω −1/2 W, M = K, B = Φ −1/2 C.If we consider ˜X as a matrix of r observations on c variables, then the coefficientsof the PCs for ˜X are given in the columns of C, and the coordinates(scores) of the observations with respect to the PCs are given by the elementsof WK (see the discussion of the biplot with α = 1 in Section 5.3).Thus, the positions of the row points given by correspondence analysis arerescaled versions of the values of the PCs for the matrix ˜X. Similarly, thecolumn positions given by correspondence analysis are rescaled versions ofvalues of PCs for the matrix ˜X ′ , a matrix of c observations on r variables.In this sense, correspondence analysis can be thought of as a form of PCAfor a transformation ˜X of the original contingency table N (or a generalizedPCA for X; see Section 14.2.1).Because of the various optimality properties of PCs discussed in Chapters2 and 3, and also the fact that the SVD provides a sequence of‘best-fitting’ approximations to ˜X of rank 1, 2,... as defined by equation(3.5.4), it follows that correspondence analysis provides coordinatesfor rows and columns of N that give the best fit in a small number ofdimensions (usually two) to a transformed version ˜X of N. This ratherconvoluted definition of correspondence analysis demonstrates its connectionwith PCA, but there are a number of other definitions that turn outto be equivalent, as shown in Greenacre (1984, Chapter 4). In particular,