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

5.2. Principal Coordinate Analysis 87But from (5.2.2), the (h, i)th element of T can be writtenn∑t hi = c hj c ij , h,i =1, 2,...,n,soj=1∆ 2 hi = t hh + t ii − 2t hi.Principal coordinate analysis then attempts to find the ‘best-fitting’ q-dimensional (q < n) approximation to the n-dimensional representationdefined above. ‘Best-fitting’ is defined here in the same way as in the geometricdefinition of PCA (Property G3 of Section 3.2), so that ‘principalcomponents’ are now found for the n ‘observations’ defined in n dimensionsby the coordinates c ij .Aq-dimensional principal coordinate representationis then given by plotting the coordinates of the observations with respectto the first q ‘PCs’. Principal coordinate analysis therefore consists of twostages, both of which involve finding eigenvalues and eigenvectors of (n×n)matrices:(i) Find the eigenvectors c 1 , c 2 ,...,c n of T, normalized to have lengthsequal to their respective eigenvalues, and represent the n observationsas points in n-dimensional space with coordinate c ij for the ithobservation in the jth dimension.(ii) Find the PCs for the ‘data set’ in n dimensions defined in (i), andcalculate coordinates of the n observations with respect to the first qPCs.If the vectors c j defined in the first stage have ∑ ni=1 c ij = 0 then thecovariance matrix that is calculated in stage (ii) will be proportional toC ′ C where C is the (n × n) matrix with jth column c j , j =1, 2,...,n.Butc ′ jc k ={τj j = k0 j ≠ k ,as the eigenvectors in the spectral decomposition (5.2.1) have the property{b ′ 1 j = kjb k =0 j ≠ kandc j = τ 1/2j b j , j =1, 2,...,n.The matrix C ′ C is therefore diagonal with diagonal elements τ j ,j =1, 2,...,n, so that the first q principal coordinates of the n observationsare∑simply the values of c ij for i =1, 2,...,n; j =1, 2,...,q. Thus whenni=1 c ij = 0, stage (ii) is unnecessary.In general, although a similarity matrix T need not lead to ∑ ni=1 c ij =0, this property can be readily achieved by replacing T by an adjusted