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

86 5. Graphical Representation of Data Using Principal Componentstween principal coordinate analysis and PCA that were noted by Gower.Like the more widely known non-metric multidimensional scaling (Kruskal,1964a,b), the technique starts with a matrix of similarities or dissimilaritiesbetween a set of observations, and aims to produce a low-dimensionalgraphical plot of the data in such a way that distances between points inthe plot are close to the original dissimilarities. There are numerous scalingtechniques; Cox and Cox (2001) provide a good overview of many of them.The starting point (an (n × n) matrix of (dis)similarities) of principalcoordinate analysis is different from that of PCA, which usually beginswith the (n × p) data matrix. However, in special cases discussed below thetwo techniques give precisely the same low-dimensional representation. Furthermore,PCA may be used to find starting configurations for the iterativealgorithms associated with non-metric multidimensional scaling (Davison,1983, Chapters 5, 6). Before showing the equivalences between PCA andprincipal coordinate analysis, we need first to describe principal coordinateanalysis in some detail.Suppose that T is an (n × n) positive-semidefinite symmetric matrix ofsimilarities among a set of n observations. (Note that it is fairly standardnotation to use A, rather than T, here. However, we have avoided theuse of A in this context, as it is consistently taken to be the matrix ofPC coefficients in the current text.) From the spectral decomposition of T(Property A3 of Sections 2.1 and 3.1 gives the spectral decomposition ofa covariance matrix, but the same idea is valid for any symmetric matrix)we haveT = τ 1 b 1 b ′ 1 + τ 2 b 2 b ′ 2 + ···+ τ n b n b ′ n, (5.2.1)where τ 1 ≥ τ 2 ≥ ··· ≥ τ n are the eigenvalues of T and b 1 , b 2 , ··· , b n arethe corresponding eigenvectors. Alternatively, this may be writtenwhereT = c 1 c ′ 1 + c 2 c ′ 2 + ···+ c n c ′ n, (5.2.2)c j = τ 1/2j b j , j =1, 2,...,n.Now consider the n observations as points in n-dimensional space withthe jth coordinate for the ith observation equal to c ij ,theith element ofc j . With this geometric interpretation of the n observations, the Euclideandistance between the hth and ith observations isn∑∆ 2 hi = (c hj − c ij ) 2=j=1n∑c 2 hj +j=1n∑n∑c 2 ij − 2 c hj c ij .j=1j=1