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

5.2. Principal Coordinate Analysis 89structed in the first stage of the principal coordinate analysis are∆ 2 hi = t hh + t ii − 2t hip∑ p∑= x 2 hj + x 2 ij − 2=j=1j=1p∑(x hj − x ij ) 2j=1= d 2 hi,p∑x hj x ijthe Euclidean distance between the observations using the original p variables.As before, the PCA in the second stage of principal coordinateanalysis gives the same results as a PCA on the original data. Note, however,that XX ′ is not a very obvious similarity matrix. For a ‘covariancematrix’ between observations it is more natural to use a row-centred, ratherthan column-centred, version of X.Even in cases where PCA and principal coordinate analysis give equivalenttwo-dimensional plots, there is a difference, namely that in principalcoordinate analysis there are no vectors of coefficients defining the axesin terms of the original variables. This means that, unlike PCA, theaxes in principal coordinate analysis cannot be interpreted, unless thecorresponding PCA is also done.The equivalence between PCA and principal coordinate analysis in thecircumstances described above is termed a duality between the two techniquesby Gower (1966). The techniques are dual in the sense that PCAoperates on a matrix of similarities between variables, whereas principal coordinateanalysis operates on a matrix of similarities between observations(individuals), but both can lead to equivalent results.To summarize, principal coordinate analysis gives a low-dimensional representationof data when the data are given in the form of a similarity ordissimilarity matrix. As it can be used with any form of similarity or dissimilaritymatrix, it is, in one sense, ‘more powerful than,’ and ‘extends,’ PCA(Gower, 1967). However, as will be seen in subsequent chapters, PCA hasmany uses other than representing data graphically, which is the overridingpurpose of principal coordinate analysis.Except in the special cases discussed above, principal coordinate analysishas no direct relationship with PCA, so no examples will be given of thegeneral application of the technique. In the case where principal coordinateanalysis gives an equivalent representation to that of PCA, nothing newwould be demonstrated by giving additional examples. The examples givenin Section 5.1 (and elsewhere) which are presented as plots with respectto the first two PCs are, in fact, equivalent to two-dimensional principalcoordinate plots if the ‘dissimilarity’ between observations h and i isproportional to the Euclidean squared distance between the hth and ithobservations in p dimensions.j=1