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

14.2. Weights, Metrics, Transformations and Centerings 391clear which of ‘sites’ and ‘species’ should be treated as ‘variables’ and whichas ‘observations.’ Another possibility is to centre with respect to sites, butnot species, in other words, carrying out an analysis with sites rather thanspecies as the variables. Buckland and Anderson (1985) analyse their datain this way.Yet another technique which has been suggested for analysing sometypes of site-species data is correspondence analysis (see, for example, Section5.4.1 and Gauch, 1982). As pointed out in Section 13.4, correspondenceanalysis has some similarity to Mandel’s approach, and hence to doublycentred PCA. In doubly centred PCA we analyse the residuals from anadditive model for row and column (site and species) effects, whereas incorrespondence analysis the residuals from a multiplicative (independence)model are considered.Both uncentred and doubly centred PCA perform eigenanalyses on matriceswhose elements are not covariances or correlations, but which canstill be viewed as measures of similarity or association between pairs ofvariables. Another technique in the same vein is proposed by Elmore andRichman (2001). Their idea is to find ‘distances’ between variables whichcan then be converted into similarities and an eigenanalysis done on theresulting similarity matrix. Although Elmore and Richman (2001) notea number of possible distance measures, they concentrate on Euclideandistance, so that the distance d jk between variables j and k is[ ∑ n(x ij − x ik ) 2] 1 2.i=1If D is largest of the p 2 d jk , the corresponding similarity matrix is definedto have elementss jk =1− d jkD .The procedure is referred to as PCA based on ES (Euclidean similarity).There is an apparent connection with principal coordinate analysis (Section5.2) but for ES-based PCA it is distances between variables, ratherthan between observations, that are analysed.The technique is only appropriate if the variables are all measured in thesame units—it makes no sense to compute a distance between a vector oftemperatures and a vector of heights, for example. Elmore and Richman(2001) report that the method does better at finding known ‘modes’ in adata set than PCA based on either a covariance or a correlation matrix.However, as with uncentred and doubly centred PCA, it is much less clearthan it is for PCA what is optimized by the technique, and hence it is moredifficult to know how to interpret its results.