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

14.2. Weights, Metrics, Transformations and Centerings 389Standardization, in the sense of dividing each column of the data matrixby its standard deviation, leads to PCA based on the correlation matrix,and its pros and cons are discussed in Sections 2.3 and 3.3. This can bethought of a version of weighted PCA (Section 14.2.1). So, also, can dividingeach column by its range or its mean (Gower, 1966), in the latter casegiving a matrix of coefficients of variation. Underhill (1990) suggests abiplot based on this matrix (see Section 5.3.2). Such plots are only relevantwhen variables are non-negative, as with species abundance data.Principal components are linear functions of x whose coefficients aregiven by the eigenvectors of a covariance or correlation matrix or, equivalently,the eigenvectors of a matrix X ′ X. Here X is a (n × p) matrix whose(i, j)th element is the value for the ith observation of the jth variable,measured about the mean for that variable. Thus, the columns of X havebeen centred, so that the sum of each column is zero, though Holmes-Junca(1985) notes that centering by either medians or modes has been suggestedas an alternative to centering by means.Two alternatives to ‘column-centering’ are:(i) the columns of X are left uncentred, that is x ij is now the value forthe ith observation of the jth variable, as originally measured;(ii) both rows and columns of X are centred, so that sums of rows, as wellas sums of columns, are zero.In either (i) or (ii) the analysis now proceeds by looking at linear functionsof x whose coefficients are the eigenvectors of X ′ X, with X nownon-centred or doubly centred. Of course, these linear functions no longermaximize variance, and so are not PCs according to the usual definition,but it is convenient to refer to them as non-centred and doubly centredPCs, respectively.Non-centred PCA is a fairly well-established technique in ecology (TerBraak,1983). It has also been used in chemistry (Jackson, 1991, Section3.4; Cochran and Horne, 1977) and geology (Reyment and Jöreskog, 1993).As noted by Ter Braak (1983), the technique projects observations ontothe best fitting plane (or flat) through the origin, rather than through thecentroid of the data set. If the data are such that the origin is an importantpoint of reference, then this type of analysis can be relevant. However, if thecentre of the observations is a long way from the origin, then the first ‘PC’will dominate the analysis, and will simply reflect the position of the centroid.For data that consist of counts of a number of biological species (thevariables) at various sites (the observations), Ter Braak (1983) claims thatnon-centred PCA is better than standard (centred) PCA at simultaneouslyrepresenting within-site diversity and between-site diversity of species (seealso Digby and Kempton (1987, Section 3.5.5)). Centred PCA is better atrepresenting between-site species diversity than non-centred PCA, but it ismore difficult to deduce within-site diversity from a centred PCA.