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

102 5. Graphical Representation of Data Using Principal Componentsof X ′ X, and hence to PCA. The variations discussed so far relate only tochoices within this classical plot, for example the choice of α in defining g iand h j (5.3.2), the possible rescaling by a factor (n − 1) 1/2 and the formof display (axes or arrowheads, concentration ellipses).Gower and Hand (1996) describe many other variations. In particular,they look at biplots related to multivariate techniques other than PCA,including multidimensional scaling, canonical variate analysis, correspondenceanalysis and multiple correspondence analysis. Gabriel (1995a,b) alsodiscusses biplots related to multivariate methods other than PCA, in particularmultiple correspondence analysis and MANOVA (multivariate analysisof variance).A key distinction drawn by Gower and Hand (1996) is between interpolationand prediction in a biplot. The former is concerned with determiningwhere in the diagram to place an observation, given its values on the measuredvariables. Prediction refers to estimating the values of these variables,given the position of an observation in the plot. Both are straightforwardfor classical biplots—gi∗ is used for interpolation and 2˜x ij for prediction—but become more complicated for other varieties of biplot. Gower and Hand(1996, Chapter 7) describe a framework for generalized biplots that includesmost other versions as special cases. One important special case is that ofnon-linear biplots. These will be discussed further in Section 14.1, whichdescribes a number of non-linear modifications of PCA. Similarly, discussionof robust biplots, due to Daigle and Rivest (1992), will be deferreduntil Section 10.4, which covers robust versions of PCA.The discussion and examples of the classical biplot given above use anunstandardized form of X and hence are related to covariance matrix PCA.As noted in Section 2.3 and elsewhere, it is more usual, and often moreappropriate, to base PCA on the correlation matrix as in the examplesof Section 5.3.1. Corresponding biplots can be derived from the SVD of˜X, the column-centred data matrix whose jth column has been scaled bydividing by the standard deviation of x j , j =1, 2,...,p. Many aspects ofthe biplot remain the same when the correlation, rather than covariance,matrix is used. The main difference is in the positions of the h j . Recall thatif α = 0 is chosen, together with the scaling factor (n−1) 1/2 , then the lengthh ∗′ jh ∗ j approximates the variance of x j. In the case of a correlation-basedanalysis, var(x j ) = 1 and the quality of the biplot approximation to thejth variable by the point representing h ∗ j can be judged by the closeness ofh ∗ j to the unit circle centred at the origin. For this reason, the unit circle issometimes drawn on correlation biplots to assist in evaluating the quality ofthe approximation (Besse, 1994a). Another property of correlation biplotsis that the squared distance between h j and h k is 2(1 − r jk ), where r jk isthe correlation between x j and x k . The squared distance between h ∗ j andh ∗ kapproximates this quantity.An alternative to the covariance and correlation biplots is the coefficientof variation biplot, due to Underhill (1990). As its name suggests, instead