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

382 14. Generalizations and Adaptations of Principal Component Analysister 6), and a shorter description is given by Krzanowski and Marriott (1994,Chapter 8). The link between non-linear biplots and PCA is somewhattenuous, so we introduce them only briefly. Classical biplots are basedon the singular value decomposition of the data matrix X, and providea best possible rank 2 approximation to X in a least squares sense (Section3.5). The distances between observations in the 2-dimensional spaceof the biplot with α = 1 (see Section 5.3) give optimal approximations tothe corresponding Euclidean distances in p-dimensional space (Krzanowskiand Marriott, 1994). Non-linear biplots replace Euclidean distance byother distance functions. In plots thus produced the straight lines or arrowsrepresenting variables in the classical biplot are replaced by curvedtrajectories. Different trajectories are used to interpolate positions of observationson the plots and to predict values of the variables given theplotting position of an observation. Gower and Hand (1996) give examplesof interpolation biplot trajectories but state that they ‘do not yet have anexample of prediction nonlinear biplots.’Tenenbaum et al. (2000) describe an algorithm in which, as withnon-linear biplots, distances between observations other than Euclideandistance are used in a PCA-related procedure. Here so-called geodesic distancesare approximated by finding the shortest paths in a graph connectingthe observations to be analysed. These distances are then used as input towhat seems to be principal coordinate analysis, a technique which is relatedto PCA (see Section 5.2).14.2 Weights, Metrics, Transformations andCenteringsVarious authors have suggested ‘generalizations’ of PCA. We have met examplesof this in the direction of non-linearity in the previous section. Anumber of generalizations introduce weights or metrics on either observationsor variables or both. The related topics of weights and metrics makeup two of the three parts of the present section; the third is concerned withdifferent ways of transforming or centering the data.14.2.1 WeightsWe start with a definition of generalized PCA which was given by Greenacre(1984, Appendix A). It can viewed as introducing either weights or metricsinto the definition of PCA. Recall the singular value decomposition (SVD)of the (n × p) data matrix X defined in equation (3.5.1), namelyX = ULA ′ . (14.2.1)