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

372 13. Principal Component Analysis for Special Types of Datathere may be a large number of zeros in the data. If two variables x jand x k simultaneously record zero for a non-trivial number of sites, thecalculation of covariance or correlation between this pair of variables islikely to be distorted. Legendre and Legendre (1983, p. 285) argue that dataare better analysed by nonmetric multidimensional scaling (Cox and Cox,2001) or with correspondence analysis (as in Section 5.4.1), rather than byPCA, when there are many such ‘double zeros’ present. Even when suchzeros are not a problem, species abundance data often have highly skeweddistributions and a transformation; for example, taking logarithms, may beadvisable before PCA is contemplated.Another unique aspect of species abundance data is an interest in thediversity of species at the various sites. It has been argued that to examinediversity, it is more appropriate to use uncentred than column-centredPCA. This is discussed further in Section 14.2.3, together with doublycentred PCA which has also found applications to species abundance data.Large Data SetsThe problems of large data sets are different depending on whether thenumber of observations n or the number of variables p is large, with thelatter typically causing greater difficulties than the former. With large nthere may be problems in viewing graphs because of superimposed observations,but it is the size of the covariance or correlation matrix that usuallydetermines computational limitations. However, if p > n it should beremembered (Property G4 of Section 3.2) that the eigenvectors of X ′ X correspondingto non-zero eigenvalues can be found from those of the smallermatrix XX ′ .For very large values of p, Preisendorfer and Mobley (1988, Chapter 11)suggest splitting the variables into subsets of manageable size, performingPCA on each subset, and then using the separate eigenanalyses to approximatethe eigenstructure of the original large data matrix. Developmentsin computer architecture may soon allow very large problems to be tackledmuch faster using neural network algorithms for PCA (see Appendix A1and Diamantaras and Kung (1996, Chapter 8)).