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

14Generalizations and Adaptations ofPrincipal Component AnalysisThe basic technique of PCA has been generalized or adapted in many ways,and some have already been discussed, in particular in Chapter 13 whereadaptations for special types of data were described. This final chapter discussesa number of additional generalizations and modifications; for severalof them the discussion is very brief in comparison to the large amount ofmaterial that has appeared in the literature.Sections 14.1 and 14.2 present some definitions of ‘non-linear PCA’ and‘generalized PCA,’ respectively. In both cases there are connections withcorrespondence analysis, which was discussed at somewhat greater lengthin Section 13.1. Non-linear extensions of PCA (Section 14.1) include theGifi approach, principal curves, and some types of neural network, whilethe generalizations of Section 14.2 cover many varieties of weights, metrics,transformations and centerings.Section 14.3 describes modifications of PCA that may be useful whensecondary or ‘instrumental’ variables are present, and in Section 14.4 somepossible alternatives to PCA for data that are are non-normal are discussed.These include independent component analysis (ICA).Section 14.5 introduces the ideas of three-mode and multiway PCA.These analyses are appropriate when the data matrix, as well as havingtwo dimensions corresponding to individuals and variables, respectively,has one or more extra dimensions corresponding, for example, to time.The penultimate miscellaneous section (14.6) collects together someideas from neural networks and goodness-of-fit, and presents some othermodifications of PCA. The chapter ends with a few concluding remarks.