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

374 14. Generalizations and Adaptations of Principal Component Analysis14.1 Non-Linear Extensions of PrincipalComponent AnalysisOne way of introducing non-linearity into PCA is what Gnanadesikan(1977) calls ‘generalized PCA.’ This extends the vector of pvariables x to include functions of the elements of x. For example,if p = 2, so x ′ = (x 1 ,x 2 ), we could consider linear functions ofx ′ + = (x 1 ,x 2 ,x 2 1,x 2 2,x 1 x 2 ) that have maximum variance, rather thanrestricting attention to linear functions of x ′ . In theory, any functionsg 1 (x 1 ,x 2 ,...,x p ),g 2 (x 1 ,x 2 ,...,x p ),...,g h (x 1 ,x 2 ,...,x p )ofx 1 ,x 2 ,...,x pcould be added to the original vector x, in order to construct an extendedvector x + whose PCs are then found. In practice, however, Gnanadesikan(1977) concentrates on quadratic functions, so that the analysis is a procedurefor finding quadratic rather than linear functions of x that maximizevariance.An obvious alternative to Gnanadesikan’s (1977) proposal is to replacex by a function of x, rather than add to x as in Gnanadesikan’s analysis.Transforming x in this way might be appropriate, for example, if we areinterested in products of powers of the elements of x. In this case, taking logarithmsof the elements and doing a PCA on the transformed data providesa suitable analysis. Another possible use of transforming to non-linear PCsis to detect near-constant, non-linear relationships between the variables. Ifan appropriate transformation is made, such relationships will be detectedby the last few PCs of the transformed data. Transforming the data is suggestedbefore doing a PCA for allometric data (see Section 13.2) and forcompositional data (Section 13.3). Kazmierczak (1985) also advocates logarithmictransformation followed by double-centering (see Section 14.2.3)for data in which it is important for a PCA to be invariant to changes inthe units of measurement and to the choice of which measurement is usedas a ‘reference.’ However, as noted in the introduction to Chapter 4, transformationof variables should only be undertaken, in general, after carefulthought about whether it is appropriate for the data set at hand.14.1.1 Non-Linear Multivariate Data Analysis—Gifi andRelated ApproachesThe most extensively developed form of non-linear multivariate data analysisin general, and non-linear PCA in particular, is probably the Gifi(1990) approach. ‘Albert Gifi’ is the nomdeplumeof the members of theDepartment of Data Theory at the University of Leiden. As well as the1990 book, the Gifi contributors have published widely on their systemof multivariate analysis since the 1970s, mostly under their own names.Much of it is not easy reading. Here we attempt only to outline the approach.A rather longer, accessible, description is provided by Krzanowski