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

14.2. Weights, Metrics, Transformations and Centerings 387M in (3.9.1)) is equal to Γ −1 (see Besse (1994b)). Futhermore, it can beshown (Besse, 1994b) that Γ −1 is approximately optimal even without theassumption of multivariate normality. Optimality is defined here as findingQ for which E[ 1 ∑ nn i=1 ‖z i − ẑ i ‖ 2 A ] is minimized, where A is any givenEuclidean metric. The matrix Q enters this expression because ẑ i is theQ-orthogonal projection of x i onto the optimal q-dimensional subspace.Of course, the model is often a fiction, and even when it might be believed,Γ will typically not be known. There are, however, certain types ofdata where plausible estimators exist for Γ. One is the case where the datafall into groups or clusters. If the groups are known, then within-groupvariation can be used to estimate Γ, and generalized PCA is equivalentto a form of discriminant analysis (Besse, 1994b). In the case of unknownclusters, Caussinus and Ruiz (1990) use a form of generalized PCA as aprojection pursuit technique to find such clusters (see Section 9.2.2). Anotherform of generalized PCA is used by the same authors to look foroutliers in a data set (Section 10.1).Besse (1988) searches for an ‘optimal’ metric in a less formal manner. Inthe context of fitting splines to functional data, he suggests several familiesof metric that combine elements of closeness between vectors with closenessbetween their smoothness. A family is indexed by a parameter playinga similar rôle to λ in equation (12.3.6), which governs smoothness. Theoptimal value of λ, and hence the optimal metric, is chosen to give themost clear-cut decision on how many PCs to retain.Thacker (1996) independently came up with a similar approach, whichhe refers to as metric-based PCA. He assumes that associated with a setof p variables x is a covariance matrix E for errors or uncertainties. If Sis the covariance matrix of x, then rather than finding a ′ x that maximizesa ′ Sa, it may be more relevant to maximize a′ Saa ′ Ea. This reduces to solvingthe eigenproblemSa k = l k Ea k (14.2.6)for k =1, 2,...,p.Second, third, and subsequent a k are subject to the constraints a ′ h Ea k =0forh