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

388 14. Generalizations and Adaptations of Principal Component AnalysisMetric-based PCA, as defined by Thacker (1996), corresponds to thetriple (X, E −1 , 1 n I n), and E plays the same rôle as does Γ in the fixedeffects model. Tipping and Bishop’s (1999a) model (Section 3.9) can befitted as a special case with E = σ 2 I p . In this case the a k are simplyeigenvectors of S.Consider a model for x in which x is the sum of a signal and an independentnoise term, so that the overall covariance matrix can be decomposedas S = S S + S N , where S S , S N are constructed from signal and noise,respectively. If S N = E, then S S = S − E andS S a k = Sa k − Ea k = l k Ea k − Ea k =(l k − 1)Ea k ,so the a k are also eigenvectors of the signal covariance matrix, using themetric defined by E −1 . Hannachi (2000) demonstrates equivalences between• Thacker’s technique;• a method that finds a linear function of x that minimizes the probabilitydensity of noise for a fixed value of the probability density ofthe data, assuming both densities are multivariate normal;• maximization of signal to noise ratio as defined by Allen and Smith(1997).Diamantaras and Kung (1996, Section 7.2) discuss maximization of signalto noise ratio in a neural network context using what they call ‘orientedPCA.’ Their optimization problem is again equivalent to that of Thacker(1996). The fingerprint techniques in Section 12.4.3 also analyse signal tonoise ratios, but in that case the signal is defined as a squared expectation,rather than in terms of a signal covariance matrix.Because any linear transformation of x affects both the numerator anddenominator of the ratio in the same way, Thacker’s (1996) technique shareswith canonical variate analysis and CCA an invariance to the units ofmeasurement. In particular, unlike PCA, the results from covariance andcorrelation matrices are equivalent.14.2.3 Transformations and CenteringData may be transformed in a variety of ways before PCA is carried out,and we have seen a number of instances of this elsewhere in the book.Transformations are often used as a way of producing non-linearity (Section14.1) and are a frequent preprocessing step in the analysis of specialtypes of data. For example, discrete data may be ranked (Section 13.1)and size/shape data, compositional data and species abundance data (Sections13.2, 13.3, 13.8) may each be log-transformed before a PCA is done.The log transformation is particularly common and its properties, with andwithout standardization, are illustrated by Baxter (1995) using a numberof examples from archaeology.