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

384 14. Generalizations and Adaptations of Principal Component Analysisa form of ‘weighted PCA,’ where different variables can have differentweights, φ 1 ,φ 2 ,...,φ p , and different observations can have different weightsω 1 ,ω 2 ,...,ω n . Cochran and Horne (1977) discuss the use of this type ofweighted PCA in a chemistry context.It is possible that one set of weights, but not the other, is present. Forexample, if ω 1 = ω 2 = ··· = ω n but the φ j are different, then only thevariables have different weights and the observations are treated identically.Using the correlation matrix rather than the covariance matrix isa special case in which φ j =1/s jj where s jj is the sample variance ofthe jth variable. Deville and Malinvaud (1983) argue that the choice ofφ j =1/s jj is somewhat arbitrary and that other weights may be appropriatein some circumstances, and Rao (1964) also suggested the possibility ofusing weights for variables. Gower (1966) notes the possibility of dividingthe variables by their ranges rather than standard deviations, or by theirmeans (for positive random variables, leading to an analysis of coefficientsof variation), or ‘even [by] the cube root of the sample third moment.’ Inother circumstances, there may be reasons to allow different observationsto have different weights, although the variables remain equally weighted.In practice, it must be rare that an obvious uniquely appropriate set ofthe ω i or φ j is available, though a general pattern may suggest itself. Forexample, when data are time series Diamantaras and Kung (1996, Section3.5) suggest basing PCA on a weighted estimate of the covariance matrix,where weights of observations decrease geometrically as the distance in timefrom the end of the series increases. In forecasting functional data, Aguileraet al. (1999b) cut a time series into segments, which are then treatedas different realizations of a single series (see Section 12.3.4). Those segmentscorresponding to more recent parts of the original series are givengreater weight in forming functional PCs for forecasting than are segmentsfurther in the past. Both linearly decreasing and exponentially decreasingweights are tried. These are examples of weighted observations. An exampleof weighted variables, also from functional PCA, is presented by Ramsayand Abrahamowicz (1989). Here the functions are varying binomial parameters,so that at different parts of the curves, weights are assigned that areinversely proportional to the binomial standard deviation for that part ofthe curve.An even more general set of weights than that given in (14.2.4) isproposed by Gabriel and Zamir (1979). Here X is approximated byminimizingn∑ p∑w ij ( mˆx ij − x ij ) 2 , (14.2.5)i=1 j=1where the rank m approximation to X has elements mˆx ij of the form mˆx ij =∑ mk=1 g ikh jk for suitably chosen constants g ik ,h jk ,i= 1, 2,...,n, j =1, 2,...,p, k =1, 2,...,m. This does not readily fit into the generalized