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

12.3. Functional PCA 317A key reference for functional PCA is the book by Ramsay and Silverman(1997), written by two researchers in the field who, together withBesse (see, for example, Besse and Ramsay, 1986), have been largely responsiblefor bringing these ideas to the attention of statisticians. We drawheavily on this book in the present section. However, the ideas of PCAin a continuous domain have also been developed in other fields, such assignal processing, and the topic is an active research area. The terminology‘Karhunen-Loève expansion’ is in common use in some disciplines to denotePCA in a continuous domain. Diamantaras and Kung (1996, Section 3.2)extend the terminology to cover the case where the data are discrete timeseries with a theoretically infinite number of time points.In atmospheric science the Karhunen-Loève expansion has been used inthe case where the continuum is spatial, and the different observationscorrespond to different discrete times. Preisendorfer and Mobley (1988,Section 2d) give a thorough discussion of this case and cite a number ofearlier references dating back to Obukhov (1947). Bouhaddou et al. (1987)independently consider a spatial context for what they refer to as ‘principalcomponent analysis of a stochastic process,’ but which is PCA for a (twodimensionalspatial) continuum of variables. They use their approach toapproximate both a spatial covariance function and the underlying spatialstochastic process, and compare it with what they regard as the less flexiblealternative of kriging. Guttorp and Sampson (1994) discuss similar ideasin a wider review of methods for estimating spatial covariance matrices.Durbin and Knott (1972) derive a special case of functional PCA in thecontext of goodness-of-fit testing (see Section 14.6.2).12.3.1 The Basics of Functional PCA (FPCA)When data are functions, our usual data structure x ij ,i=1, 2,...,n; j =1, 2,...,pis replaced by x i (t), i=1, 2,...,n where t is continuous in someinterval. We assume, as elsewhere in the book, that the data are centred,so that a mean curve ¯x = 1 ∑ nn i=1 ˜x i(t) has been subtracted from each ofthe original curves ˜x i (t). Linear functions of the curves are now integralsinstead of sums, that is z i = ∫ a(t)x i (t)dt rather than z i = ∑ pj=1 a jx ij .In‘ordinary’ PCA the first PC has weights a 11 ,a 21 ,...,a p1 , which maximizethe sample variance var(z i ), subject to ∑ pj=1 a2 j1 = 1. Because the data arecentred,var(z i1 )= 1n − 1n∑i=1z 2 i1 = 1n − 1n∑ [ ∑p ] 2.a j1 x iji=1j=1Analogously for curves, we find a 1 (t) which maximizes1n∑zi1 2 = 1 n∑[∫2a 1 (t)x i (t) dt],n − 1 n − 1i=1i=1