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

12.3. Functional PCA 319The sample covariance between x(s) andx(t) can be written1n − 1 x′ (s)x(t) = 1n − 1 φ′ (s)C ′ Cφ(t).Any eigenfunction a(t) can be expressed in terms of the basis functionsas a(t) = ∑ Gg=1 b gφ g (t) = φ ′ (t)b for some vector of coefficientsb ′ =(b 1 ,b 2 ,...,b G ). The left-hand-side of equation (12.3.1) is then∫∫1S(s, t)a(t) dt =n − 1 φ′ (s)C ′ Cφ(t)φ ′ (t)b dt= 1[∫]n − 1 φ′ (s)C ′ C φ(t)φ ′ (t) dt b.The integral is a (G×G) matrix W whose (g, h)th element is ∫ φ g (t)φ h (t)dt.If the basis is orthogonal, W is simply the identity matrix I G . Hence choosingan orthogonal basis in circumstances where such a choice makes sense,as with a Fourier basis for periodic data, gives simplified calculations. Ingeneral (12.3.1) becomes1n − 1 φ′ (s)C ′ CWb = lφ ′ (s)bbut, because this equation must hold for all available values of s, it reducesto1n − 1 C′ CWb = lb. (12.3.2)When ∫ a 2 (t)dt = 1 it follows that∫∫1= a 2 (t) dt = b ′ φ(t)φ ′ (t)b dt = b ′ Wb.If a k (t) is written in terms of the basis functions as a k (t) = ∑ Gg=1 b kgφ g (t),with a similar expression for a l (t), then a k (t) is orthogonal to a l (t) ifb ′ k Wb l = 0, where b ′ k =(b k1,b k2 ,...,b kG ), and b ′ lis defined similarly.In an eigenequation, the eigenvector is usually normalized to have unitlength (norm). To convert (12.3.2) into this form, define u = W 1 2 b. Thenu ′ u = 1 and (12.3.2) can be written1n − 1 W 1 2 C ′ CW 1 2 u = lu. (12.3.3)Equation (12.3.3) is solved for l and u, b is obtained as W − 1 2 u, and finallya(t) =φ ′ (t)b = φ ′ (t)W − 1 2 u.The special case where the basis is orthogonal has already been mentioned.Here W = I G ,sob = u is an eigenvector of1n−1 C′ C. Anotherspecial case, noted by Ramsay and Silverman (1997), occurs when thedata curves themselves are taken as the basis. Then C = I n and u is1an eigenvector ofn−1 W.