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Example text

Then the trace of Σy , denoted tr (Σy ), is maximized by taking B = Aq , where Aq consists of the first q columns of A. 12 2. Properties of Population Principal Components Proof. Let β k be the kth column of B; as the columns of A form a basis for p-dimensional space, we have p cjk αj , βk = k = 1, 2, . . , q, j=1 where cjk , j = 1, 2, . . , p, k = 1, 2, . . , q, are appropriately defined constants. 3) p = λ j cj cj j=1 where cj is the jth row of C. Therefore p tr(B ΣB) = λj tr(cj cj ) j=1 p = λj tr(cj cj ) j=1 p = λ j cj cj j=1 p q λj c2jk .

And the smallest eigenvalue, respectively. Furthermore, var[αk x] = λk for k = 1, 2, . . , p. 2) and Diamantaras and Kung (1996, Chapter 3) give alternative derivations that do not involve differentiation. ’ This usage, though sometimes defended (see Dawkins (1990), Kuhfeld (1990) for some discussion), is confusing. It is preferable to reserve the term ‘principal components’ for the derived variables αk x, and refer to αk as the vector of coefficients or loadings for the kth PC. Some authors distinguish between the terms ‘loadings’ and ‘coefficients,’ depending on the normalization constraint used, but they will be used interchangeably in this book.

Xn , and ak is the corresponding eigenvector for k = 1, 2, . . , p. ˜ and Z ˜ to have (i, k)th elements equal to Define the (n × p) matrices X ˜ and the value of the kth element x ˜ik of xi , and to z˜ik , respectively. Then Z ˜ ˜ ˜ X are related by Z = XA, where A is the (p × p) orthogonal matrix whose kth column is ak . ˜ X. 1. Optimal Algebraic Properties of Sample Principal Components 31 where x ¯j = 1 n n x ˜ij , j = 1, 2, . . , p. 1) will be very useful in this and subsequent chapters.

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