MAXIMIZATION OF SUMS OF QUOTIENTS OF QUADRATIC-FORMS AND SOME GENERALIZATIONS

Monotonically convergent algorithms are described for maximizing six ( constrained) functions of vectors x, or matrices X with columns x(1),. .., x(r). These functions are h(1)(x) = Sigma(k) (x'A(k)x)(x'C(k)x)(-1 ), H-1(X) = Sigma(k) tr (X'A(k)X)(X'C(k)X)(-1), (h) over tilde(1)(X) = Sigma(k) Sigma(l)(x'(l)A(k)x(l))(x'(l)C(k)x(l))(-1) with X constraine d to be columnwise orthonormal, h(2)(x) = Sigma(k) (x'A(k)x)(2)(x'C(k) X)(-1) subject to x'x = 1, H-2(X) = Sigma(k) tr (X'A(k)X)(X'A(k)X)'(X' C(k)X)(-1) subject to X'X = I, and (h) over tilde(2)(X) = Sigma(k) Sig ma(l) (x'(l)A(k)x(l))(2)(x'(l)C(k)x(l))(-1) subject to X'X = I. In the se functions the matrices C-k are assumed to be positive definite. The matrices A(k) can be arbitrary square matrices. The general formulati on of the functions and the algorithms allows for application of the a lgorithms in various problems that arise in multivariate analysis. Sev eral applications of the general algorithms are given. Specifically, a lgorithms are given for reciprocal principal components analysis, bino rmamin rotation, generalized discriminant analysis, variants of genera lized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general a lgorithms.