Least-squares approximation of an improper by a proper correlation matrix using a semi-infinite convex program

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based on a solution for C. I. Mosier's oblique Procrustes rotation problem offered by J. M. F. ten Berge and K. Nevels (1977). It is shown that the minimization problem belongs to a certain class of convex programs in optimization theory. A necessary and sufficient condition for a solution to yield the unique global minimum of the least-squares function is derived from a theorem by A. Shapiro (1985). A computer program was implemented to yield the solution of the minimization problem with the proposed algorithm. This empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely

Simplicity of core arrays in three-way principal component analysis and the typical rank of p×q×2 arrays

AbstractInterpreting the solution of a Principal Component Analysis of a three-way array is greatly simplified when the core array has a large number of zero elements. The possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand, and by mathematical analysis on the other. In the present paper, it is shown that a p×q×2 array, with p>q⩾2, can almost surely be transformed to have all but 2q elements zero. It is also shown that arrays of that form have three-way rank p at most. This has direct implications for the typical rank of p×q×2 arrays, also when p=q. When p⩾2q, the typical rank is 2q; when q<p<2q it is p, and when p=q, the rank is typically (almost surely) p or p+1. These typical rank results pertain to the decomposition of real valued three-way arrays in terms of real valued rank one arrays, and do not apply in the complex setting, where the typical rank of p×q×2 arrays is also min[p,2q] when p>q, but it is p when p=q

Three-mode principal component analysis and perfect congruence analysis for sets of covariance matrices

FSW - Gezinsopvoeding - Ou

Three-mode principal component analysis and perfect congruence analysis for sets of covariance matrices

FSW - Gezinsopvoeding - Ou

An efficient algorithm for TUCKALS3 on data with large numbers of observational units

Datatheorie voor analyse van individuele verschille

Some additional results on principal component analysis of three-mode data by means of alternating least squares algorithms

FSW - Gezinsopvoeding - Ou