A not sign-preserving iteration algorithm for the ‘Improved Normalized Squared Differences’ matrix adjustment model

Abstract

Estimating the elements of a matrix, when only the margins (row and column sums) are known, but a supposedly similar ‘reference matrix’ is available, is a standard problem in many disciplines. After discussing the main types, issues and applications of these two-directional matrix adjustment problems the paper concentrates on the case of negative matrix elements and models with quadratic objective functions. The solution of the Improved Normalized Squared Differences (INSD) model is proved to be the same as the result of that iteration algorithm which is presented in the paper. It is also argued that if the sign-preservation requirement is dropped then the iteration procedure suggested by Huang et al. (Econ Syst Res 20(1):111–123, 2008) boils down to the same algorithm. Using the numerical example of the earlier literature it is also demonstrated that even in this not sign-preserving case, which even requires sign-flips for some elements, the INSD-model produces good fit in mathematical terms.