Enhancing the Distance Minimization Methods of Matrix Updating within a Homothetic Paradigm
Matrix updating methods are used for constructing the target matrix with the prescribed row and column marginal totals that demonstrates the highest possible level of its structural similarity to initial matrix given. A concept of structural similarity has a vague framework that can be slightly refined under considering a particular case of strict proportionality between row and column marginal totals for target and initial matrices. Here the question arises: can we accept the initial matrix homothety as optimal solution for proportionality case of matrix updating problem?
In most practical situations an affirmative answer to the question is almost obvious. It is natural to call this common notion by homothetic paradigm and to refer its checking as homothetic testing. Some well-known methods for matrix updating serve as an additional instrumental confirmation to validity of homothetic paradigm. It is shown that RAS method and Kuroda’s method pass through the homothetic test successfully.
Homothetic paradigm can be helpful for enhancing a collection of matrix updating methods based on constrained minimization of the distance functions. Main attention is paid to improving the methods with weighted squared differences (both regular and relative) as an objective function.
As an instance of a incorrigible failure in the homothetic testing, the GRAS method for updating the economic matrices with some negative entries is analyzed in details. A collection of illustrative numerical examples and some recommendations for method’s choice are given.