Enhancing the distance minimization methods of matrix updating within a homothetic paradigm

Vladimir Motorin

doi:10.1186/s40008-017-0094-7

Abstract

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.

Highlights

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
The 3 × 4-dimensional initial matrix A combines the entries in intersections of the columns “Agriculture,” “Industry,” “Services,” “Final d.” with the rows “Agriculture,” “Industry,” “Services” in “Table 1: Input–output data for year 0.”
The row marginal total vector u of dimension 3 × 1 is the proper part of the column “Output” in “Table 2: Input–output data for year 1,” and the column marginal total vector v′ of dimension 1 × 4 involves the proper entries of the row “Total” in the near-mentioned data source

Summary

Introduction

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. In most practical situations an affirmative answer to this question is almost obvious As it is shown below, the well-known and widely used RAS and Kuroda’s methods for matrix updating serve as an additional instrumental confirmation to such an answer. In this connection, we will call this rather common notion by homothetic paradigm and will refer examining the property “if u = kuA and v = kvA X = kA” as a homothetic test for the matrix updating methods. It is advisable to propose that a successful passing through homothetic test were to be appreciated as a positive feature of any matrix updating method

Objectives

Findings

Conclusion