A Comparative Study of Algorithms for Matrix Balancing

Abstract

The problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics, urban planning, statistics, demography, and stochastic modeling; these problems are called Matrix Balancing Problems. We describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing—matrix scaling and nonlinear optimization. The algorithms we study are the RAS algorithm, a diagonal similarity scaling algorithm, and a truncated Newton algorithm for network optimization. We present results from computational experiments with large-scale problems based on producing consistent estimates of Social Accounting Matrices for developing countries.

Offense-Defense Approach to Ranking Team Sports

“…In a general case the rows and columns of A do not have to add to the same scalar, see Schneider and Zenios (1994) for details.…”

“…The matrix scaling (see Schneider and Zenios, 1994, Kalantari et al, 1993, and Rothblum et al, 1994 procedure in the Sinkhorn-Knopp theorem is one of the methods for matrix balancing and in some papers is referred to as a RAS method. Matrix scaling and its convergence has received increased attention in the past several decades.…”

Integrating Expenditure and Income Data: What to Do with the Statistical Discrepancy?

“…Schneider and Zenios (1990) credit a Russian mathematician Bregman for this result, although the fact that the first-order conditions for the minimization problem yield the RAS iterative solution is not difficult to illustrate; see for example Günlük-Şenesen and Bates (1988). One could therefore weight the entropy kernel to allow for tuning parameters, though this would complicate the iterative technique to arrive at a solution.…”

A Flexible Modeling Framework to Estimate Interregional Trade Patterns and Input-Output Accounts

“…These applications include the estimation of input-output tables (Bachem and Korte, 1981;Harrigan and Buchanan, 1984;Miller and Blair, 1985;Kaneko, 1988;Antonello, 1990) and inter-regional trade flows in regional science (Batten, 1982;Byron et al, 1993), balancing of social/national accounts in economics (Byron, 1978;Van der Ploeg, 1982Zenios, Drud, and Mulvey, 1989;Nagurney, Kim, and Robinson, 1990), estimating interregional migration in demography (Plane, 1982), the analysis of voting patterns in political science (Johnson, Hay, and Taylor, 1982), the treatment of census data and estimation of contingency tables in statistics (Friedlander, 1961), the estimation of transition probabilities in stochastic modeling (Theil and Rey, 1966), and the projection of traffic within telecommunication and transportation networks (Florian, 1986;Klincewicz, 1989). A comprehensive survey can be found in Schneider and Zenios (1990).…”