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Maximum-Entropy Representations in Convex Polytopes: Applications to Spatial Interaction

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  • P B Slater

    (Community and Organization Research Institute, University of California, Santa Barbara, CA 93106, USA)

Abstract

Of all representations of a given point situated in a convex polytope, as a convex combination of extreme points, there exists one for which the probability or weighting distribution has maximum entropy. The determination of this multiplicative or exponential distribution can be accomplished by inverting a certain bijection—developed by Rothaus and by Bregman—of convex polytopes into themselves. An iterative algorithm is available for this procedure. The doubly stochastic matrix with a given set of transversals (generalized diagonal products) can be found by means of this method. Applications are discussed of the Rothaus -Bregman map to a proof of Birkhoff's theorem and to the calculation of trajectories of points leading to stationary or equilibrium values of the generalized permanent, in particular in spatial interaction modeling.

Suggested Citation

  • P B Slater, 1989. "Maximum-Entropy Representations in Convex Polytopes: Applications to Spatial Interaction," Environment and Planning A, , vol. 21(11), pages 1541-1546, November.
    • Handle: RePEc:sae:envira:v:21:y:1989:i:11:p:1541-1546
    DOI: 10.1068/a211541
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    References listed on IDEAS

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    1. Paul Slater, 1981. "Combinatorial procedures for structuring internal migration and other transaction flows," Quality & Quantity: International Journal of Methodology, Springer, vol. 15(2), pages 179-202, April.
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    Cited by:

    1. Slater, Paul B., 1992. "Minimum relative entropies of low-dimensional spin systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 182(1), pages 145-154.

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