A Probability Model for Central Place Locations

A probability model for the locational properties of central place theory is formulated as a type of two-dimensional stochastic point process obtained by combining a random element with the hexagonal point lattice of classical central place theory. Lattice points identify equilibrium positions, and the realized position of each central place point is a random displacement from its equilibrium position. A stochastic process generating this type of point pattern is identified and the resulting pattern is described by the spacing between neighboring points. Both the general form of spacing measures and special cases related to the normal probability law are examined. For the Gaussian model, it is shown that under mild restrictions the spacing between neighboring points is distributed as an order statistic from the non-central chi-square distribution. Because some integrals are not evaluated, a Monte Carlo simulation of the model is also described. The Gaussian model is applied to an urban system in the state of Iowa, and the level of correspondence between observed and calculated values is judged sufficiently close to recommend further study of stochastic central place models.