A Method of Fitting the Gravity Model Based on the Poisson Distribution*

Abstract

"In this paper, [the authors] suggest an alternative method for fitting the gravity model. In this method, the interaction variable is treated as the outcome of a discrete probability process, whose mean is a function of the size and distance variables. This treatment seems appropriate when the dependent variable represents a count of the number of items (people, vehicles, shipments) moving from one place to another. It would seem to have special advantages where there are some pairs of places between which few items move. The argument will be illustrated with reference to data on the numbers of migrants moving in 1970-1971 between pairs of the 126 labor market areas defined for Great Britain...."

Fitting the Lognormal Gravity Model to Heteroscedastic Data

“…When the flows are constrained to be integers, as in the case of migration models, the Poisson model described in Flowerdew and Aitkin (1982) is more appropriate. A comparison between the Poisson and iterative weighted lognormal models shows a substantial difference in parameter estimates and a clear superiority in goodness of fit for the Poisson.…”

“…Flowerdew and Aitkin (1982) have described this method and discussed its advantages over the lognormal approach. The Poisson method avoids all four problems mentioned in the Introduction, and is more appropriate where the amount of interaction must be a discrete quantity.…”

“…It has been mentioned above that the unweighted lognormal model produces different results according to how zero flows are treated (see Flowerdew and Aitkin 1982). Equation ( …”

“…Flowerdew and Aitkin (1982) have considered the problems of fitting gravity models of this type in contexts, such as migration, where interaction is necessarily measured in whole numbers. These problems include the bias introduced into the model by using a logarithmic scale; as Haworth and Vincent (1979) show, an unbiased estimate of the mean of a random variable on a logarithmic scale becomes a biased estimate when antilogarithms are taken.…”

Spatial Interaction Models and the Role of Geographic Information Systems

“…Let Tij denote a flow of persons from i to j, Tij might be considered to be the outcome of a Poisson process if it is assumed that there is a constant probability of any individual in i moving to j, that the population of i is large, and the number of individuals interacting is an independent process (Flowerdew andAitkin 1982, Fotheringham andO'Kelly 1989). Then the probability that Tij is the number of people recorded as moving from i to j is given by (19) where Tij is the expected outcome of the Poisson process and T'ij the observed value which is subject to sampling and measuring errors and thus fluctuates around the expected value.…”