Reducing aggregation error in spatial interaction models by location sampling

Models of spatial interaction such as transport, migration, commuting and trade usually partition space into zones, to represent the receiving and sending end of the interaction. When zones encompass multiple locations, the partitioning causes an aggregation error (Hillsman and Rhoda 1978). The aggregation error increases with the size of zones. Aggregation errors can cause bias (Goodchild 1979; Openshaw 1984) and when zones are larger than a (generally unknown) threshold, models become invalid (Tobler 1989). It therefore seems obvious to make zones smaller whenever possible. In practice, however, zones often remain large for a number of reasons, including data availability, parsimony and computational complexity. There are different aspects to the aggregation error; there is the information loss associated with averaging variables and the loss of spatial precision – typically by conceptually concentrating all of a zone in its centroid. Both types of error are amplified when non-linear functions are applied on the aggregated variables, which can lead to a further model bias. One domain where non-linear use of aggregated variables causes a risk of bias is Discrete Choice Modelling where the utility of an alternative is typically an exponential function of descriptive variables. It is therefore well-recognized that aggregation of alternatives must account for the effect of size and variability of those alternatives. However size and variability are often imperfectly understood and the analysis has to depend on judgment, experience and proxy variables (Ben-Akiva and Lerman 1985 p. 252-275). In recent years (micro)simulation has been established as a method for aggregation that circumvents many of the complications of analytical solutions (Train 2009). The location variation however, is not usually considered in simulation applications. For instance Train (2009 p. 55) suggests that alternatives with a geographical dimension require utility parameters specified in a log function to facilitate analytical aggregation. This paper intends to follow the simulation approach and extent it to the issue of geographical aggregation.