Spatial flows and spatial patterns

The misspecification of gravity spatial interaction models has recently been described by the author. The bias in parameter estimates that results from such misspecification appears to produce the 'map pattern effect' or 'spatial structure bias' in estimated distance-decay parameters. A further aspect of the misspecification bias in gravity parameter estimates is explored here. The severity of the bias is shown to vary in a predictable manner with variations in spatial structure. In particular, the bias is shown to be dependent upon the pattern of accessibility that exists within a spatial system. The relevant aspects of this pattern are discussed for intraurban and interurban (or interregional) flow matrices. It is shown that from an examination of the spatial structure of centres in a spatial system it is possible, a priori, to identify whether significant bias will arise in the calibration of a gravity model. Certain configurations of centres are shown theoretically to produce maximal bias. The author thus answers the question, "why do gravity parameter estimates appear to be biased in some systems but not in others?" 1 Introduction In spatial interaction modelling, distance-decay can be defined as the rate at which the volume of interaction between centres decreases as the distance between the centres increases, ceteris paribus. An estimated distance-decay parameter obtained in the calibration of a gravity interaction model is assumed to measure this rate and, hence, is usually given a behavioural interpretation. The parameter estimate is traditionally assumed to measure the perceived deterrence of distance to interaction. However, I have recently shown (Fotheringham, 1983a; 1983b) that gravity interaction models are misspecifications of reality because they fail to model accurately relationships between destinations, and this misspecification produces bias in parameter estimates. Relationships between destinations may arise either from forces of competition or from forces of agglomeration. If the former are present, then, as the proximity of a destination to other destinations increases, interaction to that destination decreases, ceteris paribus. If the latter are present, then, as the proximity of a destination to other destinations increases, interaction to that destination increases, ceteris paribus. Spatial interaction models that fail to model competition or agglomeration forces accurately are henceforth described as misspecified, meaning that the models are misspecifications of reality. The bias in parameter estimates that results from such misspecification is related to the spatial arrangement of centres in the system under investigation so that such estimates cannot be given any behavioural …