The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

This paper presents a methodology for deriving best statistics for the calibration of spatial interaction models, and several procedures for finding best parameter values are described. The family of spatial interaction models due to Wilson is first outlined, and then some existing calibration methods are briefly reviewed. A procedure for deriving best statistics based on the principle of maximum-likelihood is then developed from the work of Hyman and Evans, and the methodology is illustrated using the example of a retail gravity model. Five methods for solving the maximum-likelihood equations are outlined: procedures based on a simple first-order iterative process, the Newton—Raphson method for several variables, multivariate Fibonacci search, search using the Simplex method, and search based on quadratic convergence, are all tested and compared. It appears that the Newton—Raphson method is the most efficient, and this is further tested in the calibration of disaggregated residential location models.

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Optimal Zoning Systems for Spatial Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a090169 ◽

1977 ◽

Vol 9 (2) ◽

pp. 169-184 ◽

Cited By ~ 134

Author(s):

S Openshaw

Keyword(s):

Spatial Model ◽

Model Building ◽

Spatial Interaction ◽

Empirical Studies ◽

Model Performance ◽

Interaction Models ◽

Spatial Interaction Models ◽

Parameter Values ◽

Zoning Systems

The design of zoning systems for spatial interaction models is a major problem which affects both the interpretation and acceptability of these models. This paper demonstrates that zoning-system effects on parameter values and model performance are nontrivial, and that their magnitude is far larger than was previously thought likely. An approach which is most appropriate in an applied context, where there is also the problem of poor model performance, is to identify a zoning system which will approximately optimise model performance. The paper gives details of how this may be achieved. This method is demonstrated by a series of empirical studies. Finally, there is a brief discussion of the general implications for spatial model building.

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A New Set of Spatial-Interaction Models: The Theory of Competing Destinations

Environment and Planning A Economy and Space ◽

10.1177/0308518x8301500103 ◽

1983 ◽

Vol 15 (1) ◽

pp. 15-36 ◽

Cited By ~ 133

Author(s):

A S Fotheringham

Keyword(s):

Spatial Interaction ◽

Model Misspecification ◽

Distance Decay ◽

Gravity Models ◽

Parameter Estimates ◽

Interaction Models ◽

Spatial Interaction Models ◽

The Family ◽

Decay Parameters ◽

Alternative Set

Members of the family of spatial-interaction models commonly referred to as gravity models are shown to be misspecified. One result of this misspecification is the occurrence of an undesirable ‘spatial-structure effect’ in estimated distance-decay parameters and this effect is examined in detail. An alternative set of spatial-interaction models is formulated from which more accurate predictions of interactions and more accurate parameter estimates can be obtained. These new interaction models are termed competing destinations models, and estimated distance-decay parameters obtained in their calibration are shown to have a purely behavioural interpretation. The implications of gravity-model misspecification are discussed.

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Proximate Aggregation-Estimation of Spatial Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a160467 ◽

1984 ◽

Vol 16 (4) ◽

pp. 467-486 ◽

Cited By ~ 10

Author(s):

M Batty ◽

P K Sikdar

Keyword(s):

Spatial Interaction ◽

Gravity Models ◽

New Method ◽

Entropy Measure ◽

Interaction Models ◽

Information Measures ◽

Spatial Interaction Models ◽

Entropy Measures ◽

Log Linear ◽

Parameter Values

In this paper the authors introduce a method of approximating the parameter values of gravity models from measures of information or entropy associated with the observed pattern of spatial interaction. The method builds on the previous work of the authors in which parameter values were estimated in a two-stage process which involved utilising the log-linear properties of entropy models through the canonical form of entropy, together with other approximations based on Kirby's method. Here the method is elaborated by adopting a consistent set of information measures to which the parameters of the model are related and this negates the need for other approximations. The original framework is first reviewed and then elaborated through the introduction of a weighted entropy measure. The traditional family of spatial interaction models is sketched and the new method developed for each of these models. The models are then applied to various aggregations of trip data in the Reading (United Kingdom) subregion, and estimates of parameter values based on the old, new, and conventional methods are compared. The new method is demonstrably superior to the old method and various extensions through the spatial disaggregation of entropy measures are noted in conclusion.

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A Note on the Calibration of Maximum-Performance Spatial-Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a101151 ◽

1978 ◽

Vol 10 (10) ◽

pp. 1151-1154

Author(s):

M J Baxter

Keyword(s):

Spatial Interaction ◽

Model Performance ◽

Maximum Performance ◽

Interaction Models ◽

Spatial Interaction Models ◽

The Family

Evidence is presented to show that the improvement in model performance achieved by the family of maximum-performance spatial-interaction models developed by Openshaw and Connolly (1977) may be explicable solely for statistical reasons and that there is no need for a geographical or behavioural explanation, as was suggested.

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A Note on the Treatment of Flows across System Boundaries in Spatial-Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a110447 ◽

1979 ◽

Vol 11 (4) ◽

pp. 447-454 ◽

Cited By ~ 6

Author(s):

I Masser

Keyword(s):

Spatial Interaction ◽

System Boundaries ◽

Interaction Models ◽

Spatial Interaction Models ◽

The Family ◽

Constrained Model

Some problems that must be resolved by the analyst in connection with the treatment of flows across system boundaries to and from external zones are discussed in this paper. These problems make it necessary to make modifications to the conventional formulations both of the doubly and of the singly constrained members of the family of spatial-interaction models. None of the possible modifications wholly satisfies theoretical requirements in terms of the doubly constrained model, and the advantages and limitations of various approaches can only be assessed in an operational situation. For this reason some of the findings from a study of the Amersfoort region are presented which help to throw light on this problem.

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On the Relationship between Alonso's Theory of Movement and Wilson's Family of Spatial-Interaction Models

Environment and Planning A Economy and Space ◽

10.1068/a130217 ◽

1981 ◽

Vol 13 (2) ◽

pp. 217-224 ◽

Cited By ~ 15

Author(s):

J Ledent

Keyword(s):

Input Data ◽

Spatial Interaction ◽

System Of Equations ◽

Interaction Models ◽

Spatial Interaction Models ◽

Standard Family ◽

Original Contribution ◽

Standard Models ◽

The Relationship

This paper compares the system of equations underlying Alonso's theory of movement with that of Wilson's standard family of spatial-interaction models. It is shown that the Alonso model is equivalent to one of Wilson's four standard models depending on the assumption at the outset about which of the total outflows and/or inflows are known. This result turns out to supersede earlier findings—inconsistent only in appearance—which were derived independently by Wilson and Ledent. In addition to this, an original contribution of this paper—obtained as a byproduct of the process leading to the aforementioned result—is to provide an exact methodology permitting one to solve the Alonso model for each possible choice of the input data.

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